Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris ...

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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

6197

Guillaume Hanrot François Morain Emmanuel Thomé (Eds.)

Algorithmic Number Theory 9th International Symposium, ANTS-IX Nancy, France, July 19-23, 2010 Proceedings

13

Volume Editors Guillaume Hanrot LIP/ENS-Lyon, 46, allée d’Italie 69364 Lyon Cedex 07, France E-mail: [email protected] François Morain LIX/École polytechnique 91128 Palaiseau Cedex, France E-mail: [email protected] Emmanuel Thomé INRIA Nancy, projet CARAMEL 615 rue du jardin botanique 54602 Villers-lès-Nancy Cedex, France E-mail: [email protected]

Library of Congress Control Number: 2010930653 CR Subject Classification (1998): F.2, G.2, E.3, I.1 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-642-14517-5 Springer Berlin Heidelberg New York 978-3-642-14517-9 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180

Preface

ANTS-IX was the ninth edition of the biennial International Symposium on Algorithmic Number Theory. The ﬁrst edition of this symposium was held at Cornell University in 1994. ANTS-IX was held July 19-23, 2010 at INRIA in Nancy, France. The ANTS-IX Program Committee consisted of 12 members whose names are listed on the next page. The selection of the accepted papers among the submissions was made from mid-January to end of March 2010. Each paper was thoroughly reviewed by at least two experts, including a Program Committee member. The Program Committee selected 25 high-quality articles, which are excellent representatives of the current state of the art in various areas of algorithmic number theory. The Selfridge Prize in computational number theory was awarded to the authors of the best contributed paper presented at the conference. We gratefully thank the authors of all submitted papers for their hard work which made the selection of a varied program possible. We also thank the authors of the accepted papers for their cooperation in the timely production of the revised versions. Each submitted paper was presented by one of its co-authors at the conference. Besides contributed papers, the conference included ﬁve invited talks by Henri Darmon (McGill University), Jean-Fran¸cois Mestre (Universit´e Paris 7), Gabriele Nebe (RWTH Aachen), Carl Pomerance (Dartmouth College), and Oded Regev (Tel-Aviv University). We thank the invited speakers for having been able to provide abstracts of their talk, which are reproduced in this volume. This list of invited speakers originally included Fritz Grunewald (HHU D¨ usseldorf), who unfortunately passed away on March 21, 2010, four months before the conference. A special lecture was held to honor his memory. The conference organizers wish to thank all the people who made the conference possible. In particular, we gratefully acknowledge the support of the funding institutions. May 2010

Guillaume Hanrot Fran¸cois Morain Emmanuel Thom´e

Organization

Organizing Committee Anne-Lise Charbonnier J´er´emie Detrey Pierrick Gaudry (Chair) Emmanuel Thom´e Paul Zimmermann

INRIA, Nancy, France INRIA, Nancy, France CNRS, Nancy, France INRIA, Nancy, France INRIA, Nancy, France

Program Committee Nigel Boston University of Wisconsin, USA John Cremona Warwick Mathematics Institute, UK Claus Fieker University of Sydney, Australia ´ Guillaume Hanrot (PC Chair) Ecole Normale Sup´erieure, Lyon, France Kevin Hare University of Waterloo, Canada ´ Thorsten Kleinjung Ecole Polytechnique F´ed´erale de Lausanne, Switzerland Kamal Khuri-Makdisi American University of Beirut, Lebanon ´ Fran¸cois Morain (PC Chair) Ecole Polytechnique, France Takakazu Satoh Tokyo Institute of Technology, Japan Igor Shparlinski Macquarie University, Australia Alice Silverberg University of California at Irvine, USA Frederik Vercauteren Katholieke Universiteit Leuven, Belgium

Poster Session Benjamin Smith

´ INRIA Saclay, Ecole Polytechnique, France

Sponsoring Institutions Institut National de Recherche en Informatique et Automatique (INRIA) Laboratoire Lorrain de Recherche en Informatique et Applications (LORIA) ´ Ecole Polytechnique Centre National de la Recherche Scientiﬁque (CNRS) Microsoft Research, USA Nancy Universit´e Groupement de Recherches en Informatique Math´ematique (GDR IM) Communaut´e Urbaine du Grand Nancy Conseil R´egional de Lorraine

VIII

Organization

Conference Website The names of the winners of the Selfridge Prize, material supplementing the contributed papers, and errata for the proceedings (if relevant), as well as the abstracts of the posters and the posters presented at ANTS-IX, can be found at http://ants9.org/.

Table of Contents

Invited papers Putting the Hodge and Tate Conjectures to the Test . . . . . . . . . . . . . . . . . Henri Darmon Curves of Genus 3 With a Group of Automorphisms Isomorphic to S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Fran¸cois Mestre

1

2

Learning with Errors over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oded Regev

3

Lattices and Spherical Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriele Nebe

4

Fixed Points for Discrete Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mariana Levin, Carl Pomerance, and K. Soundararajan

6

Contributed papers Explicit Coleman Integration for Hyperelliptic Curves . . . . . . . . . . . . . . . . Jennifer S. Balakrishnan, Robert W. Bradshaw, and Kiran S. Kedlaya

16

Smallest Reduction Matrix of Binary Quadratic Forms: And Cryptographic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aurore Bernard and Nicolas Gama

32

Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Fran¸cois Biasse and Michael J. Jacobson Jr.

50

On the Use of the Negation Map in the Pollard Rho Method . . . . . . . . . . Joppe W. Bos, Thorsten Kleinjung, and Arjen K. Lenstra

66

An O(M (n) log n) Algorithm for the Jacobi Symbol . . . . . . . . . . . . . . . . . . Richard P. Brent and Paul Zimmermann

83

New Families of ECM Curves for Cunningham Numbers . . . . . . . . . . . . . . ´ Eric Brier and Christophe Clavier

96

Visualizing Elements of Sha[3] in Genus 2 Jacobians . . . . . . . . . . . . . . . . . . Nils Bruin and Sander R. Dahmen

110

X

Table of Contents

On Weil polynomials of K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas-Stephan Elsenhans and J¨ org Jahnel

126

Class Invariants by the CRT Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas Enge and Andrew V. Sutherland

142

Short Bases of Lattices over Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . Claus Fieker and Damien Stehl´e

157

On the Complexity of the Montes Ideal Factorization Algorithm . . . . . . . David Ford and Olga Veres

174

Congruent Number Theta Coeﬃcients to 1012 . . . . . . . . . . . . . . . . . . . . . . . William B. Hart, Gonzalo Tornar´ıa, and Mark Watkins

186

Pairing the Volcano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sorina Ionica and Antoine Joux

201

A Subexponential Algorithm for Evaluating Large Degree Isogenies . . . . David Jao and Vladimir Soukharev

219

Huﬀ’s Model for Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marc Joye, Mehdi Tibouchi, and Damien Vergnaud

234

Eﬃcient Pairing Computation With Theta Functions . . . . . . . . . . . . . . . . . David Lubicz and Damien Robert

251

Small-Span Characteristic Polynomials of Integer Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James McKee

270

Decomposition Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koh-ichi Nagao

285

Factoring Polynomials over Local Fields II . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Pauli

301

On a Problem of Hajdu and Tengely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samir Siksek and Michael Stoll

316

Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures . . . . . . . . . . . . . . Jonathan P. Sorenson On the Extremality of an 80-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . Damien Stehl´e and Mark Watkins

331

340

Table of Contents

Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Voight

XI

357

Improved Primality Proving with Eisenstein Pseudocubes . . . . . . . . . . . . . Kjell Wooding and H.C. Williams

372

Hyperbolic Tessellations Associated to Bianchi Groups . . . . . . . . . . . . . . . . Dan Yasaki

385

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397

Putting the Hodge and Tate Conjectures to the Test Henri Darmon Department of Mathematics, McGill University, Burnside Hall, Montreal, QC, Canada [email protected]

The Hodge conjecture asserts that the presence of algebraic cycles on a (smooth, projective) variety over the complex numbers can be detected in its Betti cohomology equipped with the Hodge structure arising from its relation with complex deRham cohomology. The Tate conjecture makes a similar assertion with -adic cohomology replacing Betti cohomology. One of the diﬃculties with these conjectures is that the predictions that they make are often hard to test numerically, even in speciﬁc concrete instances. Unlike closely related parts of number theory (a case in point being the Birch and Swinnerton-Dyer conjecture) the study of algebraic cycles has therefore not been as strongly aﬀected by the growth of the experimental and computational community as it perhaps could be. In this lecture, I will describe some numerical experiments that are designed to “test” the Hodge and Tate conjectures for certain varieties (of arbitrarily large dimension) which arise from elliptic curves with complex multiplication and theta series of CM Hecke characters.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 1, 2010. c Springer-Verlag Berlin Heidelberg 2010

Curves of Genus 3 with a Group of Automorphisms Isomorphic to S3 Jean-Fran¸cois Mestre Centre de Math´ematiques de Jussieu Projet Th´eorie des Nombres [email protected]

In this talk, we construct curves of genus 3 with automorphism group equal to S3 ; we give some applications of this construction to the problem of optimal curves, i.e. of curves over a finite field Fq having a number of points equal to the Serre-Weil bound Mq ; in particular, we prove that there exists infinitely many fields F3n having optimal curves; we prove also that there exists an integer C such that, for any finite field F7n , there exists a curve of genus 3 defined over having at least Mq − C points.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 2, 2010. c Springer-Verlag Berlin Heidelberg 2010

Learning with Errors over Rings Oded Regev Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel [email protected]

The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather ineﬃcient due to an inherent quadratic overhead in the use of LWE. After a short introduction to the area, we will discuss recent work on making LWE and its applications truly eﬃcient by exploiting extra algebraic structure. Namely, we will deﬁne the ring-LWE problem, and prove that it too enjoys very strong hardness guarantees. Based on joint work with Vadim Lyubashevsky and Chris Peikert.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 3, 2010. c Springer-Verlag Berlin Heidelberg 2010

Lattices and Spherical Designs Gabriele Nebe Lehrstuhl D f¨ ur Mathematik, RWTH Aachen University, Germany [email protected]

A lattice is a ﬁnitely generated discrete subgroup of Euclidean space. Lattices are an important algorithmic tool in number theory, integral representation theory, geometry, information theory, cryptography, crystallography and have various other applications within mathematics and beyond. Any lattice has only ﬁnitely many vectors of a given length, they form the layers of the lattice, which are ﬁnite subsets of spheres in the underlying Euclidean space. A spherical design of strength t is a ﬁnite set X = ∅ in the Euclidean 1 sphere for which the mean value |X| x∈X f (x) equals the integral of f over the sphere for all polynomials f of degree up to t. This condition is equivalent to x∈X f (x) = 0 for all non-constant harmonic polynomials of degree ≤ t. Spherical designs hence consist of well distributed points on a sphere and are relevant for numerical integration, in information theory, geometry, statistics and have applications for instance in medicine. Boris Venkov combined these two concepts in a very fruitful way that allows to use lattices to classify spherical designs and to use designs for ﬁnding good lattices. An introduction to this subject as well as some applications are given in “R´eseaux euclidiens, designs sph´eriques et formes modulaires”, Enseignement Math., Geneva, 2001. There Venkov introduces the notion of a strongly perfect lattice, which is a lattice whose minimal vectors form a spherical 4-design. Using the characterization by Korkine, Voronoi and Zolotarev one shows that strongly perfect lattices realise local maxima of the sphere packing density function on the space of all similarity classes of n-dimensional lattices (in fact in the space of all periodic packings as proved by Sch¨ urmann). All local maxima of this function are known up to dimension 8. In dimension 8 Dutour, Sch¨ urmann, Vallentin and Riener proved that there are 2408 local maxima. The densest lattice sphere packings are known up to dimension 8 and, thanks to recent results by Elkies and Kumar, in dimension 24, where the Leech lattice is the densest lattice. Combining number theory and geometry with combinatorial methods allows classify strongly perfect lattices, where a full classiﬁcation up to dimension 12 is obtained in joined work with Venkov. With one exception all known strongly perfect lattices Λ have the additional property that also the dual lattice Λ∗ is strongly perfect. Such lattices are called dual strongly perfect, the classiﬁcation of dual strongly perfect lattices in small dimension has been completed in dimension 14 and is an ongoing PhD project by Elisabeth Nossek in Aachen. There are two general approaches to study and construct strongly perfect lattices: by modular forms and by invariant theory of ﬁnite groups. Both concepts usually allow to show that all non-empty layers of the lattice form spherical G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 4–5, 2010. c Springer-Verlag Berlin Heidelberg 2010

Lattices and Spherical Designs

5

4-designs. Such lattices are called universally perfect and play a role in Riemannian geometry. If Λ is a universally perfect lattice then the torus Rn /Λ∗ deﬁned by the dual lattice Λ∗ provides a strict local minimum of the height function on the set of all n-dimensional ﬂat tori. R. Coulangeon also shows that universally perfect lattices Λ achieve local minima of Epstein’s zeta function, they are so called ζ-extreme lattices. The question to ﬁnd ζ-extreme lattices has a long history going back to Sobolev’s work on numerical integration and to work of Delon´e. Universally perfect lattices are dual strongly perfect. The relation with modular forms arises, because the condition that the minimal vectors of the lattice form a 4-design means the annihilation of certain coeﬃcients in its theta series with harmonic coeﬃcients. In this way one can prove the strong perfectness of many extremal lattices of small level. For example there are more than 106 even unimodular lattices without roots in dimension 32 (by work of Oliver King) and the theory of modular forms shows that all of them are universally perfect; this is the only known method to prove that all these lattices are locally densest lattices. If a lattice Λ has a big automorphism group G := Aut(Λ) which has no invariant harmonic polynomials of degree 2 and 4, a condition easily expressed in terms of the character of G ≤ O(n), then Λ is universally perfect. There are many interesting lattices such as the Barnes-Wall lattices, the 248-dimensional Thompson-Smith lattice and others which are strongly perfect by this reason. Tiep and others used representation theory to classify certain matrix groups G for which all orbits form spherical 4-designs. On the other hand lattices are an important tool to ﬁnd and classify good spherical designs. Fixing the strength t and the dimension n, one tries to ﬁnd spherical t-designs X ⊂ S n−1 of minimal possible cardinality. If t = 2m is even, then n−1+m n−2+m |X| ≥ + m m−1 and if t = 2m + 1 is odd then n−1+m |X| ≥ 2 . m A t-design X for which equality holds is called a tight t-design. Tight t-designs in Rn with n ≥ 3 are very rare. Bannai has shown that such tight designs only exist if t ≤ 5 and t = 7, 11. The tight t-designs with t = 1, 2, 3 as well as t = 11 are completely classiﬁed whereas their classiﬁcation for t = 4, 5, 7 is still an open problem. It is conjectured that there are just seven tight t-designs of dimension n ≥ 3 and strength 4, 5, 7, namely in dimensions 6,22 (t=4), 3,7,23 (t=5) respectively 8,23 (t=7); each of these is known to be unique. One possible approach to prove that there are no further tight designs X is to investigate the Euclidean lattice Λ generated by X and to obtain properties of Λ (such as its determinant or its minimum) from the design properties of X and then prove the non existence of such a lattice Λ. This strategy has been successfully applied by Bannai, Munemasa and Venkov to show that there are no further tight designs up to dimension 103.

Fixed Points for Discrete Logarithms Mariana Levin1 , Carl Pomerance2, and K. Soundararajan3 1

Graduate Group in Science and Mathematics Education University of California Berkeley, CA 94720, USA [email protected] 2 Department of Mathematics Dartmouth College Hanover, NH 03755, USA [email protected] 3 Department of Mathematics Stanford University Stanford, CA 94305, USA [email protected]

Abstract. We establish a conjecture of Brizolis that for every prime p > 3 there is a primitive root g and an integer x in the interval [1, p − 1] with logg x = x. Here, log g is the discrete logarithm function to the base g for the cyclic group (Z/pZ)× . Tools include a numerically explicit “smoothed” version of the P´ olya–Vinogradov inequality for the sum of values of a Dirichlet character on an interval, a simple lower bound sieve, and an exhaustive search over small cases.

1

Introduction

If g is an element in a group G and t ∈ g, there is some integer n with g n = t. Finding a valid choice for n is known as the discrete logarithm problem. Note that if g has ﬁnite order m, then n is actually a residue class modulo m. We write logg t = n (or logg t ≡ n (mod m)) in analogy to usual logarithmic notation. Thus, the problem in the title of this paper does not seem to make good sense, since if logg x = x, then the ﬁrst x is a member of the group g and the second x is either an integer or a residue class modulo m. However, sense is made of the equation through the traditional conﬂation of members of the ring Z/kZ with least nonnegative members of residue classes.

The work for this paper was begun at Bell Laboratories in 2001 while the first author was a summer student working with the second author. A version of this work was presented as the 2003 Master’s Thesis of the first author at U. C. Berkeley, see [3]. The second author was supported in part by NSF grant DMS-0703850. The third author was supported in part by NSF grant DMS-0500711.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 6–15, 2010. c Springer-Verlag Berlin Heidelberg 2010

Fixed Points for Discrete Logarithms

7

In particular, suppose G = (Z/pZ)× , where p is a prime number. This is known to be a cyclic group of order p − 1. Suppose g is a cyclic generator of this group, known as a primitive root for p. A ﬁxed point for the discrete logarithm modulo p to the base g is then an integer x in the interval [1, p − 1] such that logg x = x, that is, g x ≡ x (mod p). (Note that if x is not restricted to the interval [1, p − 1] it is easy to ﬁnd ﬁxed points. Namely, if x is a solution to the Chinese remainder problem x ≡ 1 (mod p − 1), x ≡ g (mod p), then g x ≡ x (mod p).) Brizolis (see Guy [6, Section F9]) made the conjecture that for every prime p > 3 there is a primitive root g and an integer x in [1, p − 1] with logg x = x, that is, g x ≡ x (mod p). In this paper we prove this conjecture in a somewhat stronger form. Brizolis had noticed that if there is a primitive root x for p with x in [1, p − 1] and gcd(x, p − 1) = 1, then with y the multiplicative inverse of x modulo p − 1 and g = xy , we would have that g is a primitive root for p as well, and g x ≡ xxy ≡ x (mod p), that is, there is a solution to the ﬁxed point problem. We shall prove then the stronger result that for each prime p > 3 there is a primitive root x for p in [1, p − 1] that is coprime to p − 1. Several authors have shown that the Brizolis property holds for all suﬃciently large primes p. In particular, Zhang [12] showed the strong conjecture holds for all suﬃciently large primes p, but did not give an estimate of what “suﬃciently large” is. Cobeli and Zaharescu [4] also showed that the strong conjecture holds for suﬃciently large primes p, and gave the details that it holds for all p > 102070 , but they indicated that their method would support a bound around 1050 . Our method is similar to that of Zhang, who used the P´ olya–Vinogradov inequality for character sums on an interval. Here we introduce a numerically explicit “smoothed” version of this inequality, see §2. In addition, we combine the traditional character-sum approach with a simple lower bound sieve. There is still some need for direct calculation for smaller values of p, which are easily handled by a short Mathematica program. In particular, we directly veriﬁed the strong conjecture for each prime p < 1.25 · 109 . We mention the article by Holden and Moree [8], which considers some related problems. The total number of solutions to g x ≡ x (mod p) as p runs up to some high bound N , where either g is restricted to be a primitive root, and where it is not so restricted, is considered in Bourgain, Konyagin, and Shparlinski [2]. The smoothed version of the P´ olya–Vinogradov inequality that we introduce in the next section is quite simple and the proof is routine, so it may be known to others. We have found it to be quite useful numerically; we hope it will ﬁnd applications in “closing the gap” in other problems where character sums arise. Some notation: ω(n) denotes the number of distinct prime divisors of n.

8

2

M. Levin, C. Pomerance, and K. Soundararajan

A “Smoothed” P´ olya–Vinogradov Inequality

Let χ be a non-principal Dirichlet character to the modulus q. The P´olya– Vinogradov inequality (independently discovered by P´ olya and Vinogradov in 1918) asserts that there is a universal constant c such that ≤ c√q log q χ(a) (1) M≤a≤M+N for any choice of numbers M, N . Let N (p) denote the numer of primitive roots g for p with g ∈ [1, p − 1] and gcd(g, p − 1) = 1. Using (1) one can show (see Zhang [12] and Campbell [3]) that N (p) =

ϕ(p − 1)2 + O(p1/2+ ), p−1

for every ﬁxed > 0, and so N (p) > 0 for all suﬃciently large p. The aim of this paper is to close the gap and ﬁnd the complete set of primes p with N (p) > 0. Towards this end it would be useful to have a numerically explicit version of (1). In [3], the theorem of Bachman and Rachakonda [1] was used (plus a small unpublished improvement on a secondary term in their inequality due to the second author of the present paper). Recently, elaborating on the work in an early paper of Landau [10], plus an idea of Bateman as mentioned in Hildebrand [7], the second author in [11] proved a stronger numerically explicit version of (1). Using this simpliﬁes the approach in [3]. However, we have found a way to simplify even further by using a “smoothed” version of (1). In this section we prove the following theorem. Theorem 1. Let χ be a primitive Dirichlet character to the modulus q > 1 and let M, N be real numbers with 0 < N ≤ q. Then √ a − M N − 1 ≤ q − √ . χ(a) 1 − N q M≤a≤M+2N Proof. We use Poisson summation, see [9, §4.3]. Let H(t) = max{0, 1 − |t|}. We wish to estimate |S|, where S :=

χ(a)H

a∈Z

a−M −1 . N

Towards this end we use the identity q−1

1 χ(a) = χ(j)e(aj/q), ¯ τ (χ) ¯ j=0

Fixed Points for Discrete Logarithms

9

where τ (χ) ¯ is the Gauss sum for χ ¯ and e(x) := e2πix . Thus, q−1

S=

1 χ(j) ¯ e(aj/q)H τ (χ) ¯ j=0 a∈Z

a−M −1 . N

The Fourier transform of H is ∞ 1 − cos 2πs ˆ ˆ H(s) = H(t)e(−st) dt = when s = 0, H(0) = 1, 2π 2 s2 −∞ which is nonnegative for s real. By a change of variables in the integral, we see that the Fourier transform of e(jt/q)H((t − M )/N − 1) is ˆ (s − j/q)N . N e − (M + N )(s − j/q) H Hence, by Poisson summation, we have q−1 N ˆ (n − j/q)N . χ(j) ¯ e − (M + N )(n − j/q) H S= τ (χ) ¯ j=0 n∈Z

Estimating trivially (that is, taking the absolute value of each term) and using ˆ nonnegative and χ(0) = 0, we have H q−1 N ˆ kN N ˆ . H (n − j/q)N = √ H |S| ≤ √ q j=1 q q n∈Z

k∈Z\qZ

ˆ Since (N/q)H(sN/q) is the Fourier transform of H(qt/N ), from the last calculation we have

N kN N ˆ √ N ˆ kN √ ˆ |S| ≤ q ≤ q − H(0) + H H q q q q q k∈Z k∈Z\qZ

N √ √ N ql √ N = − √ + qH(0) = q − √ , = q − + H q N q q l∈Z

by another appeal to Poisson summation and the deﬁnition of H. This completes the proof of the theorem. In our application we will need a version of Theorem 1 with the variable a satisfying a coprimality condition. We deduce such a result below. Corollary 2. Let k be a square-free integer and let χ be a primitive character to the modulus q > 1. For 0 < N ≤ q, we have √ a 2ω(k) q always ≤ χ(a) 1 − − 1 ω(k)−1 √ N 2 q if k is even. 0≤a≤2N (a,k)=1

10

M. Levin, C. Pomerance, and K. Soundararajan

Proof. Since d|(k,a) μ(d) gives 1 if (a, k) = 1 and 0 otherwise, the sum in question equals ad − 1 μ(d)χ(d) χ(a) 1 − N d|k

a≤2N/d

√ and using Theorem 1 this is bounded in size by 2ω(k) q as desired. If (k, q) is even, then χ(d) = 0 for even divisors d of k, so that we achieve the bound √ 2ω(k)−1 q, again as desired. Suppose now that k is even and q is odd. For each odd divisor d of k, we group together the contribution from d and 2d, and so we may write the sum in question as ad − 1 . μ(d)χ(d) χ(a) 1 − N d|k/2

a≤2N/d a odd

We replace a in the inner sum by q + a, and since q is now odd, the condition that a is odd may be replaced with the condition that q + a = 2b is even. Thus, the above sum becomes 2d(b − q/2) μ(d)χ(d)χ(2) χ(b) 1 − − 1 , N d|k/2

q/2≤b≤q/2+N/d

and appealing again to Theorem 1 we obtain the Corollary in this case. Though we will not need it for our proof, we record the following corollary of Theorem 1. Corollary 3. Let χ be a primitive Dirichlet character to the modulus q > 1 and let M, N be real numbers with N > 0. Then, with θ the fractional part of N/q, q 3/2 a − M − 1 ≤ θ(1 − θ). χ(a) 1 − N N M≤a≤M+2N

3

A Criterion for the Brizolis Property

Let us write the largest square-free divisor of p − 1 as uv where u and v will be chosen later. We shall assume that u is even, and have in mind the situation that u is composed of the small prime factors of p − 1, and that v is composed of the large prime factors; we also allow for the possibility that v = 1. For the rest of the paper, the letter will denote a prime number. Let S denote the set of primitive roots in [1, p − 1] that are coprime to p − 1. Thus, an integer g ∈ [1, p − 1] is in S if and only if for each prime | p − 1 we have both g and g is not an -th power (mod p). Let S1 denote the set of integers in [1, p − 1] that are coprime to u and which are not equal to an -th power (mod p) for any prime dividing u. Let S2 denote the set of integers in S1 which are divisible by some prime which divides v. Let S3 denote the set

Fixed Points for Discrete Logarithms

11

of integers in S1 which equal an -th power (mod p) for some prime dividing v. Now S ⊂ S1 , and the elements in S1 that are not in S are precisely those that, for some prime | v, are either divisible by or are an -th power (mod p). Thus, S = S1 \(S2 ∪ S3 ). We seek a positive lower bound for N :=

g∈S

2g − 1 , 1− p−1

since if N > 0, then S = ∅. By our observation above we have N ≥ N1 − N2 − N3 , where, for j = 1, 2, 3, Nj =

g∈Sj

2g − 1 . 1− p−1

If d is a square-free divisor of p − 1 and g is an integer in [1, p − 1], let Cd (g) be 1 if g is a d-th power (mod p) and 0 otherwise. Thus, Cd (g) =

C (g) =

|d

⎛ 1 ⎝ = 1+ d |d

1 |d

χ of order

Note that

χ(g)

χ =χ0

⎞

1 χ(g)⎠ = d

χ(g).

m|d χ of order m

μ(d)Cd (g)

d|u

is 1 if, for each | u, g is not an -th power (mod p), and is 0 otherwise. By the above calculation, this expression is μ(d) d|u

d

χ(g) =

m|d χ of order m

χ(g)

m|u χ of order m

μ(nm) . nm

n|u/m

The inner sum here is (ϕ(u)/u)μ(m)/ϕ(m), so that N1 =

ϕ(u) u

1≤g≤p−1 (g,u)=1

μ(m) 2g − 1 1− p−1 ϕ(m) m|u

χ(g).

(2)

χ of order m

Let m | u with m > 1. Using Corollary 2, the terms above contribute an amount bounded in magnitude by ϕ(u) ω(u)−1 √ 2 p, u

12

M. Levin, C. Pomerance, and K. Soundararajan

so the total contribution over all m | u with m > 1 has magnitude at most ϕ(u) ω(u) √ 2 − 1 2ω(u)−1 p. u The sum over g in (2) with m = 1 (and so χ = χ0 ) is ϕ(u) u

1≤g≤p−1 (g,u)=1

ϕ(u) 2g − 1 = 1− μ(d) p−1 u d|u

h≤(p−1)/d

2dh − 1 . 1− p−1

The inner sum over h can be evaluated explicitly: it equals (p−1)/(2d) if (p−1)/d is even, and it equals (p − 1)/(2d) − d/(2(p − 1)) if (p − 1)/d is odd. It follows that the contribution when m = 1 is 2 ϕ(u) p − 1 ϕ(u) 1 − dμ(d) u 2 u 2(p − 1) ≥

ϕ(u) u

We conclude that 2 ϕ(u) N1 ≥ u 2 ϕ(u) > u

2

d|u (p−1)/d odd

p−1 ϕ(u)2 − ≥ 2 u(p − 1)

ϕ(u) u

2

p ϕ(u) − . 2 u

p ϕ(u) ϕ(u) ω(u) √ − − 2 − 1 2ω(u)−1 p 2 u u p ϕ(u) ω(u) √ − 4 p. 2 2u

Next we turn to N2 . Since an element in S2 must be divisible by some prime |v we have that ϕ(u) μ(m) 2h − 1 1− N2 ≤ χ(h ). p−1 u ϕ(m) |v h≤(p−1)/ (h,u)=1

χ of order m

m|u

If v = 1, then N2 = 0, so assume v > 1. The terms with m > 1 contribute, using Corollary 2, an amount bounded in size by ϕ(u) √ ω(v) 2ω(u) − 1 2ω(u)−1 p. u The main term m = 1 above contributes (arguing as in our evaluation of the main term for N1 above) ϕ(u) u

|v h≤(p−1)/ (h,u)=1

2h ϕ(u) 2 p − 1 . 1− − 1 ≤ + p−1 u 2 v |v

Fixed Points for Discrete Logarithms

Since

|v

N2 ≤ ≤

13

≤ v, and using v > 1, we conclude that

ϕ(u) u ϕ(u) u

2 2

p−1 1 + 2

|v

p1 2

|v

+

ϕ(u) u

2 +

ϕ(u) √ ω(v) 2ω(u) − 1 2ω(u)−1 p u

ϕ(u) ω(u) √ 4 ω(v) p. 2u

Lastly we consider N3 . An element g of S3 must be an -th power for some prime |v, and the indicator function for this condition is 1 ψ =χ0 ψ(g), as seen above. Therefore we have that N3 is at most |v g≤p−1 (g,u)=1

ϕ(u) μ(m) 2g − 1 1− p−1 u ϕ(m) m|u

χ of order m

1 χ(g) ψ(g) . ψ =χ0

Appealing to Corollary 2 for the terms above with χψ = χ0 we ﬁnd that the contribution of such terms is bounded in magnitude by ϕ(u) 2ω(u)−1 √ 2 ω(v) p. u The main term χ = ψ = χ0 gives ϕ(u) ϕ(u) p − 1 2g ϕ(u) 1 ϕ(u) 1 − 1 ≤ + 1− u p−1 u u 2 p−1 |v

g≤p−1 (g,u)=1

=

|v

ϕ(u) u

Thus,

2

N3 ≤

p−1 1 + 2 v

ϕ(u) u

2

|v

1 ≤

ϕ(u) u

2

p1 . 2 |v

p 1 ϕ(u) ω(u) √ + 4 ω(v) p. 2 2u |v

Combining these bounds for N1 , N2 and N3 we obtain that N≥

ϕ(u) u

2 1 ϕ(u) p √ 1−2 − 4ω(u) (1 + 2ω(v)) p. 2 2u |v

We may conclude as follows: The Brizolis property holds for the prime p ≥ 5, if we may write the largest square-free divisor of p − 1 as uv with u even, |v 1/ < 1/2, and with √ 1 + 2ω(v) 4ω(u) u · . p> ϕ(u) 1 − 2 |v 1/

(3)

14

4

M. Levin, C. Pomerance, and K. Soundararajan

Completing the Proof

Our criterion (3) can be used in a straightforward way with v = 1 to get an upper bound for possible counterexamples to the Brizolis conjecture. Indeed, after a small calculation (using 4ω(n) < 1404n1/3 and n/ϕ(n) < 2 log log n for n larger than the product of the ﬁrst eleven primes), it is seen that the Brizolis property holds for all p > 1025 . It is not pleasant to contemplate checking each prime to this point, so instead we use (3) with v > 1. Suppose ω(p − 1) = k ≥ 10, and take v to be the product of the six largest primes dividing p − 1, and u to be the product of the other smaller primes. Since ω(p − 1) ≥ 10, the primes dividing v are all at least 11, and we have that 1−2

1 |v

≥1−2

1 1 1 1 1 1 + + + + + > 0.28. 11 13 17 19 23 29

If pj denotes the j-th prime, then 4ω(u) u/ϕ(u) ≤ k−6 j=1 (4pj /(pj − 1)), and p > k p − 1 ≥ j=1 pj . So from our criterion (3), if we have k k−6 13 4pj √ , pj ≥ 0.28 j=1 pj − 1 j=1

then the Brizolis property holds for all p with ω(p − 1) = k. We veriﬁed that the inequality above holds for k = 10. If k is increased by 1 then the LHS of our √ inequality is increased by a factor of at least 31 > 5, but the RHS is increased only by a factor of at most 4 × (11/10) = 4.4. Thus, the inequality holds for all k ≥ 10. Suppose now that k = ω(p − 1) ≤ 9. If k ≥ 4, we take u to be the product of the four smallest primes dividing p − 1, and otherwise, we take u to be the product ofall the primes dividing p − 1. Then v has at most 5 prime factors, and 1 − 2 |v 1/ ≥ 1 − 2(1/11 + 1/13 + 1/17 + 1/19 + 1/23) ≥ 0.35. Further 4 p|u 4p/(p − 1) ≤ j=1 4pj /(pj − 1) = 1120. Our criterion (3) shows that if 11 2 p ≥ 1120 × = 1,239,040,000, 0.35 then p satisﬁes the Brizolis property. Using the functions Prime[ ] and PrimitiveRoot[ ] in Mathematica, we were able to directly exhibit a primitive root g for each prime 3 < p < 1.25 · 109 with g in [1, p − 1] and coprime to p − 1. Our program runs as follows. The function Prime[ ] allows us to sequentially step through the primes up to our bound. For each prime p returned by Prime[ ], we invoke PrimitiveRoot[p] to ﬁnd the least positive primitive root r for p. We then sequentially check r2k−1 mod p for k = 1, 2, . . . until we ﬁnd a value coprime to p − 1 with 2k − 1 also coprime to p − 1. The exponent being coprime to p − 1 guarantees that the power is a primitive root, and the residue being coprime to p − 1 then guarantees that we

Fixed Points for Discrete Logarithms

15

have found a member of S. If no such primitive root exists, this algorithm would not terminate, but it did, thus verifying the Brizolis property for the given range. There are various small speed-ups that one can use to augment the program. For example, if r = 2 is a primitive root and p ≡ 1 (mod 4), then note that p − 2 is a primitive root coprime to p − 1, and so work with this prime p is complete. The augmented program ran in about 90 minutes on a Dell workstation. This completes our proof of the Brizolis conjecture. Acknowledgment. We thank Richard Crandall for some technical assistance with the Mathematica program and the referees for some helpful comments.

References 1. Bachman, G., Rachakonda, L.: On a problem of Dobrowolski and Williams and the P´ olya–Vinogradov inequality. Ramanujan J. 5, 65–71 (2001) 2. Bourgain, J., Konyagin, S.V., Shparlinski, I.E.: Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm. Int. Math. Res. Notices, art. ID rnn090, 29 (2008) (Corrigendum: ibid. 2009, No. 16, 3146–3147) 3. Campbell, M.E.: On fixed points for discrete logarithms, Master’s Thesis, U. C. Berkeley Department of Mathematics (2003) 4. Cobeli, C., Zaharescu, A.: An exponential congruence with solutions in primitive roots. Rev. Romaine Math. Pures Appl. 44, 15–22 (1999) 5. Crandall, R., Pomerance, C.: Prime numbers: a computational perspective, 2nd edn. Springer, New York (2005) 6. Guy, R.K.: Unsolved problems in number theory. Springer, Berlin (1984) 7. Hildebrand, A.: On the constant in the P´ olya–Vinogradov inequality. Canad. Math. Bull. 31, 347–352 (1988) 8. Holden, J., Moree, P.: Some heuristics and results for small cycles of the discrete logarithm. Math. Comp. 75, 419–449 (2006) 9. Iwaniec, H., Kowalski, E.: Analytic number theory. American Math. Soc., Providence (2004) 10. Landau, E.: Absch¨ atzungen von Charaktersummen, Einheiten und Klassenzahlen. Nachrichten K¨ onigl. Ges. Wiss. G¨ ottingen, 79–97 (1918) 11. Pomerance, C.: Remarks on the P´ olya–Vinogradov inequality (submitted for publication 2010) 12. Zhang, W.-P.: On a problem of Brizolis. Pure Appl. Math. 11(Suppl.), 1–3 (1995) (Chinese. English, Chinese summary)

Explicit Coleman Integration for Hyperelliptic Curves Jennifer S. Balakrishnan1, Robert W. Bradshaw2, and Kiran S. Kedlaya1 1

Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected], [email protected] 2 University of Washington, Seattle, WA 98195, USA [email protected]

Abstract. Coleman’s theory of p-adic integration ﬁgures prominently in several number-theoretic applications, such as ﬁnding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.

1

Introduction

One of the fundamental diﬃculties of p-adic analysis is that the totally disconnected topology of p-adic spaces makes it hard to introduce a meaningful form of antidiﬀerentiation. It was originally discovered by Coleman that this problem can be circumvented using the principle of Frobenius equivariance. Using this idea, Coleman introduced a p-adic integration theory ﬁrst on the projective line [9], then (partly jointly with de Shalit) on curves and abelian varieties [10], [8]. Alternative treatments have been given by Besser [3] using methods of p-adic cohomology, and by Berkovich [2] using the nonarchimedean Gel’fand transform. Although Coleman’s construction is in principle quite suitable for machine computation, this had only been implemented previously in the genus 0 case [5]. The purpose of this paper is to present an algorithm for computing single Coleman integrals on hyperelliptic curves of good reduction over Cp for p > 2, based on the third author’s algorithm for computing the Frobenius action on the de Rham cohomology of such curves [17]. We also describe an implementation of this algorithm in the Sage computer algebra system. For context, we indicate some of the many potential applications of explicit Coleman integration. Some of these will be treated, with additional numerical examples, in the ﬁrst author’s upcoming PhD thesis. (Some of these applications will require additional reﬁnements of our implementation; see Section 5.) – Torsion points on curves. Coleman’s original application of p-adic integration was to ﬁnd torsion points on curves of genus greater than 1. This could potentially be made eﬀective and automatic. – p-adic heights on curves. Investigations into p-adic analogues of the conjecture of Birch and Swinnerton-Dyer for Jacobians of hyperelliptic curves G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 16–31, 2010. c Springer-Verlag Berlin Heidelberg 2010

Explicit Coleman Integration for Hyperelliptic Curves

17

require computation of the Coleman-Gross height pairing [11]. This global p-adic height pairing can, in turn, be decomposed into a sum of local height pairings at each prime. In particular, for C a hyperelliptic curve over Qp with p a prime of good reduction and for D1 , D2 ∈ Div0 (C) with disjoint support, the Coleman-Gross p-adic height pairing at p is given in terms of the Coleman integral [10] hp (D1 , D2 ) = ωD1 , D2

–

–

–

–

for an appropriately constructed diﬀerential ωD1 associated to the divisor D1 . This pairing is eﬀectively computable by work of the ﬁrst author [1]. Using this work, it should be possible (using ideas of Besser [4]) to add in local heights away from p, and thus compute the Coleman-Gross height pairing on Jacobians of hyperelliptic curves. (In genus 1, one can then compare to an alternate computation based on work of Mazur-Stein-Tate [22] and Harvey [16].) p-adic regulators. A related topic to the previous one is the computation of p-adic regulators in higher K-theory of arithmetic schemes, which are expected to relate to special values of L-functions. Some computations in genus 0 have been made by Besser and de Jeu [5]. Rational points on curves: Chabauty’s method. For C a smooth proper curve over Z[ N1 ], the Chabauty condition on C is that rank J(C) Z N1 < dim J(C), where J(C) denotes the Jacobian of the curve. When the Chabauty condiP tion holds, there exists a 1-form ω on J(C)an with 0 ω = 0 for all points 1 P ∈ J(C) Z N . We might be able to compute C(Z[ N1 ]) if we can ﬁnd all P points P ∈ C an such that 0 ω = 0. This method has already been used in many cases, by Coleman and many others; see [23] for a survey (circa 2007). To apply Chabauty’s method in a typical case, one needs the integral of ω at some point in a residue disc, with which one can ﬁnd all zeroes of the integral in the residue disc. Several methods are suggested in [23, Remark 8.3] for doing this, including Coleman integration. However, no serious attempt has been made to use numerical Coleman integration in Chabauty’s method; it seems likely that it can handle cases where the other methods suggested in [23, Remark 8.3] for ﬁnding constants of integration prove to be impractical. Rational points on curves: nonabelian Chabauty. It may be possible to use (iterated) Coleman integration to ﬁnd rational points on curves failing the Chabauty condition, using Kim’s nonabelian Chabauty method [18]. As a demonstration of the method, Kim [19] gives an explicit double integral which vanishes on the integral points of the minimal regular model of a genus 1 curve over Q of Mordell-Weil rank 1. The erratum to [19] includes a corrected formula, together with some numerical examples computed using the methods of this paper. p-adic polylogarithms and multiple zeta values. These have been introduced recently by Furusho [13], but little numerical data exists so far.

18

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

2

Coleman’s Theory of p-adic Integration

In this section, we recall Coleman’s p-adic integration theory (for single integrals only) in the case of curves with good reduction. This theory involves some concepts from rigid analytic geometry which it would be hopeless to introduce in such limited space; some standard references are [6] and [12]. (See also [10, §1].) Let Cp be a completed algebraic closure of Qp , and let O be the valuation subring of Cp . Choose once and for all a branch of the p-adic logarithm, i.e., a homomorphism Log : C× to the disc {x ∈ Cp : |x − 1| < p → Cp whose restriction ∞ 1} is given by the logarithm series log(x) = i=1 (1−x)i /i. (The choice of branch has no eﬀect on the integrals on diﬀerentials of the second kind, i.e., everywhere meromorphic diﬀerentials with all residues zero.) We ﬁrst introduce integrals on discs and annuli within P1 . Definition 1. Let I be an open subinterval A(I) denote the of [0, +∞). Let 1 annulus (or disc) {t ∈ A1Cp : |t| ∈ I}. For i∈Z ci ti dt ∈ ΩA(I)/C and P, Q ∈ p A(I), deﬁne Q ci (Qi+1 − P i+1 ). ci ti dt = c−1 Log(Q/P ) + i+1 P i∈Z

i=−1

This is easily shown not to depend on the choice of the coordinate t. Remark 2. Note that because of the division by i + 1 in the formula for the integral, we are unable to integrate on closed discs or annuli. We next turn to curves of good reduction. Definition 3. By a curve over O, we will mean a smooth proper connected scheme X over O of relative dimension 1. Equip the function ﬁeld K(X) with the p-adic absolute value, so that the elements of K(X) of norm at most 1 constitute the local ring in X of the generic point of the special ﬁbre X of X. Let XQ denote the generic ﬁbre of X as a rigid analytic space. There is a natural specialization map from XQ to X; the inverse image of any point of X is a subspace of XQ isomorphic to an open unit disc. We call such a disc a residue disc of X. Definition 4. Let X be a curve over O. By a wide open subspace of XQ , we will mean a rigid analytic subspace of XQ of the form {x ∈ XQ : |f (x)| < λ} for some f ∈ K(X) of absolute value 1 and some λ > 1. Coleman made the surprising discovery that there is a well-behaved integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. (Note that one needs to consider wide open subspaces even to integrate diﬀerentials which are holomorphic or meromorphic on the entire curve.) In the case of hyperelliptic curves, Coleman’s construction of these integrals using Frobenius lifts will be reﬂected in our technique for computing the integrals. For the general case, see [10, §2], [3, §4], or [2, Theorem 1.6.1].

Explicit Coleman Integration for Hyperelliptic Curves

19

Theorem 5 (Coleman). We may assign to each curve X over O and each 1 wide open subspace W of XQ a map μW : Div0 (W ) × ΩW/C → Cp , subject to p the following conditions. (Here Div(W ) denotes the free group on the elements of W , and Div0 (W ) denotes the kernel of the degree map deg : Div(W ) → Z taking each element of W to 1.) 1 (a) (Linearity) The map μW is linear on Div0 (W ) and Cp -linear on ΩW/C . p (b) (Compatibility) For any residue disc D of X and any isomorphism ψ : W ∩ 1 D → A(I) for some interval I, the restriction of μW to Div0 (W ∩D)×ΩW/C p is compatible with Deﬁnition 1 via ψ. (c) (Change of variables) Let X be another curve over O, let W be a wide open subspace of X , and let ψ : W → W be any morphism of rigid spaces relative to an automorphism of Cp . Then

(1) μW (ψ(·), ·) = μW (·, ψ ∗ (·)). (d) (Fundamental theorem of calculus) For any Q = i ci (Pi ) ∈ Div0 (W ) and any f ∈ O(W ), μW (Q, df ) = i ci f (Pi ). Remark 6. One cannot expect path independence in the case of bad reduction. For instance, an elliptic curve over Cp with bad reduction admits a Tate uniformization, so its logarithm map has nonzero periods in general. In Berkovich’s theory of integration, this occurs because the nonarchimedean analytic space associated to this curve X has nontrivial ﬁrst homology.

3

Explicit Integrals for Hyperelliptic Curves

We now specialize to the situation where p > 2 and X is a genus g hyperelliptic curve over an unramiﬁed extension K of Qp having good reduction. We will assume in addition that we have been given a model of X of the form y 2 = f (x) such that deg f (x) = 2g + 1 and f has no repeated roots modulo p. (This restriction is inherited from [17], where it is used to simplify the reduction procedure. One could reduce to this case after possibly replacing K by a larger unramiﬁed extension of Qp , by performing a linear fractional transformation in x to put one root at inﬁnity, thus reducing the degree from 2g + 2 to 2g + 1.) We will distinguish between Weierstrass and non-Weierstrass residue discs of X, which respectively correspond to Weierstrass and non-Weierstrass points of X. To discuss the diﬀerentials we will be integrating, we review a core deﬁnition from [17]. Let X be the aﬃne curve obtained by deleting the Weierstrass points from X, and let A = K[x, y, z]/(y 2 − f (x), yz − 1) be the coordinate ring of X . Definition 7. The Monsky-Washnitzer (MW) weak completion of A is the ring A† consisting of inﬁnite sums of the form

∞ Bi (x) , Bi (x) ∈ K[x], deg Bi ≤ 2g , yi i=−∞

20

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

further subject to the condition that vp (Bi (x)) grows faster than a linear function of i as i → ±∞. We make a ring out of these using the relation y 2 = f (x). These functions are holomorphic on wide opens, so we will integrate 1-forms ω = g(x, y)

dx , 2y

g(x, y) ∈ A† .

(2)

Note that we only consider 1-forms which are odd, i.e., which are negated by the hyperelliptic involution. Even 1-forms can be written in terms of x alone, and so can be integrated directly as in Deﬁnition 1. (This last statement would fail if we had taken A† to be the full p-adic completion of A, rather than the weak completion. This observation is the basis for Monsky-Washnitzer’s formal cohomology, which is used in [17].) Note that the class of allowed forms includes those meromorphic diﬀerentials on X whose poles all belong to Weierstrass residue discs. For some applications (e.g., p-adic canonical heights), it is necessary to integrate meromorphic diﬀerentials with poles in non-Weierstrass residue discs. These will be discussed in [1]. Note also that for ease of exposition, we describe all of our algorithms as if it were possible to compute exactly in A† . This is not possible for two reasons: the elements of A† correspond to inﬁnite series, and the coeﬃcients of these series are polynomials with p-adic coeﬃcients. In practice, each computation will be made with suitable p-adic approximations of the truly desired quantities, so one must keep track of how much p-adic precision is needed in these estimates in order for the answers to bear a certain level of p-adic accuracy. We postpone this discussion to § 4.1. 3.1

A Basis for de Rham Cohomology

We ﬁrst note that any odd diﬀerential ω as in (2) can be written uniquely as ω = df + c0 ω0 + · · · + c2g−1 ω2g−1

(3)

with f ∈ A† , ci ∈ K, and ωi =

xi dx 2y

(i = 0, . . . , 2g − 1).

(4)

That is, the ωi form a basis of the odd part of the de Rham cohomology of A† . The process of putting ω in the form (3), using the relations y 2 = f (x), dx , d(xi y j ) = 2ixi−1 y j+1 + jxi f (x)y j−1 2y can be made algorithmic; see [17, §3]. (Brieﬂy, one uses the ﬁrst relation to reduce high powers of x, and the second to reduce large positive and negative powers of y.) Using properties from Theorem 5 (linearity and the fundamental

Explicit Coleman Integration for Hyperelliptic Curves

21

theorem of calculus), the integration of ω reduces eﬀectively to the integration of the ωi . It may be convenient for some purposes to use a diﬀerent basis of de Rham cohomology. For instance, the basis xi dx/2y 3 (i = 0, . . . , 2g − 1) is crystalline (see the erratum to [17]), so Frobenius will act via a matrix with p-adically integral entries. 3.2

Tiny Integrals

Q We refer to any Coleman integral of the form P ω in which P, Q lie in the same residue disc (Weierstrass or not) as a tiny integral. As an easy ﬁrst case, we give an algorithm to compute tiny integrals of basis diﬀerentials. Algorithm 8 (Tiny Coleman integrals). Input: Points P, Q ∈ X(Cp ) in the same residue disc (neither equal to the point at inﬁnity) and a basis diﬀerential ωi . Q Output: The integral P ωi . 1. Construct a linear interpolation from P to Q. For instance, in a nonWeierstrass residue disc, we may take x(t) = (1 − t)x(P ) + tx(Q) y(t) = f (x(t)), where y(t) is expanded as a formal power series in t. 2. Formally integrate the power series in t:

Q

P

ωi =

Q

P

xi

dx = 2y

0

1

x(t)i dx(t) dt. 2y(t) dt

Remark 9. One can similarly integrate any ω holomorphic in the residue disc containing P and Q. If ω is only meromorphic in the disc, but has no pole at P or Q, we can ﬁrst make a polar decomposition, i.e., write ω as a holomorphic diﬀerential on the disc plus some terms of the form c/(t − r)i , and integrate the latter terms directly. (If ω is everywhere meromorphic, this is achieved by a partial fractions decomposition.) 3.3

Non-Weierstrass Discs

Q We next compute integrals of the form P ωi in which P, Q ∈ X(Cp ) lie in distinct non-Weierstrass residue discs. The method of tiny integrals is not available; we instead employ Dwork’s principle of analytic continuation along Frobenius, in the form of Kedlaya’s algorithm [17] for calculating the action of Frobenius Q on de Rham cohomology. Note that we calculate the integrals P ωi for all i simultaneously. (We modify the presentation in [17] by keeping track of exact diﬀerentials, which are irrelevant for computing zeta functions.)

22

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Algorithm 10 (Kedlaya’s algorithm). Input: The basis diﬀerentials {ωi }2g−1 i=0 . Output: Functions fi ∈ A† and a 2g × 2g matrix M over K such that φ∗ (ωi ) = 2g−1 dfi + j=0 Mij ωj for a p-power lift of Frobenius φ. 1. Since K is an unramiﬁed extension of Qp , it carries a unique automorphism φK lifting the Frobenius automorphism x → xp on its residue ﬁeld. Extend φK to a Frobenius lift on A† by setting φ(x) = xp ,

1/2 φK (f )(xp ) − f (x)p φ(y) = y p 1 + f (x)p

∞ 1/2 (φK (f )(xp ) − f (x)p )i = yp , y 2pi i i=0 noting the series converges in A† because φK (f )(xp ) − f (x)p has positive valuation. (This choice of φ(y) ensures that φ(y)2 = φ(f (x)), so that the action on A† is well-deﬁned. 2. Use a Newton iteration to compute y/φ(y). Then for i = 0, . . . , 2g−1, proceed as in § 3.1 to write φ∗ (ωi ) = pxpi+p−1

2g−1 y dx = dfi + Mij ωj φ(y) 2y j=0

(5)

for some fi ∈ A† and some 2g × 2g matrix M over K. We may use Algorithm 10 to compute Coleman integrals between endpoints in non-Weierstrass residue discs, as follows. (Note that our recipe is essentially Coleman’s construction of the integrals in this case.) Algorithm 11 (Coleman integration in non-Weierstrass discs). Input: The basis diﬀerentials {ωi }2g−1 i=0 , points P, Q ∈ X(Cp ) in non-Weierstrass residue discs, and a positive integer m such that the residue ﬁelds of P, Q are contained in Fpm . Q Output: The integrals { P ωi }2g−1 i=0 . 1. Calculate the action of the m-th power of Frobenius on each basis element (see Remark 12): 2g−1 (φm )∗ ωi = dfi + Mij ωj . (6) j=0

2. By change of variables (see Remark 13), we obtain 2g−1 j=0

(M − I)ij

Q

P

ωj = fi (P ) − fi (Q) −

φm (P )

P

ωi −

Q

φm (Q)

ωi

(7)

Explicit Coleman Integration for Hyperelliptic Curves

23

(the fundamental linear system). As the eigenvalues of the matrix M are algebraic integers of Cp -norm pm/2 = 1 (see [17, §2]), the matrix M − I is Q invertible, and we may solve (7) to obtain the integrals P ωi . Remark 12. To compute the action of φm , ﬁrst perform Algorithm 10 to write φ∗ ωi = dgi +

2g−1

Bij ωj .

j=0

If we view f, g as column vectors and M, B as matrices, we then have f = φm−1 (g) + Bφm−2 (g) + · · · + BφK (B) · · · φm−2 (B)g K M = BφK (B) · · · φm−1 (B). K Remark 13. We obtain (7) as follows. By change of variables, φm (Q) Q ωi = (φm )∗ ωi φm (P )

P

Q

= P

(dfi +

2g−1

Mij ωj )

j=0

= fi (Q) − fi (P ) + Adding

φm (P ) P

Q P

ωi +

ωi =

j=0

Q

φm (Q)

φm (P )

P

2g−1

Mij

Q

P

ωj .

ωi to both sides of this equation yields

ωi +

Q

φm (Q)

ωi + fi (Q) − fi (P ) +

2g−1 j=0

Mij

Q

P

ωj ,

which is equivalent to (7). Definition 14. A Teichm¨ uller point of XQ is a point ﬁxed by some power of φ. Each non-Weierstrass residue disc contains a unique such point: if (x, y) ∈ X is a non-Weierstrass point, the Teichm¨ uller point in its residue disc has xcoordinate equal to the usual Teichm¨ uller lift of x. This leaves two choices for the y-coordinate, exactly one of which has the correct reduction modulo p. Note that Teichm¨ uller points are always deﬁned over ﬁnite unramiﬁed extensions of Qp . Remark 15. A variant of Algorithm 11 is to ﬁrst ﬁnd the Teichm¨ uller points P , Q in the residue discs of P, Q, then note that from the fundamental linear system (7), we have Q 2g−1 (M − I)ij ωj = fi (P ) − fi (Q ). (8) j=0

P

Q Q Q From (8), we obtain the integrals P ωi . Finally, write P ωi − P ωi as the Q P sum P ωi + Q ωi of tiny integrals.

24

3.4

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Weierstrass Endpoints of Integration

Suppose now that P, Q lie in diﬀerent residue discs, at least one of which is Weierstrass. Since a diﬀerential ω of the form (2) is not meromorphic over Weierstrass Q residue discs, we cannot always even deﬁne P ω, let alone compute it. We will thus assume (to cover most cases arising in applications) that ω is everywhere meromorphic, with no pole at either P or Q. We then make the following observation. Lemma 16. Let ω be an odd, everywhere meromorphic diﬀerential on X. Choose P, Q ∈ X(Cp ) which are not poles of ω, with P Weierstrass. Then for ι the hyQ Q perelliptic involution, P ω = 12 ι(Q) ω. In particular, if Q is also a Weierstrass Q point, then P ω = 0. Proof. Let I := points, we have

Q

ι(Q) P (−ω) = ι(Q) ω. Then by additivity P ω = P Q ι(Q) ω = 2I, from which the result follows.

in the end-

If P belongs to a Weierstrass residue disc while Q does not, we ﬁnd the Weierstrass point P in the disc of P , then apply Lemma 16 to write

Q

P

ω=

P

P

1 ω+ 2

Q

ι(Q)

ω.

(9)

The ﬁrst integral on the right side of (9) is tiny, while the second integral involves two points in non-Weierstrass residue discs, and so may be computed as in the previous section. The situation is even better if P, Q both belong to residue discs Q containing respective Weierstrass points P , Q : in this case, by Lemma 16, P ω Q P equals the sum P ω + Q ω of tiny integrals. Remark 17. Beware that Lemma 16 does not generalize to iterated integrals. For instance, for double integrals, if both integrands are odd, the total integrand is even, so the argument of Lemma 16 tells us nothing. It is thus worth considering alternate approaches for dealing with Weierstrass discs, which may generalize better to the iterated case. We concentrate on the case where P lies in a Weierstrass residue disc but Q does not, as we may reduce to this case by splitting Q R Q ω = P ω + R ω for some auxiliary point R in a non-Weierstrass residue P disc. In Algorithm 11, the form fi belongs to A† and so need not converge at P . However, it does converge at any point R near the boundary of the disc, i.e., in the complement of a certain smaller disc which can be bounded explicitly. We Q R Q may thus write P ωi = P ωi + R ωi for suitable R in the disc of P , to obtain an analogue of the fundamental linear system (7). Similarly, when we write

Explicit Coleman Integration for Hyperelliptic Curves

25

ω as in (3), we can ﬁnd R close enough to the boundary of the disc of P so that f Q R converges at R, use (3) to evaluate R ω, then compute P ω as a tiny integral. One defect of this approach is that forcing R to be close to the boundary of the residue disc of P forces R to be deﬁned over a highly ramiﬁed extension of Qp , over which computations are more expensive. An alternate approach exploits the fact that for P in the inﬁnite residue disc Q but distinct from the point at inﬁnity, we may compute P ω directly using Algorithm 11. This works because both the Frobenius lift and the reduction process respect the subring of A† consisting of functions which are meromorphic at inﬁnity. When P lies in a ﬁnite Weierstrass residue disc, we may reduce to the previous case using a change of variables on the x-line to move P to the inﬁnite disc. However, one still must use the approach of the previous paragraph Q Q to reduce evaluation of P ω to evaluation of the P ωi .

4

Implementation Notes and Precision

We have implemented the above algorithms in Sage [24] for curves deﬁned over Qp . In doing so, we made the following observations. 4.1

Precision Estimates

For a tiny integral, the precision of the result depends on the truncation of the power series computed. Here is the analysis for a non-Weierstrass disc; the analysis for a Weierstrass disc, using a diﬀerent local interpolation, is similar. (For points over ramiﬁed extensions, one must also account for the ramiﬁcation index in the bound, but it should be clear from the proof how this is done.) Q Proposition 18. Let P ω be a tiny integral in a non-Weierstrass residue disc, with P, Q deﬁned over an unramiﬁed extension of K and accurate to n digits of precision. Let (x(t), y(t)) be the local interpolation between P and Q deﬁned by x(t) = x(P )(1 − t) + x(Q)t = x(P ) + t(x(Q) − x(P )) y(t) = f (x(t)). Let ω = g(x, y)dx be a diﬀerential of the second kind such that h(t) = g(x(t), y(t)) belongs to O[[t]]. If we truncate h(t) modulo tm , then the computed value of the Q integral P ω will be correct to min{n, m + 1 − logp (m + 1) } digits of (absolute) precision. Proof. Let t = t(x(Q) − x(P )). As P, Q are in the same residue disc and are deﬁned over an unramiﬁed ∞extension of K, we have vp (x(Q) − x(P )) ≥ 1. If we expand g(x(t ), y(t )) = i=0 ci (t )i , then by hypothesis ci ∈ O. Thus

26

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Q

P

ω=

Q

P 1

g(x, y)dx

g(x(t), y(t))dx(t)

= 0

=

0

=

0

x(Q)−x(P )

g(x(t ), y(t ))dt

∞ x(Q)−x(P )

ci (t )i dt

i=0

∞ ci (x(Q) − x(P ))i+1 . = i + 1 i=0

The eﬀect of omitting ci (t )i from the expansion of g(x(t ), y(t )) for some i ≥ m is to change the ﬁnal sum by a quantity of valuation at least i+1− logp (i+1) ≥ m+1− logp (m+1) . The eﬀect of the ambiguity in P and Q is that the computed value of (x(Q) − x(P ))i+1 diﬀers from the true value by a quantity of valuation at least i + 1 − logp (i + 1) + n − 1 ≥ n. For Coleman integrals between diﬀerent residue discs, which we may assume are non-Weierstrass thanks to § 3.4, one must ﬁrst account for the precision loss in Algorithm 10. According to [17, Lemmas 2,3] and the erratum to [17] (or [15]), working to precision pN in Algorithm 10 produces the fi , Mij accurately modulo pN −n for n = 1 + logp max{N, 2g + 1} . We must then take into account the objects involved in the linear system (7), as follows. Q Proposition 19. Let P ω be a Coleman integral, with ω a diﬀerential of the second kind and with P, Q in non-Weierstrass residue discs, deﬁned over an unramiﬁed extension of Qp , and accurate to n digits of precision. Let Frob be the matrix of the action of Frobenius on the basis diﬀerentials. Set B = Frobt −I, Q and let m = vp (det(B)). Then the computed value of the integral P ω will be accurate to n − max{m, logp n } digits of precision. Proof. By the linear system (7), the Coleman integral is expressed in terms of tiny integrals, integrals of exact forms evaluated at points, and a matrix inversion. Suppose that the entries of B = Frobt −I are computed to precision n. Then taking B −1 , we have to divide by det(B), which lowers the precision by m = vp (det(B)). By Proposition 18, computing tiny integrals (with the series expansions truncated modulo tn−1 ) gives a result precise up to n− logp n digits. Q Thus the value of the integral P ω will be correct to n−max{m, logp n } digits of precision. 4.2

Complexity Analysis

We assume that asymptotically fast integer and polynomial multiplication algorithms are used; speciﬁcally addition, subtraction, multiplication, and divi sion take O(log N ) bit operations in Z/N Z and O(n) basering operations in

Explicit Coleman Integration for Hyperelliptic Curves

27

R[x]/xnR[x]. In particular, this allows arithmetic operations in Qp to n (rela log p). Using tive) digits of precision, hereafter called ﬁeld operations, in time O(n Newton iteration, both square roots and the Teichm¨ uller character can be com puted to n digits of precision using O(log n) arithmetic operations. (We again consider only points in non-Weierstrass discs deﬁned over unramiﬁed ﬁelds.) Q Proposition 20. Let P ω be a Coleman integral on a curve of genus g over Qp , 2g−i with ω = dfω + i=1 ci ωi a diﬀerential of the second kind and with P, Q in nonWeierstrass residue discs, deﬁned over Qp , and accurate to n digits of precision. Let Frob be the matrix of the action of Frobenius on the basis diﬀerentials, and let m = vp (det(Frobt −I)). Let F (n) be the running time of evaluating fω at P Q and Q to n digits of precision. The value of the integral P ω can be computed 2 2 to n − max{m, logp n } digits of precision in time F (n) + O(pn g + g 3 n log p). (Over a degree N unramiﬁed extension of Qp , the analysis is the same with the runtime multiplied by a factor of N .) Proof. An essential input to the algorithm is the matrix of the action of Frobenius, which can be computed by Kedlaya’s algorithm to n digits of precision 2 2 in running time O(pn g ). Inverting the resulting matrix can be (na¨ıvely) done 3 with O(g ) arithmetic operations in Qp . It remains to be shown that no other step exceeds these running times. For the tiny integral on the ﬁrst basis diﬀerential, the power series x(t)/y(t) = x(t)f (x(t))−1/2 can be computed modulo tn−1 log n) ﬁeld operations. Each other basis using Newton iteration, requiring O(n diﬀerential can be computed from the ﬁrst by multiplication by the linear poly nomial x(t) and the deﬁnite integral evaluated with O(n) ﬁeld operations, for a 2 total of O(gn ) bit operations. Computing φ(P ) and φ(Q) to n digits of preci + log p) ﬁeld sion is cheap; directly using the formula in Algorithm 10 uses O(g operations. The last potentially signiﬁcant step is computing and evaluating the fi at each P and/or Q. The coeﬃcients of the fi can be read oﬀ in the reduction phase of Kedlaya’s algorithm, and have O(png) terms each. Evaluating (or even 2 2 2 recording) all g of these forms takes O(png ) ﬁeld operations, or O(pn g ) bit operations, which is proportional to the cost of doing the reduction. 4.3

Numerical Examples

Here are some sample computations made using our Sage implementation. Additional examples will appear in the ﬁrst author’s upcoming PhD thesis. Example 21. Lepr´evost [21] showed that the divisor (1, −1) − ∞+ on the genus 2 curve y 2 = (2x − 1)(2x5 − x4 − 4x2 + 8x − 4) over Q is torsion of order 29. Consequently, the integrals of holomorphic diﬀerentials against this divisor must vanish. We may observe this vanishing numerically, as follows. Let C : y 2 = x5 +

1 33 4 3 3 3 2 1 x + x + x − x+ 16 4 8 4 16

28

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

be the pullback of Lepr´evost’s curve by the linear fractional transformation x → (1 − 2x)/(2x) taking ∞ to 1/2. The original points (1, −1), ∞+ correspond to the points P = (−1, 1), Q = (0, 14 ) on C. The curve C has good reduction at p = 11, and we compute

Q

P

ω0 =

Q

P

ω1 = O(116 ),

Q

P

ω2 = 7·11+6·112 +3·113 +114 +5·115 +O(116 ),

consistent with the fact that Q − P is torsion and ω0 , ω1 are holomorphic but ω2 is not. Example 22. We give an example arising from the Chabauty method, taken from [23, § 8.1]. Let X be the curve y 2 = x(x − 1)(x − 2)(x − 5)(x − 6), whose Jacobian has Mordell-Weil rank 1. The curve X has good reduction at 7, and X(F7 ) = {(0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, 6), (3, −6), ∞}. By [23, Theorem 5.3(2)], we know |X(Q)| ≤ 10. However, we can ﬁnd 10 rational points on X: the six rational Weierstrass points, and the points (3, ±6), (10, ±120). Hence |X(Q)| = 10. Since the Chabauty condition holds, there must exist a holomorphic diﬀerQ ential ω for which ∞ ω = 0 for all Q ∈ X(Q). We can ﬁnd such a diﬀerential by taking Q to be one of the rational non-Weierstrass points, then computing Q Q a := ∞ ω0 , b := ∞ ω1 and setting ω = bω0 − aω1 . For Q = (3, 6), we obtain a = 6 · 7 + 6 · 72 + 3 · 73 + 3 · 74 + 2 · 75 + O(76 ) b = 4 · 7 + 2 · 72 + 6 · 73 + 4 · 75 + O(76 ). We then verify that

R Q

ω vanishes for each of the other rational points R.

Remark 23. It is worth pointing out some facts not exposed by Example 22. For instance, since ω is already determined by a single rational non-Weierstrass point, we could have used it instead of a brute-force search to ﬁnd other rational points. More seriously, in other examples, the integral ω may vanish at a point deﬁned over a number ﬁeld which has a rational multiple in the Jacobian. Such points may be diﬃcult to ﬁnd by brute-force search; it may be ∗ easier to reconstruct them from p-adic approximations, obtained by writing ∞ ω as a function of a linear parameter of a residue disc, then ﬁnding the zeroes of that function.

5

Future Directions

Here are some potential extensions of our computation of Coleman integrals.

Explicit Coleman Integration for Hyperelliptic Curves

5.1

29

Iterated Integrals

Coleman’s theory of integration is not limited to single integrals; it gives rise to an entire class of locally analytic functions, the Coleman functions, on which antidiﬀerentiation is well-deﬁned. In other words, one can deﬁne integrals Q ωn · · · ω1 P

which behave formally like iterated path integrals 1 t1 tn−1 ··· fn (tn ) · · · f1 (t1 ) dtn · · · dt1 . 0

0

0

These appear in several applications of Coleman integration, e.g., p-adic regulators in K-theory, and the nonabelian Chabauty method. As in the case of a single integral, one can use Frobenius equivariance to compute iterated Coleman integrals on hyperelliptic curves. One obtains a linear system expressing all n-fold integrals of basis diﬀerentials in terms of lower order integrals. Note that the number of such n-fold integrals is (2g)n , so this is only feasible for small n. The cases n ≤ 4 are already useful for applications, but ideas for reducing the combinatorial explosion for larger n would also be of interest. (One must be slightly careful in dealing with Weierstrass residue discs; see Remark 17.) We have made some limited experiments with double Coleman integrals in Sage. The Fubini identity Q

P

ω2 ω1 +

Q

P

ω1 ω2 =

Q

P

ω1

Q

P

ω2

turns out to be a useful consistency check for both single and double integrals. 5.2

Beyond Hyperelliptic Curves

It should be possible to convert other algorithms for computing Frobenius actions on de Rham cohomology, for various classes of curves, into algorithms for computing Coleman integrals on such curves. Candidate algorithms include the adaptation of Kedlaya’s algorithm to superelliptic curves by Gaudry and G¨ urel [14], or the general algorithm for nondegenerate curves due to Castryck, Denef, and Vercauteren [7]. It should also be possible to compute Coleman integrals using Frobenius structures on Picard-Fuchs (Gauss-Manin) connections, extending Lauder’s deformation method for computing Frobenius matrices [20]. 5.3

Heights After Harvey

We noted earlier that our algorithms for Coleman integration over Qp have linear runtime dependence on the prime p, arising from the corresponding dependence

30

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

in Kedlaya’s algorithm. In [15], Harvey gives a variant of Kedlaya’s algorithm with only square-root dependence on p (but somewhat worse dependence on other parameters), by reorganizing the computation so that the dominant step is ﬁnding the p-th term of a linear matrix recurrence whose coeﬃcients are polynomials in the sequence index. Harvey demonstrates the practicality of his algorithm for primes greater than 250 , which may have some relevance in cryptography for ﬁnding curves of low genus with nearly prime Jacobian orders. It should be possible to use similar ideas to obtain square-root dependence on p for Coleman integration, by constructing a recurrence that computes not just the entries of the Frobenius matrix but also the values fi (P ) and fi (Q). However, this is presently a purely theoretical question, as we do not know of any applications of Coleman integration for very large p. Acknowledgments. The authors thank William Stein for access to his computer sage.math.washington.edu (funded by NSF grant DMS-0821725), and Robert Coleman and Bjorn Poonen for helpful conversations. Balakrishnan was supported by a National Defense Science and Engineering Graduate Fellowship and an NSF Graduate Research Fellowship. Bradshaw was supported by NSF grant DMS-0713225. Kedlaya was supported by NSF CAREER grant DMS0545904, the MIT NEC Research Support Fund, and the MIT Cecil and Ida Green Career Development Professorship. Some development work was carried out at the 2006 MSRI Summer Graduate Workshop on computational number theory, and the 2007 Arizona Winter School on p-adic geometry.

References 1. Balakrishnan, J.S.: Local heights on hyperelliptic curves (2010) (in preparation) 2. Berkovich, V.G.: Integration of one-forms on p-adic analytic spaces. Annals of Mathematics Studies, vol. 162. Princeton University Press, Princeton (2007) 3. Besser, A.: Coleman integration using the Tannakian formalism. Math. Ann. 322(1), 19–48 (2002) 4. Besser, A.: On the computation of p-adic height pairings on Jacobians of hyperelliptic curves, Sage Days 5 (2007), http://wiki.sagemath.org/days5/sched 5. Besser, A., de Jeu, R.: Li(p) -service? An algorithm for computing p-adic polylogarithms. Math. Comp. 77(262), 1105–1134 (2008) 6. Bosch, S., G¨ untzer, U., Remmert, R.: Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Springer, Berlin (1984) 7. Castryck, W., Denef, J., Vercauteren, F.: Computing zeta functions of nondegenerate curves. IMRP Int. Math. Res. Pap., Art. ID 72017, 57 (2006) 8. Coleman, R., de Shalit, E.: p-adic regulators on curves and special values of p-adic L-functions. Invent. Math. 93(2), 239–266 (1988) 9. Coleman, R.F.: Dilogarithms, regulators and p-adic L-functions. Invent. Math. 69(2), 171–208 (1982) 10. Coleman, R.F.: Torsion points on curves and p-adic abelian integrals. Ann. of Math. (2) 121(1), 111–168 (1985) 11. Coleman, R.F., Gross, B.H.: p-adic heights on curves. In: Algebraic Number Theory – in honor of K. Iwasawa. Advanced Studies in Pure Mathematics, vol. 17, pp. 73– 81 (1989)

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12. Fresnel, J., van der Put, M.: Rigid analytic geometry and its applications. In: Progress in Mathematics, vol. 218. Birkh¨ auser Boston Inc., Boston (2004) 13. Furusho, H.: p-adic multiple zeta values. II. Tannakian interpretations. Amer. J. Math. 129(4), 1105–1144 (2007) 14. Gaudry, P., G¨ urel, N.: An extension of Kedlaya’s point-counting algorithm to superelliptic curves. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 480–494. Springer, Heidelberg (2001) 15. Harvey, D.: Kedlaya’s algorithm in larger characteristic. Int Math Res Notices, Article ID No. rnm095, 2007, 29 (2007) 16. Harvey, D.: Eﬃcient computation of p-adic heights. LMS J. Comput. Math. 11, 40–59 (2008) 17. Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001); erratum ibid 18, 417– 418 (2003) 18. Kim, M.: The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45(1), 89–133 (2009) 19. Kim, M.: Massey products for elliptic curves of rank 1. J. Amer. Math. Soc. 23, 725–747 (2010); Erratum by Balakrishnan, J.S., Kedlaya, K.S., Kim, M., http://www.ucl.ac.uk/~ ucahmki/ 20. Lauder, A.G.B.: Deformation theory and the computation of zeta functions. Proc. London Math. Soc. 88(3), 565–602 (2004) 21. Lepr´evost, F.: Jacobiennes de certaines courbes de genre 2: torsion et simplicit´e. J. Th´eor. Nombres Bordeaux 7(1), 283–306 (1995) 22. Mazur, B., Stein, W., Tate, J.: Computation of p-adic heights and log convergence. Doc. Math. Extra, 577–614 (2006) (electronic) 23. McCallum, W., Poonen, B.: The method of Chabauty and Coleman (2007) (preprint) 24. Stein, W.A., et al.: Sage Mathematics Software (Version 4.3.5), The Sage Development Team (2010), http://www.sagemath.org

Smallest Reduction Matrix of Binary Quadratic Forms And Cryptographic Applications Aurore Bernard1 and Nicolas Gama2 1

XLIM, Limoges, France [email protected] 2 GREYC Ensicaen, Caen, France [email protected]

Abstract. We present a variant of the Lagrange-Gauss reduction of quadratic forms designed to minimize the norm of the reduction matrix within a quadratic complexity. The matrix computed by our algorithm 1 4 on the input f has norm O f 1 2 Δf , which is the square root of the best previously known bounds using classical algorithms. This new bound allows us to fully prove the heuristic lattice based attack against NICE Cryptosystems, which consists in factoring a particular subclass of integers of the form pq 2 . In the process, we set up a homogeneous variant of Boneh-Durfee-HowgraveGraham’s algorithm which ﬁnds small rational roots of a polynomial modulo unknown divisors. Such algorithm can also be used to speed-up factorization of pq r for large r.

1

Introduction

Binary quadratic forms appeared progressively in the 17-th century, when Descartes and Fermat ﬁrst introduced the concept of coordinates as a tool to algebraically solve geometric problems. Those forms have wide applications in mathematics and physics, especially in geometry, numerical analysis or algebraic topology. A binary quadratic form is a homogeneous polynomial of degree two in two variables, which can be viewed as the Cartesian equation of a surface f x, y ax2 bxy cy 2 on a given basis of R2 . Of course, this equation varies with the basis of expression, and it is natural to deﬁne an equivalence relation to regroup all these possible equations into classes. Over the real ﬁeld, there are six classes corresponding to the Sylvester’s signatures. They can be distinguished by the sign of the discriminant Δf b2 4ac, and the sign of a c. Forms of strictly negative discriminant (imaginary forms) have a unique zero at the origin, which is also their unique local and global extremum. Forms of strictly positive discriminant (real forms) represent a saddle-shape. Meanwhile, quadratic forms were also used over the integer ring by Fermat, Lagrange and Gauss to solve long standing problems from number theory. This time, binary quadratic forms are equations with integer coeﬃcients of discrete G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 32–49, 2010. c Springer-Verlag Berlin Heidelberg 2010

Smallest Reduction Matrix of Binary Quadratic Forms

33

scatter-plots on a given lattice basis of Z2 . One deﬁnes a similar equivalence relation by base change, except that transformation matrices are now unimodular, and that they preserve the value of the discriminant. Problems related to this equivalence are more complicated than on the real ﬁeld: for instance, in both real and imaginary cases, we do not know any polynomial way to compute the number of equivalence classes of a given discriminant. Deciding the equivalence of two forms is easy in the imaginary case, where each class contains a unique reduced representative computable in polynomial time. However, the problem is hard in the real case, where there are, depending on the notion of reduction, either an exponential number of polynomially computable reduced representatives, or a few representatives computable in exponential time. A reduction algorithm takes as input a quadratic form and outputs a reduced form and the reduction matrix, which is a unimodular base-change matrix used to obtain this form. The most famous polynomial time reduction algorithms are Lagrange algorithm [15] (1773) commonly known as ”Gauss reduction” algorithm [11] (1801). In [14] (1980), Lagarias modiﬁed the Gauss reduction algorithm for make it more eﬃcient. This algorithm is the one used in practice, and which we refer as the Gauss reduction algorithm, or Classical Gauss, if we need to diﬀerentiate it from new ﬂavors which we propose. The cryptanalysis of [6] shows experimental evidences that the small size of reduction matrices have important applications to the factorization of some large numbers used in public key cryptosystems, especially those of the NICE cryptosystems (see [12,13]). However the best currently known upper-bounds on the size of reduction matrices [14,1] are by an order too large, and keep all these results on the factorization heuristic. In this paper, we specially design an eﬃcient variant of the Gauss reduction algorithm to minimize the size of transformation matrix, and we prove constructive upper-bounds which are tight both in the worst case and in the average case. These bounds, combined with an improvement of the methods of [6], allows us to prove all the above mentioned heuristics of on the factorization of integers from the NICE cryptosystems.

2

Preliminaries and Notation

In this section we recall some deﬁnitions and properties concerning binary quadratic forms. For a more detailed account of the theory see [5,4,9]. Then, we summarize some results on the norm of a matrix. Quadratic Forms. A binary quadratic form f is a homogeneous polynomial of degree two in two variables f x, y ax2 bxy cy 2 with a, b, c Z3 which we abbreviate as f a, b, c. Throughout this paper the word form will be used in the sense of binary quadratic form. It is said primitive when gcd a, b, c 1. The discriminant of f is Δf b2 4ac. A discriminant Δf is called fundamental if all the forms of discriminant Δf are necessarily primitive: for example, it is the case of all odd and square-free integers. The set of all primitive forms of discriminant Δf is denoted FΔf . We impose that the discriminant is not a perfect square then a and c are always non-zero. The form f can be factored as

34

A. Bernard and N. Gama

f x, y a x yζf x yζf where ζf and ζf are the complex roots of the univariate polynomial f x, 1 which we call the aﬃne representation of f . When Δf 0, each root of f live in RQ and the form is real. In this case, ζf will denote the smallest root and ζf the largest one. When Δf 0, the roots are in CR and the form is imaginary. We note λ f min f x, y : x, y Z2 0, 0 the ﬁrst minimum of f . We note Mt the transpose of amatrix M. The a b 2 polar representation of f is the symmetric matrix b of determinant

2 c αβ Δf 4. Let M M2 Z be a 2 2 matrix with integer enγ δ tries which we often abbreviate as α, β; γ, δ . We note Id the identity matrix of M2 Z. The composition action of M on f is deﬁned as the form g x, y f αx βy, γx δy and it is noted g f.M. The coeﬃcients of g are g f α, γ , b αδ γβ 2 aαβ cγδ, f β, δ. We remark that for each Composition Action.

root ζg of g, t

αζg β γζg δ

is a root of f . Finally, the polar representation of g is

M f M which implies that Δg

det M2 Δf .

Group action. Let GL2 Z be the general linear group of matrices in M2 Z which are invertible and its subgroup SL2 Z the special linear group of matrices which have a determinant equal to one. The action deﬁned with either GL2 Z or SL2 Z on the set of primitive forms FΔf of a given discriminant is a (right) group action. Two forms f and g are equivalent if they belong to the same SL2 Z-orbit. In this case we note f g. We deﬁne Aut f the group of automorphisms of the form f FΔf as M SL2 Z, trace M 0 and f.M f . The set of all automorphisms of f is Aut f . The group Aut f is known to be cyclic, and we call its generator the fundamental automorphism of f . The largest eigenvalue of the fundamental automorphism of f is the fundamental unit. It only depends on the discriminant Δf , and will be denoted Δf .

1 0 Three specials transformations. We deﬁne the symmetry S , the 0 1 01 1h exchange E and the translation by an integer T h . They are 10 01 three (linear) transformations of GL2 Z. All matrices in GL2 Z can be written as a product of powers of these three transformations and SL2 Z is generated by the product ES and T 1. The action of these transformations on f are f.S a, b, c, f.E c, b, a f.T h a, b 2ah, f h. Note the important fact: the roots of f.S are the opposite of the roots of f and the roots of f.E are the inverse of the roots of f , and that T h subtracts h to each roots of f . Norms of matrices and forms. Let M α, β; γ, δ be a matrix in M2 Z. The Euclidean norm is M2 α2 β 2 γ 2 δ 2 , and the maximum norm is M max α, β , γ , δ . The norm M supv2 1 M.v2 is the induced Euclidean norm, which is also the square root of the largest eigenvalue of Mt M. All the norms are equivalent: M M M2 2M.

Smallest Reduction Matrix of Binary Quadratic Forms

35

Additionally, the induced norm is sub-multiplicative: if N M2 Z then MN M N and Id 1, and it is lower-bounded by the spectral radius ρ M, which is the supremum among the absolute values of the eigenvalues of M. By extension, we deﬁne the norms f ,f 2 and f of a form as the corresponding norm of its polar representation.

3

A New Reduction Algorithm for Real Quadratic Forms

A form f a, b, c is reduced if it satisﬁes two conditions simultaneously: a normalization condition, which deﬁnes the choice of the representative of b mod 2a, and a reduction condition, which often upper-bounds the size of a (or c). In the imaginary case, these conditions are very natural: a form is normal if and only if b a, a is minimal, and is reduced if additionally, a is the minimum λ f . A single translation is needed to normalize any form. However, the reduction condition takes more steps to be achieved. The classical Gauss reduction reduces a form by successive swaps SE and normalizations T b 2a (see [1]) until f is reduced. The Gauss reduction algorithm operates in quadratic time (see [1,21,18]). For each form f of discriminant Δf 4, there exists a unique reduced form g in each equivalence class, and a unique reduction matrix M SL2 Z such that f.M g. In this case Aut f Id . In the real case (Δf 0), the previous reduction conditions applied on f a, b, c are too restrictive, since the smallest integers α, β 0, 0 such that f α, β λ f are in general exponential in the size of f . No polynomial time algorithm can output an exponential reduction matrix. Thus, according to classical notions, f is classically normalized if and only if b a, a when a Δf and b Δ 2a, Δf when a Δf , and f is classically f reduced if additionally, Δf 2a b Δf . It is known that only a ﬁnite subset of forms of discriminant Δf are classically-reduced, and that they form a reduced cycle in each class. The Real-Gauss reduction algorithm, which uses the classical normalization, ﬁnds a reduced form equivalent to its input in quadratic time (see [1]). In this paper, given a normalized form f , we will bound the coeﬃcients of the smallest reduction matrix M α, β; γ, δ such that g f.M ag , bg , cg is reduced. The case of imaginary forms is eased by the uniqueness of the reduction a,c . We improve this matrix. Lemma 5.6.1 in [1] give us that M 2 max upper-bound with the following theorem:

Δf

Theorem 1 (Imaginary Bound). Let f a, b, c be a normalized imaginary form of discriminant Δf 0, and M α, β; γ, δ the reduction matrix such that g f.M ag , bg , cg , M satisﬁes these two upper-bounds: 1) M

2) αβγδ

23

14

c

ag 1 γδ 2

2 31 4

ac Δf

14

.

36

A. Bernard and N. Gama

bounded by fore γδ

α γ b 2 Δf2 , which can be lower2a 4a 4 aag 4 cc 2 γ Δf , and similarly δ 2 Δfg . There-

Proof. One has ag f α, γ aγ 2

Δf γ 2 . It follows that 4a 4 ac . The ﬁrst inequality comes from 3ag cg 3Δf

Δf ,

because

g is reduced. Unless the transformation is trivial (Id or SE), the normalization condition induces the inequalities α γ and β δ , which proves 1 1 αβγδ 4 γδ 2 . Thus, the norm of the reduction matrix is in fact basically in O

f

Δf

.

In the real case however, this proof would not apply directly, because the Δf b 2 α term

γ 2a 4a2 can be exponentially close to 0. The problem is that in the real case, each reduced cycle contains a large (often exponential) number of equivalent reduced forms, and some of them are exponentially far from f . A constructive approach is needed to build a polynomial reduction matrix. The analysis of the Gauss reduction algorithm in [1,14] basically proves that the norm of the computed reduction matrix is bounded by O f . In this paper, we study a variant of this algorithm which ﬁnds a reduction matrix of norm O

f

Δf and we verify that it is tight even in the average case. We deﬁne new relaxed notions of reduction and normalization, and express them according to the roots of the forms, which is more intuitive than the classical conditions on the coeﬃcients: Definition 1. A real binary quadratic form f is: – primary normalized if 0 ζf 1 and primary reduced if also ζf 1 – secondary normalized if 1 ζf 0 and secondary reduced if also 1 ζf .

Finally f is largely reduced if it is either primary or secondary reduced. Both primary and secondary notions are exchanged by the action of S, which negates the roots. Furthermore, primary and secondary reductions are exchanged by E, which inverts the roots. As usual, primary and secondary normalization can always be achieved by the action of some T h. Note that a classically normalized form, which has by deﬁnition at least one root in the interval 1, 1, is either primary or secondary normalized. Similarly, a classically reduced form a, b, c is a largely-reduced form satisfying b 0, which can again be ensured by the action of S. Our main contribution is to solve the following problems, which are equivalent. Lemma 1. The two problems are equivalent: 1. Smallest SL2 Z matrix Given a classically-normalized real form f , ﬁnd M SL2 Z such that f.M is classically-reduced and M is minimal. 2. Smallest GL2 Z matrix Given a primary-normalized real form f , ﬁnd M GL2 Z such that f.M is largely-reduced and M is minimal.

Smallest Reduction Matrix of Binary Quadratic Forms

37

Proof. From a solution M GL2 Z of Problem 2, one deduces a solution of Problem 1 by left-multiplication by Id or S to make the normalization of the input correspond, followed by a right-multiplication by Id or S to force the coeﬃcient b of the reduced form to be positive, followed by a right multiplication by Id or E so that the determinant is 1. The reduction of Problem 2 to Problem 1 is similar. Since Id, S and E are permutation matrices, they do not modify these norms or . Remark that, reducing a problem to the other also preserves the absolute value of the product of the coeﬃcients in each row of the reduction matrices. Lemma 1 motivates the search of a reduction algorithm solving the less restrictive Problem 2, since we can use the above permutation matrices to return to classical notions in SL2 Z. 3.1

Algorithm and Analysis

Let f be a real form. We deﬁne the two integers h f and hf as hf hf

ζf

and

ζf . It is easy to show that h f and hf are respectively the unique integers

such that f.T h f is primary-normalized, and f.T hf is secondary-normalized. Among the two integers hf , h f the one of smallest absolute value is noted h f : that is h f hf if hf hf , and h f hf otherwise. In other words, h f is the shortest normalization of f . As a comparison, there is only a single integer νf in the classical case such that f.T νf is classically-normalized, νf being one of the integers hf , h f but not necessarily the one with the smallest absolute value. Our reduction algorithm, is a variant of the Gauss reduction which operates in GL2 Z. It alternates exchange E and the shortest normalization T h f at each loop, and terminates on a largely-reduced form. As we will see later, any kind 1 of normalization by hf or h f would make a reduction algorithm terminate , but the choice of the shortest normalization h f instead of the classical νf (especially during the last steps) is the key element to minimize the reduction matrix. The main result of the section is the following theorem on the quality of the output of our algorithm, which is the real-case analogue of Theorem 1. Algorithm 1. RedGL2 Input: f a, b, c a primary-normalized form Output: f.M a largely-reduced form and M GL2 Z 1: M Id 2: while f not largely-reduced do 3: f f.E and M ME 4: f f.T hf and M MT hf 5: end while 6: return f and M

1

Exchange step Normalization step

The original Gauss algorithm of 1801 used actually the largest normalization at each step. The number of reduction steps is exponential on some entries. Lagarias introduced the classical normalization to obtain a quadratic complexity

38

A. Bernard and N. Gama

Theorem 2 (Real bound). Let f a, b, c be a primary-normalized form of logdiscriminant Δ 0. Given f as input, RedGL2 terminates after at most

a Δ 4 iterations where ω 15 is the gold number. Its output M 2 log ω 2 α, β; γ, δ and fr

1) M 4 2) αβγδ

14

a

f.M ar , br , cr satisﬁes:

ar

γδ 2 1

21

a

Δ.

Before proving this theorem, we remark that the best known upper-bounds achieved by the classical Gauss algorithm under the conditions same (see theo1 rem 4.4 of [1]) are M a 1 1 Δ and γδ 2 a Δ 1 1 Δ. They are basically the square of the upper-bounds of RedGL2. Figure 1 and 2 illustrate respectively the families of forms Fn n, b, 1 and Gn n, n, 1 with n N and b 2n 3 2 3 , which are families of forms where the Gauss reduction algorithm outputs reduction matrices Δ times larger than our variant RedGL2. Finally, note that a multiplicative triangular inequality on the norms of the polar representations of f fr .M 1 yields f fr 2M, which conﬁrms the optimality of Theorem 2 in average. The analysis of Gauss reduction algo log a Δ rithm in [1] upper-bounds the number of iterations by 2 log 2 2 reduction steps. Our upper-bound on the number of iterations of RedGL2 is tight in the worst case, and is only by a multiplicative factor around 1.4 larger than the maximum number of iterations of the Gauss reduction algorithm. However the primary goal of RedGL2 is the minimization of the reduction matrix. 3.2

Proof of Theorem 2

To prove Theorem 2, we ﬁrst study the termination cases, characterized by the presence of integers between the roots of f.E, and where the choice of the shortest normalization is of greatest importance. Eventually, we shall treat the general case and the complexity. Termination cases. We ﬁrst study the two cases where the algorithm terminates in a single step of reduction. The ﬁrst one deals with normal form f containing exactly one integer between its roots. This is the only case where hf h f , so all notions of normalizations (classical, primary, secondary, shortest) coincide. Lemma 2. Let f a, b, c be a real form satisfying 1 ζf 0 ζf 1, and h h f.E . The form fr f.ET h ar , br , cr is largely-reduced, and its coeﬃcients satisfy ar c, cr a, and h2 ar a. Proof. The reduction matrix from f to fr is ET h 0, 1; 1, h. Consider the parabola p x cx2 bx a which is the aﬃne representation of g f.E. Then we have h h g , and ζg hg 1 1 h g ζg , cr p h g b and p 0 a. By deﬁnition of h we have two cases: if 2c 0 then we have h hg 0 b 2c, else we have b 2c 0 h h g . In both cases we

Smallest Reduction Matrix of Binary Quadratic Forms px=cx2 +bx+a

h

px=cx2+bx+a

a

a

h

ζg ζg

0 νg

b

2c

Gauss

ζg

h=hg νg

cr

39

h-1

ζg h =hg

cr

h-1

Red GL2

Fig. 1. Illustration of Lemma 2 This figure illustrates the convexity inequalities of Lemma 2. In this case, the shortest normalization chosen by RedGL2 is hg ζg , which can be O Δ smaller than the classical normalization ν g ζg in Gauss Algorithm. It is clear that cr is in the interval 0, a . Comparison of heights of the two rectangles on the same convex and decreasing branch of the parabola, gives ch2 a .

Fig. 2. Illustration of Lemma 3 This figure is the analogue for Lemma 3. In this case, the shortest normalization chosen by RedGL2 is hg ζg , which can be O Δ smaller than the classical normalization in Gauss algorithm is ν g ζg . Inequality on the slopes of p before and after ζg gives cr a . Comparison of heights of the two rectangles on the same convex and decreasing branch of the parabola, gives ch 12 a .

graphically verify that cr p h p 0 a (see Figure 1). A convexity inequality on p between 0, h and b 2c, b 2c h shows a cr ch2. Since a and cr have the same sign and a is larger, then a ar h2 . Theorem 2 holds in this termination case: the reduction matrix is M 0, 1; 1, h. By Lemma 2, its norm satisﬁes M h a ar . Since f fr .M 1 , its ﬁrst coeﬃcient is a ar h2 br h cr , thus br h a cr ar h2 and br h2 4ar cr h2 Δ h2 a2 c2r a2r h4 2acr 2aar h2 2ar cr h2 a cr ar h2 2 9a2 , which proves the second point of Theorem 2. The second case of single-step termination concerns normalized form f such that at least two integers lie between the roots of f.E (namely hf.E h f.E ). We just write a proof for primary-normalized forms, but it can be easily extended to secondary-normalized forms. Lemma 3. Let f a, b, c be a real form satisfying 0 ζf ζf 1, and such that hf.E h f.E . If h h f.E , then fr f.ET h ar , br , cr is secondaryreduced, and its coeﬃcients satisfy ar c, cr a, and h2 ar 4a. Proof. The proof of this lemma is also based on convexity inequalities. Let g f.E, of aﬃne representation p x cx2 bx a. Note that h ζg 2. Again, one has p 0 a, p h cr . It follows from the deﬁnition that fr is

40

A. Bernard and N. Gama

secondary-reduced. The reduction matrix is M 0, 1; 1, h, which proves ar c. Application of a convexity inequality (see Figure 2) on p in the two intervals b b 0; h 1 and 2c h 1; 2c of same length yields ar h 12 a p h 1 a, therefore ar h2 4ar h 12 4a. Finally, another convexity inequality centered on ζg gives p

h cr .

p ζg

p 0

0

ζg

p ζg , so a p 0 h ζg

ζg

p h

h ζg

Once again, Theorem 2 holds in this termination case, but this time, M 2 h 2 a ar and Δ h2 a cr ar h2 2 6a .

General case. We now prove the general case of Theorem 2. We call fi ai , bi , ci the successive values of f at the beginning of the while loop of Algorithm 1, and hi h fi .E . We suppose that the primary-normalized form f0 does not have any integer between its roots (otherwise it would either already be reduced or as in Lemma 2). Thus 0 ζf0 ζf0 1. For each iteration i in the loop, if there is at least one integer between the roots of fi .E, then we set m i 1 and the algorithm reaches one of the two termination cases above. Otherwise the shortest normalization hi is the primary one hi h fi .E hfi .E . Thus fi is also primary-normalized and 0 ζfi ζfi 1. Note that the

distance between the roots strictly increases ζfi

ζf

i1

1

ζf

i1

i1

ζf

ζf E ζf E

ζf ζf ζf . Such process can not hold for- ever, otherwise the integer sequence of the ﬁrst coeﬃcients ai Δ ζf ζf i1

ζf

i1

i

i1

i1

i1

i

i

would be strictly decreasing. This proves the termination of the algorithm. The integer m is the smallest index, such that fm 1 .E contains at least one integer between its roots. The shortest normalization hm 1 hfm1 .E h fm1 E is in this case secondary, and satisﬁes hm 1 2. We eventually use the following lemma to conclude the proof of Theorem 2. Lemma 4. Let f a, b, c and g ag , bg , cg be two real forms and M α, β; γ, δ GL2 Z such that f.M g. If all the roots of g are positive and γ 0 and δ 1 then ag δ 2 a. Proof. If γ 0, then M is triangular, so α δ 1 and ag a. We now αζ β suppose γ 0. Let ζg be a root of g, then ζf γζgg δ is a root of f . We have 2 α γ ζf 1 γ ζg γδ 1 γδ thanks to the positivity conditions. Since this

bound holds for both roots of f , ag γ 2 a α γ ζf

α γ ζ a δ2 . f

We continue the proof of Theorem 2 by applying this lemma to the main loop of RedGL2. Note that for each i 1; m, the reduction matrix from f0 to fi is Mi

0 1 1 h0

0 1 0 1 ... 1 h1 1 hi 1

αi βi . γi δi

(1)

Smallest Reduction Matrix of Binary Quadratic Forms

41

Their coeﬃcients are all positive, and satisfy these recurrence equalities for i 2: γi1 δi hi 1 δi 1 δi 2 and δ0 , δ1 1, h1 αi1 βi hi 1 βi 1 βi 2 and β0 , β1 0, 1 Since all the hj j 0..i are greater than 1, it follows that αi min βi , γi max βi , γi δi and Mi δi ω i 2 by induction and comparison to the Fibonacci sequence 2 . Applying Lemma 4 on f0 and fm 1 implies that Mm 1 2 a0 am 1 . At iteration m, Lemma 4 can be applied to fm .T 1, which has positive roots and shares its ﬁrst coeﬃcient am with fm . The transformation matrix Mm T 1 Mm 1 0, 1; 1, hm 1 1 still satisﬁes the conditions of Lemma 4 because hm 1 2. We obtain Mm T 12 a0 am, and ﬁnally Mm 2 4a0 am after a backwards translation by T 1. We already know that fm is secondary-normalized and that the largest root of fm is positive. There are two cases: 1. If the largest root of fm is strictly greater than 1, then r m, fr is secondaryreduced, and the reduction matrix is Mm αm , βm ; γm , δm . One already 2 has Mm 2 4a0 ar . From f0 fr .M 1 , we draw a0 ar δm br δm γm 2 2 2 2 2 2 2 2 2 cr γm , so Δδm γm br δm γm 4ar cr δm γm a0 ar δm cr γm . Since 2 2 2 by construction γm δm 1 Mm 1 and by Lemma 3 applied on fm 1 2 2 and fr , cr am 1 , one ﬁnds Δδm γm 6 a0 2 . 2. If the second root of fm is strictly lower than 1, then by Lemma 2, fm1 is αr βr 0 1 Mm , and reduced. The matrix of reduction is M 1 hm γr δr r m 1. Thus M2 Mm 2 1 hm2 4a0 am 4h2m 16a0 ar . One still has Δδr2 γr2 a0 ar δr2 cr γr2 2 21a0 2 , because cr am by Lemma 2. This concludes the proof of items 1 and 2 of Theorem 2. It remains the complexity issue, proved in the following paragraph. Complexity. We now prove the number of iterations performed by RedGL2. Two steps before the end, at iteration r 2 of RedGL2, we know that the form fr 2 ar 2 , br 2 , cr 2 satisﬁes Δ ar 2 , because the distance between the roots of fm1 is smaller than 1. By Lemma 4 we have ω r 4

a0 ar 2

M r 2

a0

Δ. It follows that r 4 is upper-bounded by

log a Δ steps where ω 12 5 . 2 log ω The worst case complexity of algorithm RedGL2 is reached when all the normalizations occurring in the algorithm until the index r 2 are by h 1. For instance, we experimentally verify that it is the case on this family of inputs g. T 1E n where g is reduced and n grows. 2

The ith number of the sequence of Fibonacci numbers is bigger than ω i2 .

42

4

A. Bernard and N. Gama

Proof of Heuristic Cryptanalysis of the NICE Cryptosystems

We propose an application of the results of the previous section to the cryptanalyses of the NICE cryptosystems. There are two variants, which are by chronological order NICE Imaginary [12] (with imaginary forms), and NICE Real [13] (with real forms). Their security relies on the intractability of factorization of the public discriminant N . They were designed for a similar level of security as RSA, but with faster decryption, since the decryption process has quadratic complexity. Both are now considered as broken. The ﬁrst one succumbed by a proved arithmetic attack in [7]. However, the more general attack against both versions of NICE (in [6]) using lattice reduction remains only experimental and relies on two heuristic assumptions. In this paper, we provide an alternative point of view on the lattice attack, which allows to avoid the use of these heuristics and to prove the attack entirely. Both variants of NICE (Real and Imaginary) have originally been described in terms of ideals of quadratic orders, and are based on a morphism between classes of primitive forms of fundamental discriminant p and classes of primitive forms of non-fundamental discriminant N q 2 p. These notions are actually not needed here to understand the lattice attack, therefore we will here give a simple description solely in term of quadratic forms. 4.1

Lifting Quadratic Orders

We summarize some important properties on the relation between the sets Fp and FN of primitives forms of discriminants respectively p and N q 2 p, using the terminology we introduced in the last section. For the cryptographic interest we restrict ourselves to the case where q is an odd prime. The following background theory can be found in [5,4,9]. Integer matrices of determinant q. We deﬁne an equivalence relation modulo SL2 Z between two integer matrices A and B M2 Z by A B M SL2 Z, AM B. The 2 2 integer matrices of determinant q correspond to matrices of rank 1 mod q, they fall into q 1 equivalence class, which are characterized by the (projective) direction from 0, 1, ..., q 1, of their image mod q. Each class contains a unique Hermite normal form: qk 10 Qk , k 0, . . . , q 1 or Q . 01 0q Lift. As we can see in [5, section 7], for each form f of discriminant N pq 2 and each M M2 Z of determinant q, there exists a (non-unique) form g Fp such that f g.M. When M Q , we deﬁne a particular function ϕ (also called lift ) which computes such g Fp from f a, b, c FN such that 2 gcd a, q 1 as follows: ϕ f a, bq2ah , ah q2bhc where h 1 q, . . . , 0 and h b 2a mod q. Note that all the divisions are exact since f is primitive of discriminant N 0 mod q 2 and q is an odd prime. It must be noted that the

Smallest Reduction Matrix of Binary Quadratic Forms

43

lift preserves the ﬁrst coeﬃcient a of the form. It is also clear that ϕ preserves primary normalization, because its action on the roots of f is a translation by h 0, q 1 followed by a division by q, which stabilizes the interval 0, 1 of the largest root. Finally, equivalence of forms is stable by lift !f, f FN , f f " ϕ f ϕ f . The converse is in general false. Given a form g Fp and U its fundamental automorphism, there are exactly q p q primitive forms (in FN ) among

g.Q0 , . . . , g.Qq 1 , g.Q where p q denotes the Legendre symbol. These forms split into q p q sq sets of sq equivalent forms (see [5] theorem 7.4), where sq is the order of U modulo q. The fundamental unit N is equal to the power s p q . These q p q sq diﬀerent classes of equivalence are the only ones to be lifted to the class of g. Reduced cycle. Let g FΔ be a classically-reduced form of discriminant Δ 0, the right neighbour of g is the classical normalization of g.SE. If we note H g the largest normalization of g (by the integer among hg , h g of largest absolute value), then the right neighbour of g is g.SET H g.SE . Successive iterations of the right neighbour enumerates all the reduced forms equivalent to g, and deﬁne the reduced cycle of the class of g. The cardinality of such reduced cycle is in O log Δ where Δ is the fundamental unit. Principal cycle, and q-belt. The principal class of a discriminant Δ 0 is the class containing 1, 1, #. The principal form is the classical-normalization of this form, and the principal cycle ½Δ is the reduced cycle of the principal class. Note that the principal class is the only class containing a form of ﬁrst (or last) coeﬃcient equals to 1. We deﬁne the q-belt of a discriminant N pq 2 as the set of all primary normalized forms q 2 , kq, # of the principal class. Necessarily, k p, 2q p. There are exactly sq 1 forms in the q-belt of N : let g0 be the principal form 1, #, # of FN and f ϕ g0 is (necessarily) the principal form of Fp . Let U be the fundamental automorphism of f , we set by induction k0 and ki the unique integer such that U Qki1 Qki for i 1. Note that Qki U i Qk0 , and that the order of U mod q is precisely sq , therefore the sequence ki is periodic and ksq k0 . Finally, the q-belt of N is the set g1 f.Qk1 , . . . , gk f.Qksq 1 . They are indeed primary-normalized and equivalent by construction. A transformation matrix from gi to gi 1 is by construction Qki1 U Qki1 SL2 Z, because U Qki1 Qki . 4.2

Cryptosystem Real NICE

We now describe the NICE Real encryption and decryption. The public key is a composite integer N pq 2 and the secret key p, q with p and q two distinct primes of the same size, satisﬁes two conditions: – p is a Schinzel prime [19] which is a positive squarefree integer of the form p A2 x2 2Bx C with A, B, C, x Z, A 0 and B 2 4AC dividing 4 gcd A2 , B 2 . Such special primes implies a very low number of reduced

44

A. Bernard and N. Gama

forms in each class, namely there are O log p reduced forms in Fp in each equivalence class ([8] and [22, theorem 5.8, p. 52]). It is therefore practical to enumerate every reduced form equivalent to a given one. With a generic discriminant, the number of reduced forms per cycle would be exponential, around O p) (see [3]). To avoid any confusion, please note that even for a Schinzel prime, the number of classes in Fp remains exponential. – q is such that sq is linear in q. This imply that the number of reduced forms of discriminant N q 2 p in each equivalence class is at least linear in q and upper-bounded by O q log p, which is exponential. The encryption of a message m works as follows: m is embedded into a (usually prime) integer a p 2 which satisﬁes some low-probability pattern, and such 2 that q p is a square modulo a. This integer is expanded into a quadratic form fs a, b , c of discriminant q 2 p (which is not printed). The ciphertext is a random reduced form fc equivalent to fs (there are exponentially many). It can be generated from fs by successive multiplications by random unimodular matrices and reductions. The decryption algorithm lifts the ciphertext in Fp and enumerate all the reduced forms equivalent to ϕ fc , looking for the pattern. Of course, the knowledge of q is needed to compute ϕ. There are only O log p of them. It will necessarily ﬁnd it, because the (unknown) lift of fs fc is an equivalent form ϕ fs a, #, #, whose normalization a, #, # is reduced due to the small size of a, and it satisﬁes the pattern by construction. Due to the small number of reduced forms, it is likely the only one of the small reduced cycle to satisfy the pattern, and the plaintext m is eventually extracted from a. 4.3

Cryptanalysis

The cryptanalysis of NICE Real presented in [6] works as follows. The authors present an algorithm inspired of Coppersmith methods (see [10,17]), which solves in polynomial time the equation au2 buv xv 2 0 mod q 2 in the variables u, v, q where N pq 2 is known and max u, v O N 1 9 . They call this algorithm Homogeneous-Coppersmith in [6]. Their cryptanalysis of NICE Real is: Pick3 a form g of the principal cycle, and try to solve the equation g u, v 0 mod q 2 with Homogeneous-Coppersmith. Repeat this until it ﬁnds a solution u, v, q and return the private key q. The proof of the attack of [6] relies on this heuristic assumption: Assumption 1. The cardinality of the set A g 1 O N 9 and g u, v 0 mod q 2 is linear in sq . 3

½N ,

u, v max u, v

The authors of [6] enumerates the forms sequentially, until it ﬁnds a solvable one. They need an assumption not only on the large number of such forms, but also on their regular repartition on the principal cycle. Randomizing the enumeration avoids to prove the assumption on regular repartition (Heuristic 2 in [6]), which is feasable using the distance introduced in Theorem 3, but is beyond the scope of this paper.

Smallest Reduction Matrix of Binary Quadratic Forms

45

The authors of [6] experimentally verify this assumption. Namely, if g¯k denotes the reduction of the form gk q 2 , #, # of the q-belt by Classical Gauss reduction. The bottom two coeﬃcients of the reduction matrix satisfy g¯k δ, γ q 2 . Homogeneous-Coppersmith experimentally recovers δ, γ for most of the g¯k and even a few of their direct left or right neighbours on the principal cycle. This indicates that the norm of the reduction matrix is in general upper-bounded by O N 1 9 . However we also found rare cases of g¯k where the norm of reduction matrix was by an order greater than N 1 9 , and on which Homogeneous-Coppersmith algorithm cannot ﬁnd any solution. We call these particular forms unbalanced, because they have in general an unusually small coeﬃcient. The main three diﬃculties which prevented the authors of [6] to prove Assumption 1 were to justify that that the proportion of unbalanced forms is negligible among the set of g¯k , that the reduction matrix using Classical Gauss reduction is bounded by O N 1 9 , and that Classical Gauss is injective on a large enough subset of the q-belt, which prevents g¯k from being too small. Our ﬁrst improvement in their analysis is to replace the Classical Gauss reduction algorithm with RedGL2. This allows to square-root the upper-bounds on the reduction matrix as of Theorem 2. Thus we deﬁne gˆk as the reduction by RedGL2 of the q-belt form gk for each k. We ensure that gˆk is classically reduced and that the reduction matrix has determinant 1 using Lemma 1. The ﬁrst point of Theorem 2 implies that the norm of the reduction matrix is in O N 1 9 as soon as the smallest coeﬃcient of gˆk is greater than N 4 9 . We can either prove that this condition is satisﬁed by a large proportion of the gk , or we can also circumvent this limitation by using the second point of Theorem 2, which indicates that the size of the product uv is always upper-bounded by O N 1 6 . We therefore improve the Homogeneous-Coppersmith algorithm so that it also ﬁnds unbalanced solutions: namely, we design a rational variant of Boneh-DurfeeHowgraveGraham algorithm [2] which in particular solves g u, v au2 buv cv 2 0 mod q 2 on u, v, q as soon as the product uv is in O N 2 9 . Our new polynomial attack on Nice Real is the following: Randomly select a form g on the principal cycle ½N , and try to solve g u, v 0 mod q 2 in u, v, q using Rational-BonehDurfeeHowgraveGraham. Repeat until it ﬁnds a solution, and return q. The proof of this attack works in two steps: ﬁrst, we prove (in Theorem 3) that the above-deﬁned gˆk represent a non-negligible proportion of the principal cycle, and second, we prove (in Section 4.4) that Rational-BonehDurfeeHowgraveGraham ﬁnds q from any of the gˆk in polynomial time. Definition 2 (distance). we deﬁne a notion of distance between two equivalent forms f g as dist f, g min log M, M SL2 Z and f.M g . Let f, g, h be three equivalent forms in FΔ , the distance function satisﬁes the following properties: 1. dist f, g dist g, f 0 2. dist f, g 0 f g or f

g.SE

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3. dist f, h dist f, g dist g, h 4. if M SL2 Z satisﬁes f.M g and log M.

M

Δ , then dist f, g

Proof. The ﬁrst three points follow from basic properties of the induced norm, and the fact that only isometries have a unit norm. To prove the fourth statement, let U be the fundamental automorphism of f , the eigenvalues of U are Δ and Δ1 . Any non-trivial automorphism V of f satisﬁes V Δ , because V is a non-zero power of U , and its spectral radius is a positive power of Δ . The matrix M of the fourth point is necessarily the smallest transformation matrix from f to g, otherwise any matrix X SL2 Z such that f.X g and X M would produce a non-trivial automorphism MX 1 of f of too small norm MX 1 Δ , which is impossible. One of the greatest advantage of this distance is the fourth statement, which in general indicates that any polynomial transformation matrix is necessarily the smallest one. This allows to eﬃciently lower-bound a distance. As shown in the proof, it is essential that the group of automorphism is cyclic, the fourth statement would be false on GL2 Z. The authors of [6] used another distance between f, g , which could have been formalized as the smallest k N such that k there exists h1 , . . . , hk such that i1 SET hi transforms f into g or g.SE. Inside the reduced cycle, this corresponds to Shanks distance [20]. Unfortunately, it does not satisfy any equivalent of the fourth point: there is no way to eﬃciently verify that a given distance, as small as it could be, is correct. All the variants we found of this distance, which aims to approximate this statement, based either on the logarithms of the hi or some maximum norms, break the positive deﬁniteness or the triangular inequality. This explains why we do not base our proof on Shanks distance and introduce our own instead. Theorem 3. Given a NICE modulus N pq 2 , the set A gˆk RedGL2 gk , k 1, . . . sq 1 of the reduced of the q-belt has at least K.sq elements for some constant K 0. Proof. We now call Up the fundamental automorphism of the principal form of Fp . We verify that Upj 2 jp p j and that for all i, j, Qki1 Upj Qkij transforms gi into gij . Its norm is bounded by 1q Qki U j Qkij 4q jp p j . Due to point 4, for all j 1, sq 2 2, the distance dist gi , gij log Qki1 Upj Qkij is greater than j log p log 2q . By Theorem 2, the norm 2 of the reduction matrix from a gi to gˆi is upper-bounded by 2 21q N 42q p, and it follows that dist gˆi , gˆij j log p log 3528q 3 p. For this reason, if j log 3528q 3 p log p , then dist gˆi , gˆij 0 and gˆi gˆij . Using the NICE parameters, one has log 3528q 3p log p 3, thus the forms gˆ1 , gˆ4 , gˆ7 , . . . , gˆ3n1 are distinct (with n sq 6.

Smallest Reduction Matrix of Binary Quadratic Forms

4.4

47

Rational Improvement of the Boneh-Durfee-HowgraveGraham’s Algorithm

In this section, we describe our Rational-BonehDurfeeHowgraveGraham algorithm as a variant of Boneh Durfee Howgrave-Graham algorithm [2] solving rational linear equations u v C 0 mod q in the variables u, v, q when a multiple N pq r is known. The description of Rational-BonehDurfeeHowgraveGraham is summarized in Algorithm 2. Among others, it can be used to solve all the equations gˆk u, v au2 buv cv 2 0 mod q 2 of discriminant pq 2 of the previous section, because they are equivalent to u v b 2a 0 mod q. Since the solution we are looking for satisﬁes uv O N 1 6 , the following Theorem 4 proves that Rational-BonehDurfeeHowgraveGraham ﬁnds all solutions uv O N 2 9 , and concludes the proof of our new attack on Nice Real. More generally, given a polynomial P , the technique due to Boneh Durfee Howgrave-Graham transforms the equation P u v 0 mod q, into a lattice L of dimension m and bounded determinant, and whose short vectors are orthogonal to the integer vector S um , um 1 v, ..., uv m 1 , v m . The solutions u and v can be extracted from any of those short lattice vectors. This lattice is described by a basis B, whose rows contain the coeﬃcients of m 1-degree polynomials having u v as a root modulo a power of q. When u and v have approximately the same size (like in Homogeneous-Coppersmith of [6]), the celebrated LLL reduction algorithm on B outputs directly the desired vector orthogonal to S. Otherwise, when u and v are unbalanced, say for instance that u is 1000 times larger than v, one ﬁrst needs to re-balance the lattice by multiplying each i-th column by C i , where C is close to 1000, and only then reduce the basis. The original Boneh-Durfee-HowgraveGraham’s algorithm, which interests in integer solutions (arbitrary u and v 1), follows the above rule: the lattice basis which is actually LLL-reduced is the basis of Homogeneous-Coppersmith where each i-th column has been multiplied by X i , where X is a power of 2 just larger than the solution u. More generally, if we don’t know the relative balance between u and v but only know that the size of uv is n-bits, then we can test the n possible powers of two sequentially within a linear-factor overhead. Besides, we remark that instead of multiplying the columns of the input Homogeneous-Coppersmith basis by 1, 2, 4, ..., 2m, we describe the exact same lattice by multiplying the columns of the LLL-reduced basis, and the second one is almost reduced (LLL terminates in a very few steps). Thus after the reduction of the ﬁrst Homogeneous-Coppersmith basis, one obtains all the other possible balances of u and v for free. Theorem 4. Given any integer N pq r (where p and q are unknown), and a r bound β 14 q log q log N , Algorithm 2 terminates in polynomial time, and ﬁnds a solution (if it exists) of the equation uv c mod q where u, v are unknown integers satisfying uv β. u Proof. Let U, V R2 such

that

U and v 1 log qr . ters m N 0 and t mlog

N

V . We use the same parame-

We denote by Rm X, Y the span of homogeneous polynomials of degree m, and we deﬁne the isomorphism ϕ : Rm X, Y $ Rm1 which computes

48

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Algorithm 2. Rational Boneh-Durfee-HowgraveGraham Input: An integer N N of the form pq r (p and q are unknown), an integer c logq r log N 0, N 1 and a bound β 14 q Output: u, v N3 such that uv c mod q and u v β if it exists 1: Choose the smallest m such that

m1logq r logN

1

1

1

N 2 8r m 1 2

m11

1.5, and set t

. tk

2: Compute the family Pk X, Y N r X cY k Y mk for k 0..m 3: for l 0 to log2β do 4: U 2l ; V β 2l 5: Express (or update) the family Pk k0..m on the monomial k mk Y X , and form a matrix B Mm1 Z U k V mk k0..m 6: LLL-reduce B, and call α0 , . . . , αm the ﬁrst vector αk k 7: for each rational root uv of RX m k0 U k V mk X 0 do 8: if uv β and gcdu cv, N is non-trivial return u, v 9: end for 10: end for k

basis

mk

Y the coordinates of a polynomial on the basis X U k V mk k0..m . For instance, k m k k m k U V ek where ek is the k-th canonical basis vector. Let ϕX Y tk Pk k0..m be the family Pk X, Y N r X cY k Y m k Rm X, Y . By m construction, any integer linear combination R k0 Z Pk satisfy R u, v 0 mod q t and R u, v m 1 ϕ R2 (using Cauchy-Schwartz inequality). We now suppose that ϕ R is a short vector of the lattice generated by the (triangular) basis B ϕ Pk k1,m . By that, we mean ϕ R2 1.08m1 det B 1 m1 . Such a vector can be found by running the LLL algorithm on the lattice basis B (see [16]). The remainder of the proof is just a formal veriﬁcation that when m grows, det B is small enough to guaranty that R u, v q t , and therefore that R u, v 0 (in Z). Since R is homogeneous, this allows to recover u and v.

5

Conclusion

We saw that reduction algorithms are conceptually simpler to study in GL2 Z, because we mostly manipulate only positive matrices, which are easy to bound. The precision of our analysis, in the worst case and also in the average case, allows us to fully prove a lattice-based total-break attack against Nice cryptosystems [12,13], which is unusual in the history of lattice based cryptology. A further lead would be to extend these results on the reduction of the forms in higher dimension. Acknowledgements. We would like to thank Fabien Laguillaumie and Guilhem Castagnos for useful discussions and valuable comments on this paper.

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References 1. Biehl, I., Buchmann, J.: An analysis of the reduction algorithms for binary quadratic forms. In: Voronoi’s Impact on Modern Science, pp. 71–98 (1999) 2. Boneh, D., Durfee, G., Howgrave-Graham, N.A.: Factoring n pr q for large r. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, p. 326. Springer, Heidelberg (1999) 3. Buchmann, J., Thiel, C., Williams, H.: Short representation of quadratic integers. Proc. of CANT 1992, Math. Appl. 325, 159–185 (1995) 4. Buchmann, J., Vollmer, U.: Binary Quadratic Forms An Algorithmic Approach. Springer, Heidelberg (2007) 5. Buell, D.A.: Binary Quadratic Forms Classical Theory and Modern Computations. Springer, Heidelberg (1989) 6. Castagnos, G., Joux, A., Laguillaumie, F., Nguyen, P.Q.: Factoring pq 2 with quadratic forms: Nice cryptanalyses. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 469–486. Springer, Heidelberg (2009) 7. Castagnos, G., Laguillaumie, F.: On the security of cryptosystems with quadratic decryption: The nicest cryptanalysis. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 260–277. Springer, Heidelberg (2010) 8. Cheng, K.H.F., Williams, H.C.: Some results concerning certain periodic continued fractions. Acta Arith. 117, 247–264 (2005) 9. Cohen, H.: A Course in Computational Algebraic Number Theory, 2nd edn. Springer, Heidelberg (1995) 10. Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. of Cryptology 10(4), 233–260 (1997); Revised version of two articles from Eurocrypt 1996 (1996) 11. Gauss, C.F.: Disquisitiones Arithrneticae. PhD thesis (1801) 12. Hartmann, M., Paulus, S., Takagi, T.: NICE - New Ideal Coset Encryption. In: Ko¸c, C.K., ¸ Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 328–339. Springer, Heidelberg (1999) 13. Jacobson, M.J., Scheidler, R., Weimer, D.: An adaptation of the NICE cryptosystem to real quadratic orders. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 191–208. Springer, Heidelberg (2008) 14. Lagarias, J.C.: Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithm 1, 142–186 (1980) 15. Lagrange, J.L.: Recherches d’arithm´etique. Nouveaux M´emoires de l’Acad´emie de Berlin (1773) 16. Lenstra, A.K., Lenstra Jr., H.W., Lov´ asz, L.: Factoring polynomials with rational coeﬃcients. Mathematische Ann. 261, 513–534 (1982) 17. May, A.: Using LLL-reduction for solving RSA and factorization problems: A survey. In: Nguyen, P., Vallee, B. (eds.) The LLL algorithm, survey and Applications, Information Security and Cryptography, pp. 315–348 (2010) 18. Nguyen, P.Q., Stehl´e, D.: Low-dimensional lattice basis reduction revisited (extended abstract). In: Proceedings of ANTS VI. LNCS, Springer, Heidelberg (2004) 19. Schinzel, A.: On some problems of the arithmetical theory of continued fractions. Acta Arithmetica 6, 393–413 (1961) 20. Shanks, D.: The infrastructure of a real quadratic ﬁeld and its applications. In: Proc. NTC 1992, pp. 217–224 (1972) 21. Vallee, B., Vera, A.: Lattice reduction in two dimensions: Analyses under realistic probalistic models. In: Proc. of AofA 2007, DMTCS AH, pp. 181–216 (2007) 22. Weimer, D.: An Adaptation of the NICE Cryptosystem to Real Quadratic Orders, Master’s thesis. PhD thesis, Technische Universitat Darmstadt (2004)

Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields Jean-Fran¸cois Biasse1 and Michael J. Jacobson, Jr.2, 1

´ Ecole Polytechnique, 91128 Palaiseau, France [email protected] 2 Department of Computer Science, University of Calgary 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 [email protected]

Abstract. We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic ﬁeld. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein’s batch smoothness algorithm. We achieve a signiﬁcant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number ﬁeld with a 110decimal digit discriminant.

1

Introduction

Computing invariants of real quadratic ﬁelds, in particular the ideal class group and the regulator, has been of interest since the time of Gauss, and today has a variety of applications. For example, solving the well-known Pell equation is intimately linked to computing the regulator, and integer factorization algorithms have been developed that make use of this invariant. Public-key cryptosystems have also been developed whose security is related to the presumed diﬃculty of these computational tasks. See [16] for details. The fastest algorithm for computing the ideal class group and regulator in practice is a variation of Buchmann’s index-calculus algorithm [6] due to Jacobson [14]. The algorithm on which it is based has subexponential complexity in the size of the discriminant of the ﬁeld. The version in [14] includes several practical enhancements, including the use of self-initialized sieving to generate relations, a single large-prime variant (based on that of Buchmann and D¨ ullman [7] in the case of imaginary quadratic ﬁelds), and a practical version of the required linear algebra. This approach proved to work well, enabling the computation of the ideal class group and regulator of a real quadratic ﬁeld with a 101-decimal digit discriminant [15]. Unfortunately, both the complexity results of Buchmann’s algorithm and the correctness of the output are dependent on the Generalized Riemann Hypothesis (GRH). Nevertheless, for ﬁelds with large discriminants, this approach is the only one that works.

The second author is supported in part by NSERC of Canada.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 50–65, 2010. c Springer-Verlag Berlin Heidelberg 2010

Improvements in Real Quadratic Number Fields

51

Recently, Biasse [4] presented practical improvements to the corresponding algorithm for imaginary quadratic ﬁelds. These included a double large prime variant and improved algorithms for the required linear algebra. The resulting algorithm was indeed faster then the previous state-of-the-art [14], and enabled the computation of the ideal class group of an imaginary quadratic ﬁeld with 110 decimal digit discriminant. In this paper, we describe a number of practical improvements to the indexcalculus algorithm for computing the class group and regulator of a real quadratic ﬁeld. In addition to adaptations of Biasse’s improvements in the imaginary case, we have found some modiﬁcations designed to improve the regulator computation part of the algorithm. We also investigate applying an idea of Bernstein [3] to factor residues produced by the sieve using a batch smoothness test. Extensive computations demonstrating the eﬀectiveness of our improvements are presented, including the computation of class group and regulator of a real quadratic ﬁeld with 110 decimal digit discriminant. This paper is organized as follows. In the next section, we brieﬂy recall the required background of real quadratic ﬁelds, and give an overview of the indexcalculus algorithm using self-initialized sieving. Our improvements to the algorithm are described in Section 3, followed by numerical results in Section 4.

2

Real Quadratic Fields

We present an overview of required concepts related to real quadratic ﬁelds and the index-calculus √ algorithm for computing invariants. For more details, see [16]. Let K = Q( Δ) be the real quadratic ﬁeld of discriminant Δ, where Δ is a positive integer congruent to 0 or 1 modulo 4 with Δ or Δ/4 square-free. The integral closure of Z in K, called the maximal order, is denoted by OΔ . An interesting aspect of real quadratic ﬁelds is that their maximal orders contain inﬁnitely many non-trivial units, i.e., units that are not roots of unity. More precisely, the unit group of OΔ consists of an order 2 torsion subgroup and an inﬁnite cyclic group. The smallest unit greater than 1, denoted by εΔ , is called the fundamental unit. The regulator of OΔ is deﬁned as RΔ = log εΔ . The fractional ideals of K play an important role in the index-calculus algorithm described in this paper. In our setting, a fractional ideal is a rank 2 Z-submodule of K. Any fractional ideal can be represented as √ s b+ Δ a= aZ + Z , d 2 where a, b, s, d ∈ Z and 4a | b2 − Δ. The integers a, s, and d are unique, and b is deﬁned modulo 2a. The ideal a is said to be primitive if s = 1, and da ⊆ OΔ is integral. The norm of a is given by N (a) = as2 /d2 . Ideals can be multiplied using Gauss’s composition formulas for indeﬁnite binary quadratic forms. Ideal norm respects ideal multiplication, and the set

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IΔ forms an inﬁnite abelian group with identity OΔ under this operation. The inverse of a is √ −b + Δ d −1 a = aZ + Z . sa 2 The group IΔ is generated √ by the prime ideals of OΔ , namely those integral ideals of the form pZ + (bp + Δ)/2Z where p is a prime that is split or ramiﬁed in K. As OΔ is a Dedekind domain, the integral part of any fractional ideal can be factored uniquely as a product of prime ideals. To factor a, it suﬃces to factor N (a) and, for each prime p dividing the norm, determine whether the prime ideal p or p−1 divides a according to whether b ≡ bp or −bp modulo 2p. The ideal class group, denoted by ClΔ , is the factor group IΔ /PΔ , where PΔ ⊆ IΔ is the subgroup of principal ideals. The class group is ﬁnite abelian, and its order is called the class number, denoted by hΔ . By computing the class group we mean computing the elementary divisors m1 , . . . , ml with mi+1 | mi for 1 ≤ i < l such that ClΔ ∼ = Z/m1 Z × · · · × Z/ml Z. 2.1

The Index-Calculus Algorithm

Like other index-calculus algorithms, the algorithm for computing the class group and regulator relies on ﬁnding certain smooth quantities, those whose prime divisors are all small in some sense. In the case of quadratic ﬁelds, one searches for smooth principal ideals for which all prime ideal divisors have norm less than a given bound B1 . The set of prime ideals B = {p1 , . . . , pn } with N pi ≤ B1 is called the factor base. A principal ideal (α) = pe11 . . . penn with α ∈ K that factors completely over the factor base yields the relation (e1 , . . . , en , log |α|). The key to the index-calculus algorithm is the fact, proved by Buchmann [6], that the set of all relations forms a sublattice Λ ⊂ Zn × R of determinant hΔ RΔ provided that the prime ideals in the factor base generate ClΔ . This follows, in part, due to the fact that L, the integer component of Λ, is the kernel of the homomorphism from Zn to ClΔ given by pe11 . . . penn for (e1 , . . . , en ) ∈ Zn . If p1 , . . . , pn generate ClΔ , then this homomorphism is surjective, and the homomorphism theorem then implies that Zn /L ∼ = ClΔ . The main idea behind the index-calculus algorithm is to ﬁnd random relations until they generate the entire relation lattice Λ. Let Λ denote the sublattice of Λ generated by the relations that have been computed. To determine whether Λ = Λ, one computes an approximation h∗ of hΔ RΔ such that h∗ < hΔ RΔ < 2h∗ . The value h∗ is obtained by approximating the L-function L(1, χΔ ), where χΔ denotes the Kronecker symbol (Δ/p), and applying the analytic class number formula. If Λ ⊂ Λ, then det(Λ ) is a integer multiple of hΔ RΔ . Thus, Λ = Λ as soon as det(Λ ) < 2h∗ , because hΔ RΔ is the only integer multiple of itself in the interval (h∗ , 2h∗ ).

Improvements in Real Quadratic Number Fields

53

As described in [14], an adaptation of the strategy used in the self-initialized quadratic sieve (SIQS) factoring algorithm is used √ to compute relations. First, compute the ideal a√= pe11 . . . penn = (1/d)[aZ + (b + Δ)/2Z] with N (a) = a/d2 . Let α = (ax + (b + Δ)/2y)/d with x, y ∈ Z be an arbitrary element in a. Then √ √ b− Δ b+ Δ 1 y ax + y = (a/d2 )(ax2 + bxy + cy 2 ) N (α) = 2 ax + d 2 2 where c = (b2 − Δ)/(4a). Because ideal norm is multiplicative, there exists an ideal b with N (b) = ax2 + bxy + cy 2 such that (α) = ab. Thus, ﬁnding x and y such that N (b) factors over the norms of the prime ideals in the factor base yields a relation. Such x and y can be found by sieving the polynomial ϕ(x, y) = ax2 + bxy + cy 2 , and a careful selection of the ideals a yields a generalization of self-initialization, in which the coeﬃcients of the sieving polynomials and their roots modulo the prime ideal norms can be computed quickly. In practice, we use ϕ(x, 1) for sieving, so that the algorithm resembles the SIQS more closely. For more details, see [14] or [16]. The determinant of the relation lattice Λ is computed in two stages. The ﬁrst step is to compute the determinant of the integer part of this sublattice by ﬁnding a basis in Hermite normal form (HNF). Once Λ has full rank, the determinant of this basis is computed as the product of the diagonal elements in a matrix representation of the basis vectors. The group structure is then computed by ﬁnding the Smith normal form of this matrix. The real part of det(Λ ), a multiple of the regulator RΔ , is computed by ﬁrst ﬁnding a basis of the kernel of the matrix consisting of the integer parts of the relations. Every vector (k1 , . . . , km ) ∈ Zm in the kernel corresponds to a multiple of the regulator computed with mRΔ = k1 log |α1 | + · · · + km log |αm |. The “real gcd” of the multiples m1 RΔ , . . . , mn RΔ computed from each basis vector of the kernel, deﬁned as gcd(m1 , . . . , mn )RΔ , is then the real part of det(Λ ). An algorithm of Maurer [21] can be used to compute the real gcd eﬃciently and with guaranteed numerical accuracy given explicit representations of the αi and the kernel vectors. As mentioned in the introduction, the correctness of this algorithm depends on the truth of the Generalized Riemann Hypothesis. In fact, the GRH must be invoked in two places. The ﬁrst is to compute a suﬃciently accurate approximation h∗ of hΔ RΔ via a method due to Bach [2]. Without the GRH, an exponential number of terms in the Euler product used to approximate L(1, χΔ ) must be used (see, for example, [20]). The second is to ensure that the factor base generates ClΔ . Without the GRH, an exponential size factor base is required, whereas by a theorem of Bach [1] the prime ideals of norm less than 6 log(Δ)2 suﬃce. In practice, an even smaller factor base is often used, but in that case, the factor base must be veriﬁed by showing that every remaining prime ideal with norm less than Bach’s bound can be factored over the ideals in the factor base.

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3

Practical Improvements

In this section, we describe our practical improvements for computing the class group structure and the regulator of a the real quadratic ﬁeld. Some of these improvements, such as the double large prime variant and structured Gaussian elimination, were used in [4] for the simpler case of imaginary quadratic number ﬁelds. On the other hand, the batch smoothness test and system solving based methods for computing the regulator had never been implemented in the context of number ﬁelds before. 3.1

Relation Collection

Improving the relation collection phase allows us to speed up every other stage of the algorithm. Indeed, the faster the relations are found, the smaller the factor base can be, thus reducing the dimensions of the relation matrix and the time taken by the linear algebra phase. In addition, the veriﬁcation phase also relies on our ability to ﬁnd relations and therefore beneﬁts from improvements to the relation collection phase. Throughout the rest of the paper, M denotes the relation matrix, the matrix whose rows are the integer parts of the relations. Large prime variants. The large prime variants were developed in the context of integer factorization to speed up the relation collection phase in both the quadratic sieve and the number ﬁeld sieve. A single large prime variant was described by Buchmann and D¨ ullman [7] for computing the class group of an imaginary quadratic ﬁeld, and adapted to the real case by Jacobson [14]. Biasse [4] described how the double large prime strategy could be using in the imaginary case, and obtained a signiﬁcant speed-up. The idea is to keep relations involving one or two extra primes not in the factor base of norm less than B2 ≥ B1 . These relations thus have the form (α) = pe11 . . . penn p and (α) = pe11 . . . penn pp for pi in B, and for p, p of norm less than B2 . We will refer to these types of partial relations as 1-partial relations and 2-partial relations, respectively. Keeping partial relations only involving one large prime is the single large prime variant, whereas keeping those involving one or two is the double large prime variant which was ﬁrst described by Lenstra and Manasse [17]. We do not consider the case of more large primes, but it is a possibility that has been studied in the context of factorization [10]. Partial relations may be identiﬁed as follows. Let m be the remainder of ϕ(x, 1) after the division by all primes p ≤ B1 , and assume that B2 < B12 . If m = 1 then we have a full relation. If m ≤ B2 then we have a 1-partial relation. We can see here that detecting 1-partial relations is almost for free. If we also intend to collect 2-partial relations then we have to consider the following possibilities:

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55

m > B22 ; m is prime and m > B2 ; m is prime and m ≤ B2 ; m is composite and B12 < m ≤ B22 .

In Cases 1 and 2 we discard the relation. In Case 3 we have a 1-partial relation, and in Case 4 we have m = pp where p = N (p) and p = N (p ). Cases 1, 2, and 3 can be checked very easily, but if none are satisﬁed we need to factor m in order to determine whether Case 4 is satisﬁed. We used Milan’s implementation of the SQUFOF algorithm [22] based on the theoretical work of [12] to factor the m values produced. Even though we might have to factor the remainder, partial relations are found much faster than full relations. However, the dimensions of the resulting matrix are much larger, thus preventing us from running the linear algebra phase directly on the resulting relation matrix. In addition, we have to ﬁnd many more relations since we have to produce a full rank matrix. We will see in §3.2 how to reduce the dimensions of the relation matrix using Gaussian elimination techniques. Batch smoothness test. After detecting potential candidates for smooth integers via the SIQS, one has to certify their smoothness. In [4,14], this was done by trial division with the primes in the factor base. The time taken by trial division can be shortened by using Bernstein’s batch smoothness test [3], which uses a product tree structure and modular arithmetic to factor a batch of residues simultaneously in time O b(log b)2 log log b where b is the total number of input bits. Instead of testing the smoothness of every potential candidate as soon as they are discovered, we rather stored them and tested them at the same time using Bernstein’s method as soon their number exceeded a certain limit. This improvement has an eﬀect that is all the more important when the time spent in the trial division is long. In our algorithm, this time mostly depends on the tolerance value T, a parameter used to control the number of candidates yielded by the sieve for smoothness testing. 3.2

Structured Gaussian Elimination

As mentioned in §2.1, in order to determine whether the computed relations generate the entire relation lattice, we need to compute the HNF basis of the sublattice they generate. This can be done by putting the integer components of the relations as rows in a relation matrix, and computing the HNF. The ﬁrst step when using large primes is to compute full relations from all of the partial relations. Traditionally, rows were recombined to give full relations as follows. In the case of 1-partial relations, any pair of relations involving the same large prime p were recombined into a full relation. In the case of 2-partial relations, Lenstra [17] described the construction of a graph whose vertices were the relations and whose edges linked vertices having one large prime in common.

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Finding independent cycles in this graph allows us to recombine partial relations into full relations. In this paper, we instead follow the approach of Cavallar [8], developed for the number ﬁeld sieve, and adapted by the ﬁrst author to the computation of ideal class group structures in imaginary quadratic number ﬁelds [4], which uses Gaussian elimination on columns. The ideas is to eliminate columns using structured Gaussian strategies until the dimensions of the matrix are small enough to allow the computation of the HNF with standard algorithms. Let us recall a few deﬁnitions. First, subtracting two rows is called merging. If two relations corresponding to rows r1 and r2 share the same prime p with coeﬃcients c1 and c2 respectively, then multiplying r1 by c2 and r2 by c1 and merging is called pivoting. Finally, ﬁnding a sequence of pivots leading to the elimination of a column of Hamming weight k is a k-way merge. We aim to reduce the dimensions of the relation matrix by performing kway merges on the columns of weight k = 1, . . . , w in increasing order for a certain bound w. To limit the growth of the density and of the size of the coeﬃcients induced by these operations, we used optimized pivoting strategies. In what follows we describe an algorithm performing k-way merges to minimize the growth of both the density and the size of the coeﬃcients, thus allowing us to go deeper in the elimination process and delay the explosion of the coeﬃcients. As in [4], we deﬁne a cost function C mapping rows onto the integers. The one used in [4] satisﬁed C(r) = 1+c 1, (1) 1≤|ei |≤Q

|ej |>Q

where c and Q are positive numbers, and r = [e1 , . . . , en ] is a row corresponding to (α) = i pei i . This way, the heaviest rows are those which have a high density and large coeﬃcients. In our experiments for this work, we used a diﬀerent cost function, see §4.1. Then, to perform a k-way merge on a given column, we construct a complete graph G of size k such that – the vertices are the rows ri , and – every edge linking ri and rj has weight C(rij ), where rij is obtained by pivoting ri and rj . Finding the best sequence of pivots with respect to the chosen cost function C is equivalent to ﬁnding the minimum spanning tree T of G, and then recombining every row r with its parent starting with the leaves of T . Unlike in [4], we need to keep track of the permutations we apply to the relation matrix, and of the empty columns representing primes of norm less than 6 log2 Δ. This will be required for the regulator computation part of the algorithm described next. 3.3

Regulator Computation

As mentioned in §2.1, the usual way to compute the regulator is to ﬁnd a basis of the kernel of the relation matrix, compute integer multiples of the regulator

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from these basis vectors, and compute their real gcd using Maurer’s algorithm [21]. If det Λ > 2h∗ , then either the class number or regulator computed is too large, and we need to ﬁnd extra relations corresponding to new generators, and new kernel vectors involving them. In this section, we describe a way of taking advantage of the large number of generators involved in the diﬀerent partial relations. Indeed, the dimensions of the relation matrix before the Gaussian elimination stage is much larger than in the base scenario and thus involves more generators. Consequently, given a set of k ≤ dim(ker M ) kernel vectors (uj1 , . . . , ujn )j≤k , the probability that the corresponding elements vj := uj1 log |α1 | + . . . + ujn log |αn | , where αi is the generator of the i-th relation, can be recombined into R is much larger. On the other hand, the dimensions of the matrix prevents us from running a kernel computation directly after the relation collection phase. Thus, rather than attempting to compute the kernel, we use a method similar to that of Vollmer [24] based on solving linear systems. The ﬁrst step of our algorithm consists of putting the matrix in a pseudo-lower triangular form using a permutation obtained during the Gaussian elimination phase. Indeed, as part of this computation we obtain a unimodular matrix U ∈ Zn×n such that ⎛ ⎞ ⎜ ⎜ A ⎜ ⎜ ⎜ ⎜ ⎜ UM = ⎜ ⎜ ⎜ ⎜ ⎜ (∗) ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 (0) ⎟ ⎟. ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎠ (∗) 1 (0)

Thus, solving a linear system of the form xM = b for a vector b ∈ Zm boils down to solving a system of the form x A = b , then doing a trivial descent through the diagonal entries which equal 1 and ﬁnally permuting back the coeﬃcients using U . To solve the small linear systems, we used the algorithm certSolveRedLong from the IML library [9]. It takes a single precision dense representation of A and returns an LLL-reduced solution. Once M is in pseudo-lower triangular form, we draw a set of relations r1 , . . . rd which are not already rows of M , and for each ri , i ≤ d, we solve the system xi A = ri . We then augment M with the rows ri for i ≤ d and the vectors xi with d extra coordinates, which are all set to zero except for the i-th which is set to −1.

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⎛ ⎜ ⎜ M := ⎜ ⎝

⎞ M ri

⎟ ⎟ ⎟ xi := xi 0 . . . 0 −1 0 . . . 0 . ⎠

We clearly have xi M = 0 for i ≤ d, and the xi can be used to ﬁnd a multiple of RΔ as described in §2.1.

4

Numerical Results

In this section, we give numerical results showing the impact of our improvements. For each timing, we specify the architecture used. All the timings were obtained with our code in C++ based on the libraries GMP [11], NTL [23], IML [9] and Linbox [19]. All timings are in CPU seconds. 4.1

Comparative Timings

The state of the art concerning class group and regulator computation was established in [14], where all the timings were obtained with the SPARCStation II architecture. In addition, most of the code used at the time is unavailable now, including the HNF computation algorithm. Thus, providing a meaningful comparison between our methods and those of [14] is diﬃcult. We chose to implement the HNF computation algorithm in a way that resembles the one of [14], but takes advantage of the libraries available today for computing the determinant and the modular HNF. We used this implementation in each diﬀerent scenario. The relation collection phase is easier to compare, since our method relies on SIQS. In the following, we will refer to the base case as the strategy consisting of ﬁnding the relation matrix without using the large prime variants or the smoothness batch test, and calculating the regulator by computing its kernel with the algorithm nullspaceLong from IML library. It diﬀers from the 0 large prime case (0LP) where we use the algorithm described in §3.3 for computing the regulator, along with a relation collection phase that does not use large primes. We also denote the 1 large prime scenario by 1LP, the 2 large primes by 2LP and 2LP Batch when using batch smoothness test. Relation collection phase. In Table 1, we give the time taken to collect all necessary relations. Without large primes, we collected |B| + 100 relations, whereas when we allow large primes we need to collect enough relations to ensure that the number of rows is larger than the number of non-empty columns. We used a 2.4 GHz Opteron with 16GB of memory and took Δ = 4(10n + 3) with 40 ≤ n ≤ 70. For each discriminant, we used the optimal parameters given in [14], including the size of the factor base, even if we tend to reduce this parameter when optimizing the overall time. The only parameter we modiﬁed is the tolerance value for the SIQS, as a higher tolerance value is required for the large prime variations. In each case we took B2 = 12B1 . It is shown in [4] that the ratio B2 /B1 does not have an important impact on the sieving time.

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Table 1. Comparative table of the relation collection time n 0LP 1LP 2LP 2LP Batch 40 0.83 0.48 0.63 0.90 45 6.70 3.10 2.70 2.20 50 23.00 9.50 9.20 6.10 55 56.00 26.00 23.00 15.00 60 202.00 86.00 69.00 41.00 65 1195.00 513.00 354.00 227.00 70 4653.00 1906.00 1049.00 834.00

The timings in Table 1 correspond to the optimal value of the tolerance value in each case, found by trying values between 1.7 and 4, and keeping the optimum for each scenario. For 0LP, the optimal value is between 1.7 and 2.3 whereas it is around 2.3 for 1LP, 2.8 for 2LP and 3.0 for 2LP Batch. The latter case has a higher optimal tolerance value because using the batch smoothness test allows one to spend more time factoring the residues. When using Bernstein’s smoothness test, we took batches of 100 residues. In our experiments, this value did not seem to have an important eﬀect on the relation collection time. We observe in Table 1 that the use of the large prime variants has a strong impact on the relation collection phase, and that using the smoothness batch test strategy yields an additional speed-up of approximately 20% over the double large prime strategy. Structured Gaussian elimination. Structured Gaussian elimination allows us to reduce the time taken by the linear algebra phase by reducing the dimensions of the relation matrix. Our method minimizes the growth of the density and of the size of the coeﬃcients. To illustrate the impact of the algorithm described in §3.2, we monitor in Table 2 the evolution of the dimensions of the matrix, the average Hamming weight of its rows, the extremal values of its coeﬃcients and the time taken for computing its HNF in the case of a relation matrix corresponding to Δ = 4(1060 + 3). We keep track of these values after all i-way merges for some values of i between 5 and 170. The original dimensions of the matrix are 2000 × 1700, and the timings are obtained on a 2.4 Ghz Opteron with 32GB of memory. In [4], the ﬁrst author regularly deleted the rows having the largest coeﬃcients. To do this, we need to create more rows than in the base case. To provide a fair comparison between the two strategies, we used the same relation matrix resulting from a relation collection phase without large primes, and with as few rows as was required to use the same algorithm as in [14]. We therefore had to drop the regular row deletion. We also tuned the cost function to compensate for the resulting growth of the coeﬃcients, using C(r) = 1 + 100 |ej | , 1≤|ei |≤8

instead of (1).

|ej |>8

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The HNF computation consists of taking the GCD of the determinants of two diﬀerent submatrices of the matrix after elimination using Linbox, and using the modular HNF of NTL with this value. Indeed, this GCD (which is likely to be relatively small) is a multiple of hΔ . This method, combined with an elimination strategy due to Havas [13], was used in [14] and implemented in LiDIA [18]. As this implementation is no longer available, we instead refer to the timings of our code, which has the advantage of using the best linear algebra libraries available today. Table 2. Comparative table of elimination strategies Naive Gauss i Row Nb Col Nb Average weight max coeﬀ min coeﬀ HNF time 5 1189 1067 27.9 14 -17 357.9 10 921 799 49.3 22 -19 184.8 30 757 635 112.7 51 -50 106.6 50 718 596 160.1 81 -91 93.7 70 699 577 186.3 116 -104 85.6 90 684 562 205.5 137 -90 79.0 125 664 542 249.0 140 -146 73.8 160 655 533 282.4 167 -155 72.0 170 654 532 286.4 167 -155 222.4 With dedicated elimination strategy i Row Nb Col Nb Average weight max coeﬀ min coeﬀ HNF time 5 1200 1078 26.8 13 -12 368.0 10 928 806 42.6 20 -15 187.2 30 746 624 82.5 33 -27 100.8 50 702 580 107.6 64 -37 84.3 70 672 550 136.6 304 -676 73.4 90 656 534 157.6 1278 -1088 67.5 125 637 515 187.1 3360 -2942 63.4 160 619 497 214.6 5324 -3560 56.9 170 615 493 247.1 36761280 -22009088 192.6

Table 2 shows that the use of our elimination strategy leads to a matrix with smaller dimensions (493 rows with our method, 533 with the naive elimination) and lower density (the average weight of its rows is of 214 with our method and 282 with the naive elimination). These diﬀerences result in an improvement of the time taken by the HNF computation: 56.9 seconds with our method against 72.0 seconds with the naive Gaussian elimination. The regular cancellation of the rows having the largest coeﬃcients over the course of the algorithm would delay the explosion of the coeﬃcient size, but require more rows for the original matrix. This brutal increase in the size of the extremal values of the matrix can be seen in Table 2. At this point these higher values propagate during pivoting operations, and any further column elimination becomes counter-productive.

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Factor base verification. The improvements in the relation collection phase have an impact on the factor base veriﬁcation. The impact of the smoothness batch test is straightforward, whereas the large prime variants act in a more subtle way. Indeed, we create many more relations when using the large prime variants, and the relations created involve primes of larger norm. Therefore, a given prime not in B of norm less than 6 log2 Δ is more likely to appear in a relation, and thus not to need to be veriﬁed. Table 3 shows the impact of the large prime variants and of the batch smoothness test on the veriﬁcation time. We used a 2.4 GHz Opteron with 16GB of memory. We considered discriminants of the form Δ = 4(10n + 3) for n between 40 and 70, and we chose in every case the factor base giving the best results for the base scenario. Table 3. Comparative table of the factor base veriﬁcation time n 40 45 50 55 60 65 70

0LP 17.0 77.0 147.0 308.0 826.0 8176.0 9639.0

1LP 11.0 44.0 85.0 167.0 225.0 1606.0 4133.0

2LP 2LP Batch 11.0 6.2 30.0 18.0 52.0 43.0 134.0 110.0 282.0 274.0 1760.0 1689.0 5777.0 2706.0

Regulator computation. Our method for computing the regulator avoids computing the relation matrix kernel. Instead, we need to solve a few linear systems involving the matrix resulting from the Gaussian elimination. To illustrate the impact of this algorithm, we used the relation matrix obtained in the base case for discriminants of the form 4(10n + 3) for n between 40 and 70. The timings are obtained on a 2.4GHz Opteron with 16GB of memory. In Table 4, the timings corresponding to our system solving approach are taken with seven kernel vectors. However, in most cases only two or three vectors are required to compute the regulator. As most of the time taken by our approach Table 4. Comparative table of regulator computation time n Kernel Computation System Solving 40 15.0 6.2 45 18.0 8.3 50 38.0 20.0 55 257.0 49.0 60 286.0 103.0 65 5009.0 336.0 70 10030.0 643.0

62

J.-F. Biasse and M.J. Jacobson Table 5. Eﬀect on the overall time n 40

45

50

55

60

65

70

strategy base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch

|B| relations elimination HNF regulator veriﬁcation total 400 0.8 0.1 3.2 14.6 16.8 35.6 400 0.7 0.1 2.2 6.0 16.6 25.7 300 0.8 0.2 2.5 6.4 13.1 23.1 250 1.7 0.3 4.8 8.7 18.0 33.3 250 0.5 0.2 3.6 6.7 4.4 15.5 500 6.7 0.1 5.1 18.0 77.0 107.0 500 5.9 0.2 4.9 10.0 85.0 106.0 400 4.0 0.4 6.0 11.0 50.0 71.0 350 3.8 0.5 12.0 17.0 36.0 69.0 350 2.6 1.1 9.0 14.0 30.0 57.0 750 23.0 0.3 16.0 38.0 147.0 224.0 700 21.0 0.4 15.0 20.0 147.0 203.0 450 20.0 0.4 10.0 17.0 108.0 155.0 400 14.0 0.8 22.0 23.0 74.0 133.0 400 10.0 0.6 21.0 25.0 62.0 119.0 1200 129.0 1.9 60.0 257.0 308.0 756.0 1300 47.0 0.7 52.0 49.0 265.0 414.0 650 61.0 0.7 28.0 33.0 255.0 378.0 550 40.0 1.1 48.0 48.0 177.0 313.0 550 34.0 1.0 47.0 48.0 141.0 271.0 1700 322.0 2.9 95.0 286.0 830.0 1535.0 1700 187.0 1.3 106.0 103.0 846.0 1244.0 750 309.0 1.0 45.0 64.0 865.0 1284.0 700 143.0 2.1 152.0 137.0 365.0 799.0 700 142.0 1.8 103.0 100.0 309.0 655.0 2700 10757.0 12.0 652.0 5009.0 8176.0 24607.0 2700 1225.0 2.8 489.0 336.0 3676.0 5730.0 1900 1003.0 15.0 318.0 262.0 2984.0 4583.0 1200 753.0 4.7 525.0 398.0 1943.0 3624.0 1000 1030.0 35.0 199.0 219.0 1642.0 3125.0 3700 17255.0 24.0 1869.0 10031.0 9639.0 38818.0 3600 4934.0 19.0 1028.0 644.0 9967.0 16591.0 2500 3066.0 17.0 845.0 646.0 9005.0 13579.0 1700 2414.0 27.0 2054.0 1295.0 4590.0 10379.0 1700 2588.0 20.0 1372.0 934.0 5078.0 9991.0

is spent on system solving, we see that computing fewer kernel vectors would result in an improvement of the timings, at the risk of obtaining a multiple of the regulator. Overall time. We have studied the individual impact of our improvements on each stage of the algorithm. We now present their eﬀect on the overall time taken by the algorithm, including the factor base veriﬁcation time, for discriminants of the form Δ = 4(10n + 3) with 40 ≤ n ≤ 70 on a 2.4 GHz Opteron with 16GB of memory. We used the same parameters as in [14], except for the tolerance

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and the size of the factor base. We notice in Table 5 that the optimal size of the factor base is smaller when we use improvements for the sieving phase. For example the optimal size for the double large prime variant is half the one of the base case scenario. This results in an improvement in the HNF and regulator computation whereas the relation collection time can remain unchanged, or even increase. The tolerance value we chose varies only with the strategy, but not with the size of the discriminant. We chose 2.0 for the base case and 0LP whereas we set it to 2.3 for 1LP, 2.8 for 2LP and 3.0 for 2LP Batch. We eliminated columns of weight up to w = 150 since Table 2 indicates that further elimination is counter-productive. Table 5 shows that there is an overall speed-up of of a factor of 2 for the smallest discriminants and 4 for the largest. The base case with the largest discriminants suﬀers from the necessity of ﬁnding some relations in a more randomized way. This ensures that we can get full rank submatrices of the relation matrix after the Gaussian elimination to compute a small multiple of hΔ . Matrices produced using the large prime variants do not need this extra step, even with the largest discriminants. This naturally aﬀects the sieving time, since we cannot use SIQS for that purpose, but also aﬀects phases relying on linear algebra. Indeed, elimination produces a matrix with larger entries and dimensions. 4.2

Large Example

The improvements we described allow us to compute class groups and regulators of real number ﬁelds with larger discriminants than was previously possible. The key is to parallelize the relation collection and veriﬁcation phase, while the linear algebra has to be performed the usual way. These methods were successfully used in [4] to compute the class group structure of an imaginary quadratic ﬁeld with a 110-digit discriminant. We used a cluster with 260 2.4GHz Xeon cores to compute a relation matrix corresponding to the discriminant Δ110 := 4(10110 +3) in 4 days. We allowed two large primes, used a tolerance value of 3.0, tested batches of 100 residues, took w = 250 and set |B| = 13000 . Then, we used three 2.4 GHz Opterons with 32GB of memory each to compute determinants of full-rank submatrices of the relation matrix after the Gaussian elimination in 1 day, and one 2.4GHz Opteron to compute the HNF modulo the GCD of these determinants in 3 days. We had to ﬁnd 4018 extra relations during the veriﬁcation phase that took 4 days on 96 2.4GHz Xeon cores. We thus obtained that ClΔ110 ∼ (2) = Z/12Z × Z/2Z , and the corresponding regulator is RΔ110 ≈ 70795074091059722608293227655184666748799878533480399.6730200233 .

We estimate that it would take two weeks (4000 relations per day) to complete the relation collection for Δ120 with the same factor base as Δ110 , thus requiring a similar time for the linear algebra.

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Conclusions

Recently, our work has been extended to the problems of principal ideal testing and solving the discrete logarithm problem in the ideal class group [5]. The double large prime variant and improvements to relation generation translated directly to improvements in this context. However, HNF computations are not required for this problem, and linear system solving over Z can be used instead. The numerical results were used to give estimates for discriminant sizes that oﬀer equivalent security to recommended sizes of RSA moduli. Some possibilities for further improvements remain to be investigated. For example, a lattice sieving strategy could be used to sieve ϕ(x, y) instead of ϕ(x, 1). Factor reﬁnement and coprime factorization techniques may be a useful alternative to Bernstein’s batch smoothness test. Multiple large primes have been successfully used for integer factorization and could also be tried in our context. There is also still room for improvement to the linear algebra components. For example, a HNF algorithm that exploits the natural sparseness of the relation matrix, perhaps as a black-box algorithm, would be useful. If such an algorithm were available, we could reconsider using Gaussian elimination techniques since they induce a densiﬁcation of the matrix. We could also study the eﬀect of other dense HNF algorithms in existing linear algebra packages such as KASH, Pari, Sage and especially MAGMA which seems to have the most eﬃcient HNF algorithm for our types of matrices. In that case, we would need the elimination phase regardless of how these algorithms are aﬀected by the density and the size of the coeﬃcients of the matrix. Indeed, we cannot aﬀord manipulating a dense representation of the matrix before the Gaussian elimination phase.

References 1. Bach, E.: Explicit bounds for primality testing and related problems. Math. Comp. 55(191), 355–380 (1990) 2. Bach, E.: Improved approximations for Euler products. In: Number Theory: CMS Proc., vol. 15, pp. 13–28. Amer. Math. Soc., Providence (1995) 3. Bernstein, D.: How to ﬁnd smooth parts of integers. Mathematics of Computation (submited) 4. Biasse, J.-F.: Improvements in the computation of ideal class groups of imaginary quadratic number ﬁelds. In: Advances in Mathematics of Communications (to appear 2010) 5. Biasse, J.-F., Jacobson Jr., M.J., Silvester, A.K.: Security estimates for quadratic ﬁeld based cryptosystems. In: ACISP (to appear 2010) 6. Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds. In: S´eminaire de Th´eorie des Nombres (Paris), pp. 27–41 (1988-1989) 7. Buchmann, J., D¨ ullmann, S.: Distributed class group computation. In: Festschrift aus Anlaß des sechzigsten Geburtstages von Herrn Prof. Dr. G. Hotz, pp. 69–79. Universit¨ at des Saarlandes (1991), Teubner, Stuttgart (1992)

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8. Cavallar, S.: Strategies in ﬁltering in the number ﬁeld sieve. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 209–232. Springer, Heidelberg (2000) 9. Chen, Z., Storjohann, A., Fletcher, C.: IML: Integer Matrix Library. Software (2010), http://www.cs.uwaterloo.ca/~ astorjoh/iml.html 10. Dodson, B., Leyland, P.C., Lenstra, A.K., Muﬀett, A., Wagstaﬀ, S.: MPQS with three large primes. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 446–460. Springer, Heidelberg (2002) 11. GMP, The GNU multiple precision bignum library. Software (2010), http://gmp-lib.org/ 12. Gower, J.E., Wagstaﬀ, S.: Square form factorization. Mathematics of Computation 77, 551–588 (2008) 13. Havas, G., Majewski, B.S.: Integer matrix diagonalization. Journal of Symbolic Computing 24, 399–408 (1997) 14. Jacobson Jr., M.J.: Subexponential class group computation in quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999) 15. Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: The eﬃciency and security of a real quadratic ﬁeld based key exchange protocol. In: Public-Key Cryptography and Computational Number Theory, Warsaw, Poland, pp. 89–112. de Gruyter (2001) 16. Jacobson Jr., M.J., Williams, H.C.: Solving the Pell equation. CMS Books in Mathematics. Springer, Heidelberg (2009) ISBN 978-0-387-84922-5 17. Lenstra, A.K., Manasse, M.S.: Factoring with two large primes (extended abstract). In: Damg˚ ard, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 72–82. Springer, Heidelberg (1991) 18. LiDIA Group, LiDIA: a c++ library for computational number theory. Software, Technische Universit¨ at Darmstadt, Germany (1997), http://www.informatik.tu-darmstadt.de/TI/LiDIA 19. LinBox, Project LinBox: Exact computational linear algebra. Software (2010), http://www.linalg.org/ 20. Louboutin, S.: Computation of class numbers of quadratic number ﬁelds. Math. Comp. 71(240), 1735–1743 (2002) 21. Maurer, M.: Regulator approximation and fundamental unit computation for real quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999) 22. Milan, J.: Tifa. Software (2010), http://www.lix.polytechnique.fr/Labo/Jerome-Milan/tifa/tifa.xhtml 23. Shoup, V.: NTL: A Library for doing Number Theory. Software (2010), http://www-shoup.net/ntl 24. Vollmer, U.: An accelerated Buchmann algorithm for regulator computation in real quadratic ﬁelds. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 148–162. Springer, Heidelberg (2002)

On the Use of the Negation Map in the Pollard Rho Method Joppe W. Bos, Thorsten Kleinjung, and Arjen K. Lenstra Laboratory for Cryptologic Algorithms EPFL, Station 14, CH-1015 Lausanne, Switzerland

Abstract. The negation map can be used to speed up the Pollard rho method to compute discrete logarithms in groups of elliptic curves over ﬁnite ﬁelds. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose eﬀective alternative countermeasures. As a result, fruitless cycles can be resolved, but the best speedup we managed to achieve is√by a factor of only 1.29. Although this is less than the speedup factor of 2 generally reported in the literature, it is supported by practical evidence. Keywords: Pollard’s rho method, fruitless cycles, negation map.

1

Introduction

The diﬃculty of the elliptic curve discrete logarithm problem (ECDLP) underlies the security of cryptographic schemes based on elliptic curves over ﬁnite ﬁelds [11,13]. The best method known to solve ECDLP for curves without special properties is the parallelized [17] Pollard rho method [15]. A common optimization is to halve the search space by identifying a point with its inverse [18,9,7]. Because representatives for the equivalence classes can quickly be computed using the negation√map, this equivalence relation may result in a speedup by a factor of up to 2 when solving ECDLP. For the elliptic curves over binary extension ﬁelds F2t from [12], order t equivalence relations can be used as well, √ resulting in a speedup by a factor of up to 2t [18,9]. Usage of the negation map in the context of the Pollard rho method leads to fruitless cycles, useless cycles trapping the random walks. An analysis of their likelihood of occurrence appeared in [7]. Various methods have been proposed [18,9] to deal with them, all leading to costlier random walks and administrative overhead. The literature suggests that √ the resulting ineﬃciencies are negligible, and that a speedup by a factor of 2 is attainable [1, Section 19.5.5]. We analyze fruitless cycles and the previously published methods to avoid their ill eﬀects and show that current approaches to escape from cycles suﬀer from recurring cycles. These may have contributed to the lack of practical usage of the negation map to solve prime ﬁeld ECDLPs: it was not used for the solutions G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 66–82, 2010. c Springer-Verlag Berlin Heidelberg 2010

On the Use of the Negation Map in the Pollard Rho Method

67

[10,6] of the 79-, 89-, 97- and 109-bit prime ﬁeld Certicom challenges [5]. Neither was it used by the independent current 112-bit prime ﬁeld record [3]. We present and analyze alternative methods to deal with fruitless cycles. All our analyses are supported by experiments. We found that the negation map indeed leads to a speedup, but we have √ not been able to reach more than a factor of 1.29, somewhat short of the 2 that we had hoped for. We also found that the best attainable speedup depends on the platform one uses: for instance, if the Pollard rho method is parallelized in SIMD fashion, it is a challenge to achieve any speedup at all. This has consequences for the applicability of the negation map in large scale prime ﬁeld ECDLP solution attempts. For such efforts, all participating processors must use the same random walk deﬁnition, so one may desire to gear the implementation towards processors with the best performance/price ratio, such as graphics cards (which are SIMT, a SIMD variant). The negation map (while dealing with cycles) slows down random walks in three ways. In the ﬁrst place, on average more elliptic curve group operations are required per step of each walk. This is unavoidable and attempts should be made to minimize the number of additional operations. Secondly, dealing with cycles entails administrative overhead and branching, which cause a non-negligible slowdown when running multiple walks in SIMD-parallel fashion. Finally, the best way to counter the eﬀect of the higher average number of group operations per step is making the walks “more random” by allowing a ﬁner grained decision per step. However, the beneﬁcial eﬀects of this approach are, in most circumstances on current processors, wiped out by cache ineﬃciencies. It will be seen that it is best to strike a balance between the ﬁrst and third of these slowdowns. The second slowdown somewhat aﬀects regular PCs, but is a major obstacle to the negation map in SIMD environments. This paper is organized as follows. Section 2 recalls background on ECDLP, the Pollard rho method and fruitless cycles. Section 3 introduces recurring cycles and presents and analyzes new methods to deal with them. Section 4 compares the various cycle reduction, detection, and escape methods in practice.

2 2.1

Preliminaries The Elliptic Curve Discrete Logarithm Problem

Let Fp denote a ﬁnite ﬁeld of odd prime characteristic p. Any a, b ∈ Fp with 4a3 + 27b2 = 0 deﬁne an elliptic curve Ea,b over Fp . The additively written group of points Ea,b (Fp ) of Ea,b over Fp is deﬁned as the zero point o along with the set of pairs (x, y) ∈ Fp × Fp that satisfy the shortened Weierstrass equation y 2 = x3 + ax + b. Let p, a, b and g ∈ Ea,b (Fp ) of prime order q be such that the index [Ea,b (Fp ) : g] is small. For h ∈ g, the ECDLP is to ﬁnd an integer m such that mg = h. For curves without special properties, solving ECDLP is √ believed to require an eﬀort on the order of q. Pollard’s rho method achieves this run time, while requiring more or less constant memory.

68

2.2

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Pollard’s Rho Method

If objects are selected truly at random and with replacement from q objects, the conditional probability at step n + 1 of ﬁnding the ﬁrst duplicate (or collision) is nq (if n < q). Via straightforward arguments this leads to πq/2 for the expected number of steps until the ﬁrst collision. If random objects are selected as ug + vh ∈ g for random integer multipliers u, v, a collision corresponds to u, v, u ¯, v¯ such that ug + vh = u ¯g + v¯h. Unless v¯ ≡ v mod q, the value m = u−¯ u mod q solves the discrete logarithm problem. The expected number of steps v ¯−v of this idealized version of Pollard’s rho method [15] is πq/2. r-adding and r+s-mixed walks. Pollard’s rho method uses an approximation of a truly random walk in g. Let, for a small integer r, an index function : g → [0, r − 1] induce an r-partition g = ∪r−1 i=0 Gi of g, where Gi = {x : x ∈ g, (x) = i} and all Gi have cardinality close to qr . For random integers ui , vi , elements fi = ui g + vi h ∈ g are precomputed for 0 ≤ i < r. Starting at a random but known multiple of g, the successor of a point p of the walk is deﬁned as p + f(p) ∈ g. It is easy to keep track of the u, v such that p = ug + vh. Such an r-adding walk results in anexpected number of steps until a collision occurs that is somewhat larger than πq/2, as shown by Brent and Pollard [4] i and expanded upon in [2]. Assume that is perfectly random. Let pi = #G q . A point in the walk is said to belong to class i if its predecessor upon its ﬁrst occurrence belongs to Gi . If the nth point belongs to Gj (with probability pj ) and the (n + 1)st point produces the ﬁrst collision, the collision point cannot be of class j (this happens with probability pj ), since then the collision would have occurred in step n. Therefore, the probability that the ﬁrst collision occurs at step n + 1 is r−1 n (1 − p2j ). q j=0 With q =

1−

q r−1 j=0

p2j

this is

n q .

We get via the same arguments referred to above

πq = 2

2(1 −

πq r−1 j=0

p2j )

(1)

for the expected number of steps until the ﬁrst collision. Pollard [15] uses r = 3, f0 = h, and f2 = g, but replaces the i = 1 case by the doubling 2p. Teske [16] shows that a larger r, such as r = 20, leads to better performance on average, conform the analysis, even if none of the choices does an explicit doubling, as Pollard’s i = 1 case. Inclusion of doublings leads to r + s-mixed walks: with : g → [0, r + s − 1] q partitioning g into r + s parts of cardinality close to r+s , the next point equals p + f(p) if 0 ≤ (p) < r, but 2p if (p) ≥ r. Pollard’s walk is a 2 + 1-mixed walk. The analysis above applies again, assuming that we consider the doublings as one class, hit with probability pD . Experiments by Teske show that best performance is achieved for rs between 14 and 12 but that apart from the case r = 3 mixed

On the Use of the Negation Map in the Pollard Rho Method

69

walks are not signiﬁcantly better. The analysis and our own experiments, as reported below, suggest that the optimal ratio rs is close to zero. Per step the occurrence probability of the event p = fi (and thus a chance to solve the discrete logarithm problem) is negligible compared to the probability of a birthday collision. So, for r-adding walks doublings most likely will not occur. Parallelized random walks. Parallelization of Pollard’s rho method does not consist of running any number of random walks in parallel, until one of √ them collides: on M processors the expected speedup would be by a factor of M , so √ overall it would require M more processing power than a single processor. The proper way to parallelize Pollard’s rho method is presented in [17]. It achieves an M -fold speedup on M processors, thus requiring the same overall processing 1 power as a single process, but in M th of the time. Diﬀerent processes must be able to eﬃciently recognize if, probably at diﬀerent points in time, their walks collide. To achieve this, each process generates a single random walk, each from its own random starting point, but all using the same index function and the same fi ’s. As soon as a walk hits upon a distinguished point, this point is reported. The idea is that when two walks collide – without noticing it – they will keep taking the same steps (because they use the same walk deﬁnition) and will thus both ultimately reach the same distinguished point. This will be noticed when the colliding distinguished point is reported. The discrete logarithm can then be computed from the two, hopefully distinct, pairs of integer multipliers u, v that correspond to the same distinguished point. A distinguished point must be easy to recognize, occur with low enough probability to make it possible to store them all and to eﬃciently ﬁnd collisions, but occur often enough for every walk to hit one. The distinguishing property could be that k speciﬁc bits of the point’s x-coordinate are zero, in which case walks may hit a distinguished point once every 2k steps. The parallelized version of Pollard’s rho method requires a unique, and thus aﬃne, point representation to make the walks well-deﬁned and to recognize distinguished points. The fastest suitable type of elliptic curve group arithmetic uses the aﬃne Weierstrass point representation. Per group operation, it requires a (usually expensive) modular inversion. Its cost is amortized among the walks running in parallel per processor, at the cost of three modular multiplications per step per walk, using Montgomery’s simultaneous inversion [14]. Point doubling requires an extra modular squaring compared to regular non-doubling point addition. This makes doubling on average about 76 times slower than regular addition when parallelized walks and simultaneous inversion are used. Using automorphisms. Following [18], deﬁne an equivalence relation ∼ on g by p ∼ −p for p ∈ g and, instead of searching g of size q, search g/∼ of size about q2 . Denoting the equivalence class containing p and −p by ∼p, it may be represented by the element with y-coordinate of least absolute value. It is trivial to calculate since −(x, y) = (x, −y) for (x, √y) ∈ g. Thus, using this negation map one would expect to save a factor of 2 in the number of steps. √ For r-adding and r + s-mixed walks the speedup by a factor of 2 is slightly too pessimistic. Let the deﬁnitions of pi , pD , and of class i be as above. Assume

70

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Table 1. Number of steps required by the Pollard rho method in random elliptic curve groups of 31-bit q over prime ﬁelds of random 31-bit prime characteristic p, prime order divided by πq/2 or by πq/4 (without or with the negation map). Lowest and highest averages are over 10 measurements. Each measurement calculates the average number of steps taken until a collision occurs, over 100 000 collision searches where for each search a prime p and an elliptic curve over Fp are randomly selected until the order q of the group of points is prime. Overall average is the average of the 10 averages (thus, the average over one million searches). Expression (1) and (2) columns are the quotients as 1 expected based on expressions (1) (with pi = 1r for 0 ≤ i < r) and (2) (with pi = r+s s for 0 ≤ i < r and pD = r+s ), respectively. Those expressions are for q → ∞ and indeed for larger (smaller) q they give a better (worse) ﬁt. Without negation map Averages Expression lowest overall highest (1) 8-adding 1.079 1.083 1.085 1.069 16-adding 1.032 1.037 1.040 1.033 32-adding 1.014 1.018 1.019 1.016 16 + 4-mixed 1.041 1.043 1.044 1.043 16 + 8-mixed 1.075 1.078 1.081 1.078

With negation map Averages Expression lowest overall highest (2) 1.035 1.039 1.042 1.033 1.015 1.017 1.020 1.016 1.007 1.009 1.011 1.008 1.036 1.038 1.040 1.031 1.075 1.077 1.079 1.069

that the nth point belongs to Gj and that the (n + 1)st point produces the ﬁrst collision while hitting the representative p, directly or after negation. If this step is a doubling then the analysis is as above. This happens with probability p2D . Otherwise, we only exclude the case that, as a result of just the addition, the two predecessors hit the same point (p or −p). This happens with probability Therefore, the probability that the ﬁrst collision occurs at step n + 1 is

p2j 2 .

r−1 2 pj 2n (1 − p2D − ). q 2 j=0

As above we get

πq 4(1 −

p2D

−

1 2

r−1 j=0

p2j )

(2)

for the expected number of steps until√the ﬁrst collision. For the same parameter values this expression is more than 2 smaller than Expression (1). However, usage of the negation map requires modiﬁcations to the iteration function due to the occurrence of fruitless cycles. This disadvantage of the negation map was already pointed out in [9,18]. It is the focus of this article. The group g may admit other trivially computable maps. For Koblitz curves the √ Frobenius automorphism of a degree t binary extension ﬁeld leads to a further t-fold speedup. This does not apply to the case considered here. Small scale experiments. We checked the accuracy of predictions based on expressions (1) and (2). The results, for 31-bit primes q, are listed in Table 1.

On the Use of the Negation Map in the Pollard Rho Method

71

With all averages larger than 1, both r-adding and r + s-mixed walks on average perform worse than truly random walks. For most walks with the negation map the averages are lower than their negation-less counterparts, indicating√that the reduction factor in the expected number of steps is indeed larger than 2. This does not imply a speedup by the same factor, because to obtain the ﬁgures costly fruitless cycle detection methods had to be used. It can be seen that r + s-mixed walks are disadvantageous if s > r4 . 2.3

Fruitless Cycles

Straightforward application of the negation map to Pollard’s rho method with r-adding or r + s-mixed walks does not work due to fruitless cycles. This section describes the current state-of-the-art of dealing with those cycles. Length 2 cycles. If a random walk step goes from p to −p − fi (with probability 12 , for some i) and −p − fi ∈ Gi (with probability 1r ), then the next point after −p − fi is p again (with probability 1), thereby cancelling the eﬀect of the previous step. It follows that a fruitless 2-cycle starts from a random point with 1 probability 2r , cf. [7, Proposition 31]. This 2-cycle is denoted as (i,−)

(i,−)

p −→ −(p + fi ) −→ p. Here “(i, s)” with s ∈ {−, +} indicates that addition constant fi is added to a point p after which the result is left as is (s = +) or negated (s = −) to ﬁnd the correct representative (p + fi if s = +, or −p − fi if s = −). Any walk with two consecutive steps “(i, −)” is trapped in an inﬁnite loop. Because this happens 1 with probability 2r , all walks can be expected to end up in fruitless cycles after a moderate number of steps when the negation map is used with r-adding walks. Looking ahead to reduce 2-cycles. To reduce the occurrence of 2-cycles, Wiener and Zuccherato propose to use a more costly iteration function that results in a lower probability that two successive points belong to the same partition [18]. This can be achieved by using the ﬁrst i of (p), (p) + 1, . . ., (p) + r − 1 such that i mod r = (∼ (p + fi )), if such an index exists (here and in the sequel indices i in fi are understood to be taken modulo r). Thus, deﬁne the next point as f (p) with f : g → g deﬁned by E(p) if j = (∼(p + fj )) for 0 ≤ j < r f (p) = ∼(p + fi ) with i ≥ (p) minimal s.t. (∼(p + fi )) = i mod r. The function E : g → g may restart the walk at a new random initial point. The latter is expected to happen once every rr steps and will therefore not aﬀect the The expected cost per step of the walk is increased by a factor of r eﬃciency. 1 1 1 i=0 r i , which lies between 1 + r and 1 + r−1 . Dealing with fruitless cycles in general. Although the look-ahead technique reduces the frequency of 2-cycles, they may still occur [18]. This is elaborated upon in Section 3. Even so, it is well known that just addressing 2-cycles does

72

J.W. Bos, T. Kleinjung, and A.K. Lenstra 4.5e+06 4e+06

steps / second

3.5e+06 3e+06 2.5e+06 2e+06 1.5e+06 1e+06 500000 0

2

4

6

8

10

log2 (r)

12

14

16

18

Fig. 1. Total number of steps per second as a function of r, taken by 200 parallel r-adding walks sharing the modular inversion and not using the negation map, for Pollard’s rho method applied to a 131-bit prime ECDLP

not solve the problem of fruitless cycles, because longer cycles will occur as well. Reducing their occurrence requires additional overhead on top of what is already incurred to reduce 2-cycles. Given that fruitless cycles are unavoidable, they must be eﬀectively dealt with when they occur. In [9] a general approach is proposed to detect cycles and to escape from them: after α steps record a length β sequence of successive points and compare the next point to these β points. If a cycle is detected a cycle representative p is chosen deterministically from which the cycle is escaped. One may add f(p)+c for a ﬁxed c ∈ [2, r − 1] (the choice c = 1 is bad as it could lead to an immediate cycle recurrence). Instead one may add a distinct precomputed value f that does not depend on the escape-point, or one may add f(p) from a distinct list of r precomputed values f0 , f1 , . . . , fr−1 . In the next section we discuss fruitless cycles in greater detail and propose alternative methods that avoid problems that the method from [9] may run into.

3

Improved Fruitless Cycle Handling

The probability to enter a fruitless cycle decreases with increasing r [7]. This does not imply that it suﬃces to take r large enough to make the probability sufﬁciently low. Fig. 1 depicts the eﬀect of increasing r-values on the performance of an r-adding walk, measured as number of steps per second. The performance deterioration can be attributed to the increasing rate of cache misses during retrieval of the addition constants fi . The eﬀect varies between processors, implementations, and elliptic curves. It is worsened for more contrived walks, such as those using the negation map where cycle reduction, detection and escape methods are unavoidable. Unless the expected overall number of steps (of or√ der q) is too small to be of interest, r cannot be chosen large enough to both

On the Use of the Negation Map in the Pollard Rho Method

(i,−)

(i,−) −p−fi =q

p (i,−)

(i−1, ..)

(∼(p+fi−1 )) = i−1

73

(i−1, ..)

(∼(q+fi−1 )) = i−1

−p−fi =q

p (i−1, ..)

(i,−)

¯ p= ∼(p+fi−1 )

¯q = ∼(q+fi−1 )

(j, ..) (∼(¯p +fj )) ∈ {i−1, j}

(i−1, ..)

(k, ..) (∼(¯q +fk )) ∈ {i−1, k}

Fig. 2. 2-cycles caused by 2-cycle reduction (left) and 4-cycle reduction. The dotted steps are prevented.

avoid fruitless cycles and achieve adequate performance. Therefore, in this section we concentrate on other ways to deal with fruitless cycles. We ﬁrst discuss short-cycle reduction techniques, next discuss cycle detection methods and analyze their behavior, and ﬁnally propose alternative methods. 3.1

Short Fruitless Cycle Reduction

2-cycles. Unfortunately, the look-ahead technique to reduce 2-cycles presented above introduces new 2-cycles. The dotted lines in the left example in Fig. 2 are the steps taken by the regular iteration function, the new cycle is depicted by the solid lines which are the steps taken as a result of f (p) and f (q). This new cycle occurs with probability 2r13 . It is the most likely 2-cycle introduced by the look-ahead technique. Lemma 1. The probability to enter a fruitless 2-cycle when looking ahead to reduce 2-cycles while using an r-adding walk is 1 2r

r−1 2

1 (rr−1 − 1)2 1 1 . = = + O i 2r−1 2 3 r 2r (r − 1) 2r r4 i=1

Proof. With i as in the deﬁnition of f , the probability is r−c that i ≥ (p) + c for 0 ≤ c < r (considering the case E(p) as i = ∞), hence i = (p) + c with 1 probability r−1 r rc . We compute the probability of entering a cycle consisting of points p and q starting at p. Let j = (p) and k = (q), and let the steps from p to q and back be adding fj+c and fk+d , respectively. This implies that j + c ≡ k + d mod r and that the step from p to q involves a negation. From the deﬁnition of f it follows

74

J.W. Bos, T. Kleinjung, and A.K. Lenstra

(∼(˜ p + fk )) ∈ {i, k} (k, ..) ˜ p =∼(p + fi ) (i, ..) p

(∼(˜q + fn ) ∈ {j, n} (n, ..) ∼(−p − fj+1 + fj ) = ˜q (j + 1,−)

(i + 1,+)

(j, ..) −p − fj+1 (i + 1,+) −p − fi+1 − fj+1

p + fi+1 (j, ..)

(j + 1,−)

¯ p =∼(p + fi+1 + fj ) (l, ..) (∼(¯ p + fl )) ∈ {j, l}

(i, ..) ∼(−p − fi+1 − fj+1 + fi ) = ¯q (m, ..) (∼ (¯q + fm )) ∈ {i, m}

Fig. 3. A 4-cycle when the 4-cycle reduction method is used

that (q) ≡ j + c mod r, thus d = 0 and by symmetry c = 0. Since j is given and k is determined by j, c and d, the probabilities must be summed over all possible c and d. The probability for a c, d pair is the product of the following probabilities: • • •

r−1 1 r rc 1 2 1 r−1

•

1 rd

for the ﬁrst step being c; for the sign; for (∼(p + fj+c )) = k (we know already that (∼(p + fj+c )) ≡ j + c ≡ k mod r); for the second step being d (since (∼(q + fk+d )) ≡ k + d mod r).

This results in the probability

r−1 r−1 1 1 1 . 2r c=1 rc rd

d=1

We conclude that, even when the look-ahead technique is used, 2-cycles are still too likely to occur for relevant values of q and r. Some of the new 2-cycles are prevented by other short-cycle reduction methods, but the remaining ones must be dealt with using detection and escape methods. This is discussed below. 4-cycles. Unless the addition constants fi have been chosen poorly, 3-cycles do not occur as a direct result of the negation map, so that 4-cycles are the next type of short cycles to be considered. Excluding again that the fi have unlikely properties, a fruitless 4-cycle without proper sub-cycle is of the form (i,+)

(j,−)

(i,+)

(j,−)

p −→ p + fi −→ −p − fi − fj −→ −p − fj −→ p. The cycle may be entered at any of its four points. Hence, a fruitless 4-cycle starts from a random point with probability r−1 4r 3 . This is a lower bound for the probability of occurrence of 4-cycles when looking ahead to reduce 2-cycles.

On the Use of the Negation Map in the Pollard Rho Method

75

An extension of the 2-cycle reduction method looks ahead to the ﬁrst two successors of a point, thereby reducing the frequency of 2-cycles and 4-cycles, while still being deterministic: ⎧ E(p) if j ∈ {(q), (∼(q + f(q) ))} or (q) = (∼(q + f(q) )) ⎪ ⎪ ⎨ where q =∼(p + fj ), for 0 ≤ j < r, g(p) = q =∼(p + fi ) with i ≥ (p) minimal s.t. ⎪ ⎪ ⎩ i mod r = (q) = (∼(q + f(q) )) = i mod r. Compared to f (p), the probability that E is called increases from ( 1r )r to at least ( 2r )r because (∼(q + f(q) )) ∈ {j mod r, (q)} with probability 2r for each j. This times slower than the standard one, because iteration function is at least r+4 r with probability 2r at least two additional group operations need to be carried 1 2 out, an eﬀect that is slightly alleviated by a factor of ( r−1 r ) since the image of g is a subset of g of cardinality approximately r−1 q. The value ∼(q+f(q)) can be r stored for use in the next iteration. Usage of g reduces the occurrence of 4-cycles, and also prevents some of the 2-cycles newly introduced by the 2-cycle reduction method (such as the one depicted on the left in Fig. 2). But g introduces new types of 2-cycles and 4-cycles as well, both of which do indeed occur in practice. A newly introduced 2-cycle is shown in the right example in Fig. 2. There the 2 points ¯ p and ¯ q are ∈ Gi−1 ∪ Gi . This 2-cycle occurs with probability 2(r−2) (r−1)r 4 , which is therefore a lower bound for the probability of 2-cycles when using the 4-cycle reduction method. Fig. 3 depicts an example of a newly introduced 4cycle: the points reached via dotted lines belong to a partition diﬀerent from their predecessors. The probability that such a 4-cycle starts from a random 4 (r−1) point is at least 4(r−2) . r 11 We have not been able to design or to ﬁnd in the literature short-cycle reduction methods that do not introduce other (lower probability) short cycles. We therefore turn our attention to cycle detection and escape methods. 3.2

Cycle Detection and Escape

Recurring cycles. The cycle detection and escape method from [9] described in Section 2.3, does not prevent recurrence to the same cycle. When using f(p)+c to escape (we ﬁxed c = 4 as it worked as well as any other choice = 1), Fig. 4 depicts how the (wavy) escape from the (solid) 4-cycle recurs to the 4-cycle via one of the dotted possibilities. The probability of recurrence depends on the escape method and on which point in the cycle the walk recurs to. With f(p)+c 1 as escape, immediate recurrence to the escape point happens with probability 2r 1 when no cycle reduction is used, recurrence happens with probability at least 2r2 2

with 2-cycle reduction, and with probability at least (r−2) with 4-cycle and thus r4 2-cycle reduction. Similar recurrences occur, with lower probabilities, when f or f(p) are used to escape. Lemma 2. Lower bounds for the probabilities to enter 2-cycles or 4-cycles or to recur to cycles for three diﬀerent cycle escape methods are listed in Table 2

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J.W. Bos, T. Kleinjung, and A.K. Lenstra

−p − fi − fj (j, −)

(i, +) −p − fj

p + fi (k, −)

p

(i, +)

(k, +) (j, −)

−p − fk − fj

(k, +)

−p − fi − fk

(i, −)

(j, −)

p + fk

Fig. 4. Escaping from a fruitless 4-cycle, and recurring to it (i = j = k = i)

if no cycle reduction, or 2-cycle reduction (f ), or 4-cycle reduction (g) is used, along with a lower bound for the slowdown factor caused by f or g. Proof. The proofs for many entries of Table 2 were given earlier. We prove the entries in rows four and ﬁve. Let p be the escape point and let q be the point it escapes to. Using f or f(p) one can recur to the escape point p by entering another cycle at q and escaping from it at q again. This new cycle could be a 2-cycle. For this to happen the ﬁrst escape step to q has to involve a negation (probability 12 ), a 2-cycle has to be entered at q (probabilities in ﬁrst row, but see below), the escape point of this 2-cycle has to be q (probability 12 ), and, in the case of fi , the partition that q belongs to has to be the same as the one p belongs to (probability 1r ). In the case of 4-cycle reduction the probability to enter a 2-cycle at q is slightly lower since we do not have the information that (∼(q + f(q) )) = (q); a calculation analogous to the one done at the end of Section 3.1 produces the values listed in the table. 6-cycles. With proper fi and no sub-cycle, a common 6-cycle is of the form (i,+)

(j,−)

(k,+)

(i,+)

(j,−)

(k,+)

p −→ p+fi −→ −p−fi −fj −→ −p−fi −fj +fk −→ −p−fj +fk −→ p−fk −→ p (i = j = k = i) where with appropriate sign changes steps four and ﬁve may be swapped. It may be entered at any of its six points and occurs, when using 4-cycle reduction, with probability 4r13 + O( r14 ). A lower bound to recur to it follows by multiplying this probability with the recurring probabilities from Table 2.

On the Use of the Negation Map in the Pollard Rho Method

77

Table 2. Summary of eﬀect of cycle reduction, detection, and escape methods. With the exception of the two bold entries, all ﬁgures are lower bounds. Probability to enter

Cycle reduction method:

none

2-cycle 4-cycle

⎧ ⎪ ⎨ f(p)+c Probability to recur to escape point using f ⎪ ⎩ f(p) Slowdown factor of iteration function

3.3

2-cycle

4-cycle

1 2r r−1 4r3

1 2r 3 r−1 4r 3

2(r−2)2 (r−1)r 4 4(r−2)4 (r−1) r 11

1 2r 1 8r 1 8r 2

1 2r 2 1 8r 3 1 8r 4

(r−2)2 r4 (r−2)2 2r 5 (r−2)2 2r 6

n/a

r+1 r

r+4 r

Alternative Approaches

The purpose √ of using the negation map is to obtain a speedup, hopefully by a factor of 2. From Fig. 1 it follows that large r-values cannot be used. From Table 2 it follows that for small r-values and relevant q-values fruitless cycles are likely to occur and recur. Medium r-values look the most promising, but are not compatible with all environments. Since fruitless cycle occurrence and recurrence cannot be rooted out, alternative methods are needed if we want to make the negation map useful. In this section several possibilities are oﬀered. Heuristic. A cycle with at least one doubling is most likely not fruitless. Proof. Let p = ug + vh be a point on the cycle. The subsequent points are obtained by adding one of the fi or by doubling, and negating if needed, thus are up to sign linear combinations of the fi and a power-of-two multiple of p. If c ≥ 1 is the number of doublings in the cycle, we get a relation of the form p = ±2c p +

r−1

ci fi = ±2c p +

i=0

(1 ∓ 2c )u −

r−1 i=0

r−1 i=0

ci u i

ci u i g +

r−1

g+

ci vi h

and thus

i=0

(1 ∓ 2c )v −

r−1

ci vi

h = 0,

i=0

r−1 where ci ∈ Z. Since 1 ∓ 2c = 0, the expression (1 ∓ 2c )u − i=0 ci ui is most likely not divisible by the group order. This also holds if {fi : 0 ≤ i < r} is enlarged with f or with {fi : 0 ≤ i < r}. This concludes our heuristic argument. Cycle reduction by doubling. The regular structure required for cycles is caused by repeated addition and subtraction using the same set of constants. This structure would be broken eﬀectively by using an occasional doubling, i.e., a mixed walk. If such walks are used, the heuristics suggest that cycles occur

78

J.W. Bos, T. Kleinjung, and A.K. Lenstra

only between two doublings. If the doubling frequency is suﬃciently high, only short cycles would have to be dealt with. As borne out by expressions (1) and (2) when using the idealized values pi = 1 s r+s for 0 ≤ i < r and pD = r+s for r > 0, and as supported by the experiments reported in Table 1, an r + s-mixed walk with s > 1 always displays noticeably less random behavior than a well-partitioned r -adding walk for any r > r. Nevertheless, using properly tuned r + s-mixed walks may be a way to address the cycle problem while avoiding impractically large r-values. However, r + s-mixed walks have disadvantages caused by the underlying arithmetic. Given the relative speeds of addition and doubling, an r + s-mixed walk is r+7s/6 times slower than an r-adding walk. In a SIMD environment r+s where many walks are processed simultaneously, per step a fraction of about r r+s of the walks will do an addition, whereas the others do a doubling. If the addition and doubling code diﬀer, as is the case for the aﬃne Weierstrass representation, the two types of steps cannot be executed simultaneously. Thus, in such environments, to avoid a slowdown by a factor of more than 2 one needs to swap walks to make all parallel step-operations identical (at non-negligible overhead), or one has to settle for a suboptimal aﬃne point representation that allows identical code. SIMD-application of the negation map and the possibility of another point representation are subjects for further study. Doubling based cycle reduction and escape. Taking into account that doubling should not be used too frequently, usage could be limited to cycle reduction or escape. This would not solve the SIMD-issue, but the relative ineﬃciency and non-randomness would be addressed. If doublings are used to escape from fruitless cycles, they would not recur, as that would contradict the heuristics. Cycle reduction using doubling replaces f (p) and g(p) by f¯(p) and g¯(p), respectively, where ∼(p + f(p) ) if (p) = (∼(p + f(p) )), ¯ f (p) = ∼(2p) otherwise, g¯(p) =

q =∼(p + f(p) ) if (q) = (p) = (∼(q + f(q) )) = (q), ∼(2p) otherwise.

It follows from the heuristics that these functions avoid recurring fruitless cycles. Alternative cycle detection. Because shorter cycles are more frequent, a potentially interesting modiﬁcation of the cycle detection method from [9] (described at the end of Section 2.3) would be to occasionally compare a point to its kth successor, where k is the least common multiple of all even short cycle lengths that one wants to catch. Detecting, for instance, cycles up to length 1 12 requires only 120 th comparison per step. This can be done in several steps, recording every 12th point to catch 4- and 6-cycles, recording every 10th of these recorded points to catch 8- and 10-cycles, etc. It can be combined with the regular method with large α and β to catch longer cycles infrequently. However, if a cycle has been detected the k points need to be recorded as before, so an escape point can be chosen deterministically. This argues against

On the Use of the Negation Map in the Pollard Rho Method

79

using large k. It also suggests that an improvement can be expected only if cycles occur with low probability, and therefore that the improvement will be marginal at best (cf. α and β choices in Section 4). For this reason we did not conduct extensive experiments with this method.

4

Comparison

We implemented and compared on a traditional non-SIMD platform all previously published and newly proposed methods to deal with fruitless cycles when using the negation map. Here we report on our ﬁndings. It quickly turned out that the cycle detection methods from [9] when combined with doubling based cycle reduction and escape, are considerably more eﬃcient than r+s-mixed walks with their on average slower steps and less random behavior. Mixed walks are therefore not further discussed. Experiments with the alternative cycle detection method were quickly abandoned as well. For each combination of iteration function, escape method, and r-value a search was conducted to determine the α and β to be used for the cycle detection method from [9]. Using a heuristic argument that for β = 2k with k much smaller than r, cycles of length ≥ β occur with probability on the order of (k−1)! , values (2r)k for k that make this probability low enough resulted in good initial values for the search for close to optimal α and β. To give some examples, for “f , e,” as explained in Table 3 we used α = 31 and β = 20 for r = 16, α = 3264 and β = 12 for r = 128, and α = 52 418 and β = 10 for r = 256. For “f¯, ¯e” and the same r-values we used the same β-values but replaced the α-values by 1 618, 838 848, and 53 687 081, respectively. Each of the benchmarks presented in Table 3 was run on a single core of an AMD Phenom 2.2GHz 4-core processor, with each of the four cores processing a diﬀerent combination. A 10-bit distinguishing property was used to get a signiﬁcant amount of data in a reasonable amount of time. This somewhat aﬀects the performance, but not the cycle behavior as walks continue after hitting a distinguished point. The ﬁgures in millions as given in the table are thus an underestimate for the actual per-core yield in units when a more realistic 30-bit distinguishing property would be used (since 230 /210 = 220 ≈ 106 ). In order to be able to compare the long term yield ﬁgures, the expected number of steps must be taken into account using expressions 1 and 2. As a 1 2 result, the yields are corrected by a factor of ( r−1 r ) for the iteration functions 1 2 that do not use the negation map, and by a factor of ( 2r−1 r ) for the others, with 1 r an extra factor of ( r−1 ) 2 for g and g¯. After this correction, the best iteration function without the negation map is the one with r = 64. Comparing that one with each iteration function that uses the negation map, thus boosting the 63 12 63 12 latter’s yield ratio by a factor of C = (( 2r−1 or C = (( 2r−1 r )/( 64 )) r−1 )/( 64 )) for g and g¯, leads to the long term speedup ﬁgure given in Table 3. Note that the correction factor√C depends on the iteration function, and is close to and for some r larger than 2.

80

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Table 3. For the (iteration function, escape method, r-value) combinations speciﬁed, the non-italics entries list the long term yield (millions of distinguished points, found during the second half hour) and the long term speedup over the best r-value (r = 64) without the negation map, taking into account the correction factor C as explained in the text. Cycle detection and subsequent escape by adding f(p)+4 , f , f(p) and by doubling is indicated by “e,” “e ,” “e ” and by “¯e,” respectively. The iteration functions f (2-cycle reduction), g (4-cycle and 2-cycle reduction), f¯ (2-cycle reduction using doubling), and g¯ (4-cycle and 2-cycle reduction using doubling) are as in sections 2.3, 3.1 and 3.3. The yields are for 256 parallel walks (sharing the inversion) for a 131-bit ECDLP with a 131-bit prime order group. The yields during the ﬁrst half hour are almost consistently higher, considerably so for poorly performing combinations. They are not meaningful and are thus not listed. The italics entries are A above D, followed 9 −A) , as explained in the text. by the maximal achievable speedup factor of C(10 109 +D/6 †: This applies to “no reduction, no escape,” “just f ,” “just f¯,” “just e,” and “just e .” r = 16

r = 32

Without negation map 7.29: 0.98 7.28: 0.99 With negation map 0.00: 0.00 0.00: 0.00 † 0.00: 0.00 0.00: 0.00 just g 0.00: 0.00 0.00: 0.00 just g ¯ just e 0.00: 0.00 0.00: 0.00 3.34: 0.64 4.89: 0.95 just ¯ e 0.00: 0.00 0.00: 0.00 f, e 9 .4e8 6 .6e8 0 .0e0 }0 .48 0 .0e0 }0 .08 0.00: 0.00 3.24: 0.63 f , e 8 .0e7 3 .9e8 0 .0e0 }0 .86 0 .0e0 }1 .30 0.00: 0.00 5.34: 1.04 f , e 6 .0e7 1 .3e8 0 .0e0 }1 .22 0 .0e0 }1 .33 3.71: 0.72 6.36: 1.24 f, ¯ e 9 .2e7 6 .8e7 9 .9e5 }1 .27 2 .8e5 }1 .32 0.00: 0.00 0.01: 0.00 g, e 8 .7e8 3 .7e8 0 .0e0 }0 .19 0 .0e0 }0 .91 0.00: 0.00 0.01: 0.00 g, e 7 .8e8 3 .0e8 0 .0e0 }1 .00 0 .0e0 }0 .32 0.00: 0.00 1.09: 0.21 g, e 7 .6e8 1 .2e8 0 .0e0 }1 .27 0 .0e0 }0 .34 0.76: 0.15 5.91: 1.17 g, ¯ e 1 .7e8 3 .3e8 1 .6e5 }0 .97 6 .0e4 }1 .19 0.00: 0.00 0.00: 0.00 f¯, e 8 .7e8 4 .3e8 2 .4e6 }0 .18 1 .7e7 }0 .80 0.01: 0.0 4.24: 0.82 f¯, e 6 .8e7 2 .6e8 4 .3e7 }1 .03 2 .9e7 }1 .31 1.34: 0.26 5.80: 1.13 f¯, e 8 .9e7 5 .3e7 5 .2e7 }1 .27 2 .9e7 }1 .33 5.58: 1.06 6.14: 1.18 f¯, ¯ e 6 .1e7 3 .7e7 4 .2e7 }1 .31 3 .0e7 }1 .36 2.56: 0.51 5.80: 1.15 g ¯, e 1 .4e8 7 .9e7 9 .9e7 }1 .23 5 .6e7 }1 .31 4.74: 0.94 5.88: 1.16 g ¯, e 7 .8e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31 4.72: 0.94 5.80: 1.15 g ¯, e 7 .7e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31 4.83: 0.96 5.87: 1.16 g ¯, ¯ e 7 .9e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31

r = 64

r = 128

r = 256

r = 512

7.27: 1.00

7.19: 0.99

6.97: 0.96

6.78: 0.94

0.00: 0.00 0.00: 0.00 0.00: 0.00 0.00: 0.00 5.85: 1.14 1.52: 0.30 1 .0e8 0 .0e0 }1 .28 6.04: 1.18 4 .6e7 0 .0e0 }1 .35 6.21: 1.21 4 .2e7 0 .0e0 }1 .36 6.50: 1.27 4 .2e7 6 .5e4 }1 .36 4.89: 0.96 6 .6e7 0 .0e0 }1 .34 5.32: 1.05 6 .0e7 0 .0e0 }1 .35 5.37: 1.13 6 .0e7 0 .0e0 }1 .35 6.02: 1.18 8 .1e7 8 .1e3 }1 .32 2.70: 0.53 5 .4e7 1 .5e7 }1 .34 6.32: 1.23 3 .9e7 1 .5e7 }1 .36 6.23: 1.22 3 .9e7 1 .5e7 }1 .36 6.34: 1.23 1 .8e7 1 .5e7 }1 .39 6.02: 1.18 5 .1e7 2 .9e7 }1 .35 6.14: 1.21 5 .3e7 2 .9e7 }1 .35 6.08: 1.20 5 .3e7 2 .9e7 }1 .35 6.09: 1.20 5 .2e7 2 .9e7 }1 .35

0.00: 0.00 0.00: 0.00 0.75: 0.15 0.61: 0.12 6.10: 1.19 5.93: 1.16 3 .6e7 0 .0e0 }1 .37 6.41: 1.25 3 .3e7 0 .0e0 }1 .38 6.30: 1.23 3 .3e7 0 .0e0 }1 .38 6.57: 1.29 3 .3e7 1 .5e4 }1 .38 6.22: 1.22 4 .2e7 0 .0e0 }1 .37 6.26: 1.23 4 .1e7 0 .0e0 }1 .37 6.08: 1.20 4 .2e7 0 .0e0 }1 .37 6.25: 1.23 5 .4e7 1 .0e3 }1 .35 5.96: 1.16 1 .1e7 7 .7e6 }1 .41 6.43: 1.26 3 .2e7 7 .6e6 }1 .38 6.21: 1.22 3 .6e7 7 .5e6 }1 .37 6.42: 1.25 1 .1e7 7 .7e6 }1 .41 6.09: 1.20 4 .1e7 1 .5e7 }1 .37 6.28: 1.23 3 .9e7 1 .5e7 }1 .37 6.05: 1.19 3 .8e7 1 .5e7 }1 .37 6.16: 1.21 4 .0e7 1 .5e7 }1 .37

0.00: 0.00 0.04: 0.01 4.90: 0.96 4.94: 0.97 6.28: 1.23 6.47: 1.27 2 .9e7 0 .0e0 }1 .38 6.29: 1.23 2 .9e7 0 .0e0 }1 .38 6.20: 1.21 2 .9e7 0 .0e0 }1 .38 6.47: 1.27 2 .9e7 3 .8e3 }1 .38 6.23: 1.22 3 .3e7 0 .0e0 }1 .38 6.25: 1.23 3 .0e7 0 .0e0 }1 .38 6.06: 1.19 3 .5e7 0 .0e0 }1 .38 6.13: 1.20 4 .0e7 1 .2e2 }1 .37 6.34: 1.24 1 .0e7 3 .9e6 }1 .41 6.33: 1.24 2 .8e7 3 .8e6 }1 .38 6.15: 1.20 2 .8e7 3 .8e6 }1 .38 6.27: 1.23 1 .0e7 3 .9e6 }1 .41 6.19: 1.21 2 .6e7 7 .6e6 }1 .39 6.05: 1.19 2 .6e7 7 .6e6 }1 .39 5.91: 1.16 1 .8e7 7 .6e6 }1 .40 6.09: 1.20 2 .6e7 7 .6e6 }1 .39

0.00: 0.00 3.59: 0.70 5.90: 1.16 5.73: 1.12 6.18: 1.21 6.36: 1.25 2 .5e7 0 .0e0 }1 .39 6.21: 1.22 2 .6e7 0 .0e0 }1 .39 5.99: 1.17 2 .7e7 0 .0e0 }1 .39 6.30: 1.25 2 .7e7 9 .7e2 }1 .39 6.05: 1.19 1 .3e7 0 .0e0 }1 .41 6.11: 1.20 5 .5e7 0 .0e0 }1 .35 5.86: 1.15 4 .3e7 0 .0e0 }1 .37 6.00: 1.18 2 .7e7 9 .0e0 }1 .39 6.20: 1.21 1 .4e7 1 .9e6 }1 .40 6.20: 1.22 2 .7e7 1 .9e6 }1 .39 6.00: 1.18 2 .6e7 1 .9e6 }1 .39 6.07: 1.19 1 .4e7 1 .9e6 }1 .40 5.74: 1.13 7 .7e6 3 .9e6 }1 .41 5.80: 1.14 7 .7e6 3 .9e6 }1 .41 5.67: 1.11 7 .7e6 3 .9e6 }1 .41 5.70: 1.12 7 .7e6 3 .9e6 }1 .41

On the Use of the Negation Map in the Pollard Rho Method

81

Non-doubling 2-cycle reduction (f ) with doubling-based cycle escape (¯e) and r = 128 performed best, with an overall speedup by a factor of 1.29: although fewer distinguished points are found than for the best case without the negation map (r = 64), there is a considerable overall gain because fewer distinguished points (by a factor of C, for the relevant C) should suﬃce. For r = 16 most iteration functions with the negation map perform poorly. √ We measured to what extent our failure to achieve a speedup by a factor of 2 can be blamed on cycle detection and escape and other overheads, and which part is due to the higher average cost of the iteration function. For most combinations in Table 3 we counted the number S of useful steps performed when doing 109 group operations, while keeping track of the number D of doublings among them. Here a step is useful if it is not taken as part of a fruitless cycle, so all D doublings are useful. Without the negation map, S would be 109 and D = 0; this is the basis for the comparison. With the negation map, A = 109 − S is counted as the number of additional additions due to cycle reductions or fruitless cycles. The inherent slowdown of that iteration function is then 1 + A+D/6 , so that it S 9

−A) CS = C(10 can achieve a speedup by a factor of at most S+A+D/6 109 +D/6 , with C as deﬁned above. Based on Table 3 and Fig. 1, we√conclude that our failure to better approach the optimal speedup by a factor of 2 is due to an onset of cache eﬀects combined with various overheads. The italics ﬁgures from Table 3 make us believe that improvements may be obtained when using better implementations.

Previous results. The only publication that we know that presents practical data about Pollard’s rho method used with the negation map is [8]. Only relatively small ECDLPs were solved (42- and 43-bit prime ﬁelds) and small r-values were avoided. The adverse cycle behavior that we witnessed can therefore not be expected and we doubt if the results reported are signiﬁcant for the sizes that we consider. Only mixed walks were used, and an overall speedup by a factor of about 1.35 was reported. Cycle escaping was done by jumping to the sum of all points in a cycle, which cannot be expected to work in general because the sum may depend just on the addition constants.

5

Conclusion

With judicious application of doubling, usage of the negation map to solve ECDLPs over prime ﬁelds using Pollard’s rho method can indeed be recommended. In the best of circumstances that √ we have been able to create, however, the speedup falls short of the hoped for 2, but is with 1.29 still considerable. This conclusion does not apply to SIMD-environments where occasional doublings cause considerable delays. Alternative point representations need to be considered to assess the usefulness of the negation map for SIMD platforms, in particular because such platforms are becoming popular again. Acknowledgements. This work was supported by the Swiss National Science Foundation under grant numbers 200021-119776 and 206021-117409 and

82

J.W. Bos, T. Kleinjung, and A.K. Lenstra

by EPFL DIT. We gratefully acknowledge useful suggestions by Marcelo E. Kaihara and very insightful comments by the ANTS reviewers.

References 1. Avanzi, R.M., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC (2006) 2. Bailey, D.V., et al.: Breaking ECC2K-130. In: Cryptology ePrint Archive, Report 2009/541 (2009), http://eprint.iacr.org/ 3. Bos, J.W., Kaihara, M.E., Montgomery, P.L.: Pollard rho on the PlayStation 3. In: Workshop record of SHARCS 2009, pp. 35–50 (2009), http://www.hyperelliptic.org/tanja/SHARCS/record2.pdf 4. Brent, R.P., Pollard, J.M.: Factorization of the eighth Fermat number. Math. Comp. 36(154), 627–630 (1981) 5. Certicom. Certicom ECC Challenge (1997), http://www.certicom.com/images/pdfs/cert_ecc_challenge.pdf 6. Certicom. Press release: Certicom announces elliptic curve cryptosystem (ECC) challenge winner (2002), http://www.certicom.com/index.php/2002-press-releases/ 38-2002-press-releases/340-notre-dame-mathematician-solveseccp-109-encryption-key-problem-issued-in-1997 7. Duursma, I.M., Gaudry, P., Morain, F.: Speeding up the discrete log computation on curves with automorphisms. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 103–121. Springer, Heidelberg (1999) 8. Escott, A.E., Sager, J.C., Selkirk, A.P.L., Tsapakidis, D.: Attacking elliptic curve cryptosystems using the parallel Pollard rho method. CryptoBytes Technical Newsletter 4(2), 15–19 (1999), ftp.rsasecurity.com/pub/cryptobytes/crypto4n2.pdf 9. Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Improving the parallelized Pollard lambda search on anomalous binary curves. Math. Comp. 69(232), 1699–1705 (2000) 10. Harley, R.: Elliptic curve discrete logarithms project, http://pauillac.inria.fr/~harley/ 11. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987) 12. Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992) 13. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986) 14. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48, 243–264 (1987) 15. Pollard, J.M.: Monte Carlo methods for index computation (mod p). Math. Comp. 32, 918–924 (1978) 16. Teske, E.: On random walks for Pollard’s rho method. Math. Comp. 70(234), 809– 825 (2001) 17. van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. Journal of Cryptology 12(1), 1–28 (1999) 18. Wiener, M.J., Zuccherato, R.J.: Faster attacks on elliptic curve cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999)

An O(M (n) log n) Algorithm for the Jacobi Symbol Richard P. Brent1 and Paul Zimmermann2 1 2

Australian National University, Canberra, Australia INRIA Nancy - Grand Est, Villers-l`es-Nancy, France

Abstract. The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M (n) log n), using Sch¨ onhage’s fast continued fraction algorithm combined with an identity due to Gauss. We give a diﬀerent O(M (n) log n) algorithm based on the binary recursive gcd algorithm of Stehl´e and Zimmermann. Our implementation — which to our knowledge is the ﬁrst to run in time O(M (n) log n) — is faster than GMP’s quadratic implementation for inputs larger than about 10000 decimal digits.

1

Introduction

We want to compute the Jacobi symbol1 (b|a) for n-bit integers a and b, where a is odd positive. We give three algorithms based on the 2-adic gcd from Stehl´e and Zimmermann [13]. First we give an algorithm whose worst-case time bound is 3 ); we call this the cubic algorithm although this is pessimistic O(M (n)n2 ) = O(n since the algorithm is quadratic on average as shown in [5], and probably also in the worst case. We then show how to reduce the worst-case to O(M (n)n) = 2 ) by combining sequences of “ugly” iterations (deﬁned in Section 1.1) into O(n one “harmless” iteration. Finally, we obtain an algorithm with worst-case time O(M (n) log n). This is, up to a constant factor, the same as the time bound for the best known algorithm, apparently never published in full, but sketched in Bach [1] and in more detail in Bach and Shallit [2] (with credit to Bachmann [3]). The latter algorithm makes use of the Knuth-Sch¨ onhage fast continued fraction algorithm [9] and an identity of Gauss [6]. Although this algorithm has been attributed to Sch¨ onhage, Sch¨ onhage himself gives a diﬀerent O(M (n) log n) algorithm [10,15] which does not depend on the identity of Gauss. The algorithm is mentioned in Sch¨ onhage’s book [11, §7.2.3], but no details are given there. With our algorithm it is not necessary to compute the full continued fraction or to use the identity of Gauss for the Jacobi symbol. Thus, it provides an alternative that may be easier to implement. 1

Notation: we write the Jacobi symbol (b|a), since this is easier to typeset and as less ambiguous than the more usual ab . Also, M (n) is the time to multiply n-bit (n)) means O(f (n)(log f (n))c ) for some constant c ≥ 0. numbers, and O(f

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 83–95, 2010. c Springer-Verlag Berlin Heidelberg 2010

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It is possible to modify some of the other fast GCD algorithms considered by M¨ oller [8] to compute the Jacobi symbol, but we do not consider such possibilities here. At best they give a small constant factor speedup over our algorithm. We recall the main identities satisﬁed by the Jacobi symbol: (bc|a) = (b|a)(c|a); 2 (2|a) = (−1)(a −1)/8 ; (b|a) = (−1)(a−1)(b−1)/4 (a|b) for a, b odd; and (b|a) = 0 if (a, b) = 1. Note that all our algorithms compute (b|a) with b even positive and a odd positive. For the more general case where b is any integer, we can reduce to b even and positive using (b|a) = (−1)(a−1)/2 (−b|a) if b is negative, and (b|a) = (b+a|a) if b is odd. We ﬁrst describe a cubic algorithm to compute the Jacobi symbol. The quadratic algorithm in Section 2 is based on this cubic algorithm, and the subquadratic algorithm in Section 3 uses the same ideas as the quadratic algorithm but with an asymptotically fast recursive implementation. For a ∈ Z, the notation ν(a) denotes the 2-adic valuation ν2 (a) of a, that is the maximum k such that 2k |a, or +∞ if a = 0. 1.1

Binary Division with Positive Quotient

Throughout the paper we use the binary division with positive quotient deﬁned by Algorithm 1.1. Compared to the “centered division” of [13], it returns a quotient in [1, 2j+1 − 1] instead of in [1 − 2j , 2j − 1]. Note that the quotient q is always odd. Algorithm 1.1. BinaryDividePos Input: a, b ∈ N with ν(a) = 0 < ν(b) = j Output: q and r = a + qb/2j such that 0 < q < 2j+1 , ν(b) < ν(r) 1: q ← −a/(b/2j ) mod 2j+1 q is odd and positive 2: return q, r = a + qb/2j .

With this binary division, we deﬁne Algorithm CubicBinaryJacobi, where the fact that the quotient q is positive ensures that all a, b terms computed remain positive, and a remains odd, thus (b|a) remains well-deﬁned.2 Theorem 1. Algorithm CubicBinaryJacobi is correct (assuming it terminates). Proof. We prove that the following invariant holds during the algorithm, if a0 , b0 are the initial values of a, b: (b0 |a0 ) = (−1)s (b|a). This is true before we enter the while-loop, since s = 0, a = a0 , and b = b0 . For each step in the while loop, we divide b by 2j , swap a and b = b/2j , replace a 2

M¨ oller says in [8]: “if one tries to use positive quotients 0 < q < 2k+1 , the [binary gcd] algorithm no longer terminates”. However, with a modiﬁed stopping criterion as in Algorithm CubicBinaryJacobi, the algorithm terminates (we prove this below).

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Algorithm 1.2. CubicBinaryJacobi Input: a, b ∈ N with ν(a) = 0 < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: b ← b/2j 4: (q, r) ← BinaryDividePos(a, b) 2 5: s ← (s + j(a2 − 1)/8 + (a − 1)(b − 1)/4 + j(b − 1)/8) mod 2 j 6: (a, b) ← (b , r/2 ), j ← ν(b) 7: if a = 1 then return (−1)s else return 0

by r = a + qb , and divide r by 2j . The Jacobi symbol is modiﬁed by a factor 2 (−1)j(a −1)/8 for the division of b by 2j , by a factor (−1)(a−1)(b −1)/4 for the 2 interchange of a and b , and by a factor (−1)j(b −1)/8 for the division of r by 2j . At the end of the loop, we have gcd(a0 , b0 ) = a; if a = 1, since (b|1) = 1, we have (b0 |a0 ) = (−1)s , otherwise (b0 |a0 ) = 0. Lemma 1. The quantity a + 2b is non-increasing in Algorithm CubicBinaryJacobi. Proof. At each iteration of the “while” loop, a becomes b/2j , and b becomes (a + qb/2j )/2j . In matrix notation 0 1/2j a a ← . (1) b b 1/2j q/22j Therefore a + 2b becomes b 2a b a + qb/2j = j + (1 + 2q/2j ) j . +2 j j 2 2 2 2

(2)

Since j ≥ 1, the ﬁrst term is bounded by a. In the second term, q ≤ 2j+1 − 1, thus the second term is bounded by (5/2j − 2/22j )b, which is bounded by 9b/8 for j ≥ 2, and equals 2b for j = 1. If j ≥ 2, then a + 2b is multiplied by a factor at most 9/16. If j = q = 1 then a + 2b decreases, but by a factor which could be arbitrarily close to 1. The only case where a + 2b does not decrease is when j = 1 and q = 3; in this case a + 2b is unchanged. This motivates us to deﬁne three classes of iterations: good, bad, and ugly. Let us say that we have a good iteration when j ≥ 2, a bad iteration when j = q = 1, and an ugly iteration when j = 1 and q = 3. Since q is odd and 1 ≤ q ≤ 2j+1 − 1, this covers all possibilities. For a bad iteration, (a, b) becomes (b/2, a/2 + b/4), and for an ugly iteration, (a, b) becomes (b/2, a/2 + 3b/4). We denote the matrices corresponding to good, bad and ugly iterations by G, B and U respectively. Thus

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G = Gj,q =

0 1/2j 1/2j q/4j

, B=

0 1/2 1/2 1/4

, U=

0 1/2 1/2 3/4

.

The eﬀect of m successive ugly iterations is easily seen to be given by the matrix 1 1 + 4(−1/4)m 2 − 2(−1/4)m m U = . (3) 5 2 − 2(−1/4)m 4 + (−1/4)m Assume we start from (a, b) = (a0 , b0 ), and after m > 0 successive ugly iterations we get values (am , bm ). Then, from Equation (3), 5am = (a + 2b) + 2(2a − b)(−1/4)m , m

5bm = 2(a + 2b) − (2a − b)(−1/4) .

(4) (5)

We can not have 2a0 = b0 or the algorithm would have terminated. However, am must be an integer. This gives an upper bound on m. For a0 , b0 of n bits, the number of successive ugly iterations is bounded by n/2 + O(1) (a precise statement is made in Lemma 2). If there were no bad iterations, this would prove that for n-bit inputs the number of iterations is O(n2 ), since each sequence of ugly iterations would be followed by at least one good iteration. Bad iterations can be handled by a more complicated argument which we omit, since they will be considered in detail in §2 when we discuss the complexity of the quadratic algorithm (see the proof of Theorem 2). Since the number of iterations is O(n2 ) from Theorem 2, and each iteration costs time O(M (n)), the overall time for Algorithm CubicBinaryJacobi is 3 ). Note that this worst-case bound is almost certainly too O(n2 M (n)) = O(n pessimistic (see §4).

2

A Provably Quadratic Algorithm

Suppose we have a sequence of m > 0 ugly iterations. It is possible to combine the m ugly iterations into one harmless iteration which is not much more expensive than a normal (good or bad) iteration. Also, it is possible to predict the maximal such m in advance. Using this trick, we reduce the number of iterations (good, 2 ). Without loss bad and harmless) to O(n) and their cost to O(M (n)n) = O(n of generality, suppose that we start from (a0 , b0 ) = (a, b). Lemma 2. If μ = ν(a−b/2), then we have exactly μ/2 ugly iterations starting from (a, b), followed by a good iteration if μ is even, and by a bad iteration if μ is odd. Proof. We prove the lemma by induction on μ. If μ = 0, a − b/2 is odd, but a is odd, so b/2 is even, which yields j ≥ 2 in BinaryDividePos, thus a, b yield a good iteration. If μ = 1, a − b/2 is even, which implies that b/2 is odd, thus we have j = 1. If we had q = 3 in BinaryDividePos, this would mean that

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a + 3(b/2) = 0 mod 4, or equivalently a − b/2 = 0 mod 4, which is incompatible with μ = 1. Thus we have q = 1, and a bad iteration. Now assume μ ≥ 2. The ﬁrst iteration is ugly since 4 divides a − b/2, which implies that b/2 is odd. Thus j = 1, and a − b/2 = 0 mod 4 implies that q = 3. After one ugly iteration (a, b) becomes (b/2, a/2 + 3b/4), thus a − b/2 becomes −(a − b/2)/4, and the 2-valuation of a − b/2 decreases by 2. From the above, we see that, for a sequence of m ugly iterations, a0 , a1 , . . . , am satisfy the three-term recurrence 4ai+1 − 3ai − ai−1 = 0 for 0 < i < m, and similarly for b0 , b1 , . . . , bm . It follows that ai = a mod 4, and similarly bi = b mod 4, for 1 ≤ i < m. We can modify Algorithm CubicBinaryJacobi to consolidate m consecutive ugly iterations into one harmless iteration, using the expressions (4)–(5) for am and bm (we give an optimised evaluation below). It remains to modify step 5 of CubicBinaryJacobi to take account of the m updates to s. Since j = 1 for each ugly iteration, we have to increment s by an amount 2 a2i − 1 bi − 1 ai − 1 bi − 1 δ= + + mod 2, 8 8 2 2 0≤i<m

where we write bi for bi /2. However, ai+1 = bi for 0 ≤ i < m, so the terms involving division by 8 “collapse” mod 2, leaving just the ﬁrst and last terms. The terms involving two divisions by 2 are all equal to (a − 1)/2 · (b − 1)/2 mod 2, using the observation that ai mod 4 is constant for 0 ≤ i < m. Thus 2 a0 − 1 a 1 − 1 a0 − 1 a2m − 1 + +m mod 2. δ= 8 8 2 2 One further simpliﬁcation is possible. Since a0 = a1 mod 4, and a0 is odd, we can replace a1 by a0 in the last term, and use the fact that x2 = x mod 2 to obtain 2 a0 − 1 a0 − 1 a2m − 1 + +m mod 2. (6) δ= 8 8 2 We can economise the computation of am and bm from (4)–(5) by ﬁrst computing d = a − b , m = ν(d) div 2, c = (d − (−1)m (d/4m ))/5, where the divisions by 4m and by 5 are exact; then am = a − 4c, bm = b + 2c. From these observations, it is easy to modify Algorithm CubicBinaryJacobi to obtain Algorithm QuadraticBinaryJacobi. In this algorithm, steps 7–11 implement a harmless iteration equivalent to m > 0 consecutive ugly iterations; steps 13–14 implement bad and good iterations, and the remaining steps are common to both. Step 5 of Algorithm CubicBinaryJacobi is split into three steps 4, 13 and 15. In the case of a harmless iteration, the computation of δ satisfying (6) is implicit in steps 4, 10 and 15.

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Algorithm 2.1. QuadraticBinaryJacobi Input: a, b ∈ N with ν(a) = 0 < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: b ← b/2j 4: s ← (s + j(a2 − 1)/8) mod 2 5: (q, r) ← BinaryDividePos(a, b) 6: if (j, q) = (1, 3) then 7: d ← a − b 8: m ← ν(d) div 2 9: c ← (d − (−1)m d/4m )/5 10: s ← (s + m(a − 1)/2) mod 2 11: (a, b) ← (a − 4c, b + 2c) 12: else 13: s ← (s + (a − 1)(b − 1)/4) mod 2 14: (a, b) ← (b , r/2j ) 15: s ← (s + j(a2 − 1)/8) mod 2, j ← ν(b) 16: if a = 1 then return (−1)s else return 0

harmless iteration

good or bad iteration

Theorem 2. Algorithm QuadraticBinaryJacobi is correct and terminates after O(n) iterations of the “while” loop (steps 2–15) if the inputs are positive integers of at most n bits, with 0 = ν(a) < ν(b). Proof. Correctness follows from the equivalence to Algorithm CubicBinaryJacobi. To prove that convergence takes O(n) iterations, we show that a + 2b is multiplied by a factor at most 5/8 in each block of three iterations. This is true if the block includes at least one good iteration, so we need only consider harmless and bad iterations. Two harmless iterations do not occur in succession, so the block must include either (harmless, bad) or (bad, bad). In the ﬁrst case, the corresponding matrix is BU m = BU · U m−1 for some m > 0. We saw in §1.1 that the matrix U leaves a + 2b unchanged, so U m−1 also leaves a + 2b unchanged, and we need only consider the eﬀect of BU . Suppose that (a, b) is transformed into ( a, b) by BU . Thus a a 1/4 3/8 a . b = BU b = 1/8 7/16 b We see that a + 2b =

a 5b 5 + ≤ (a + 2b). 2 4 8

The case of two successive bad iterations is similar – just replace BU by B 2 in the above, and deduce that a + 2b ≤ (a + 2b)/2. We conclude that the number of iterations of the while loop is at most cn + O(1), where c = 3/ log2 (8/5) ≈ 4.4243.

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Remarks 1. A more complicated argument along similar lines can reduce the constant c to √ 2/ log2 (1/ρ(BU )) = 2/ log2 ((11 − 57)/2) ≈ 2.5424. Here ρ denotes the spectral radius: ρ(A) = limk→∞ ||Ak ||1/k . 2. In practice QuadraticBinaryJacobi is not much (if any) faster than CubicBinaryJacobi. Its advantage is simply the better worst-case time bound. A heuristic argument suggests that on average only 1/4 of the iterations of CubicBinaryJacobi are ugly. 3. Our implementations of CubicBinaryJacobi and QuadraticBinaryJacobi are slower than GMP’s O(n2 ) algorithm (which is based on Stein’s binary gcd, as in Shallit and Sorenson [12]). However, in the next section we use the ideas of our QuadraticBinaryJacobi algorithm to get an O(M (n) log n) algorithm. We do not see how to modify the algorithm of Shallit and Sorenson to do this.3

3

An O(M (n) log n) Algorithm

Algorithm HalfBinaryJacobi below is a modiﬁcation of Algorithm Half-GB-gcd from [13]. (Algorithm Half-GB-gcd is a subquadratic right-to-left gcd algorithm; for more on the general structure of subquadratic gcd algorithms, we refer the reader to M¨ oller [8].) The main diﬀerences between Half-GB-gcd and our algorithm are the following: 1. binary division with positive (not centered) quotient is used; 2. the algorithm returns an integer s such that if a, b are the inputs, c, d the output values deﬁned by Theorem 3, then (b|a) = (−1)s (d|c); 3. at steps 4 and 27, we reduce mod 22k1 +2 (resp. 22k2 +2 ) instead of mod 22k1 +1 (resp. 22k2 +1 ), so that we have enough information to correctly update s0 at steps 10, 17, 21 and 25; 4. we have to “cut” some harmless iterations in two (step 15). Remarks. The matrix Q occurring at step 19 is just 22m U m , where U m is given by Equation (3). Similarly, the matrix Q occurring at step 23 is 22j0 Gj0 ,q . In practice, steps 13–20 can be omitted (so the algorithm becomes a fast version of CubicBinaryJacobi) – this variant is simpler and slightly faster on average. We now state our main theorem. Its proof is based on comparing the GB sequence of a, b and that of a1 , b1 , where a1 = a mod 22k1 +2 and b1 = b mod 22k1 +2 . The GB — which stands for Generalized Binary division, see [13] — sequence of a, b is the sequence of remainders we obtain by applying the binary division iteratively. Two GB sequences match if they produce the same binary quotients qi . 3

In Algorithm Binary Jacobi of [12], it is necessary to know the sign of a − n (b − a in our notation) to decide whether to perform an interchange. This makes it diﬃcult to construct an recursive O(M (n) log n) algorithm along the lines of Algorithm HalfBinaryJacobi.

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Algorithm 3.1. HalfBinaryJacobi Input: a ∈ N, b ∈ N ∪ {0} with 0 = ν(a) < ν(b), and k ∈ N Output: two integers s, j and a 2 × 2 matrix R 1: if ν(b) > k then 10 2: Return 0, 0, 01 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28:

b = 0 is possible

k1 ← k/2 a1 ← a mod 22k1 +2 , b1 ← b mod 22k1 +2 s1 , j1 , R ← HalfBinaryJacobi(a1 , b1 , k1 ) a ← 2−2j1 (R1,1 a + R1,2 b), b ← 2−2j1 (R2,1 a + R2,2 b) j0 ← ν(b ) if j0 + j1 > k then Return s1 , j1 , R 2 s0 ← j0 (a − 1)/8 mod 2 q, r ← BinaryDividePos(a , b ) b ← b /2j0 if (j0 , q) = (1, 3) then d ← a − b m ← min(ν(d) div 2, k − j1 ) c ← (d − (−1)m d/4m )/5 s0 ← s0 + m(a − 1)/2 mod 2 harmless iteration (a2 , b2) ← (a − 4c, 2(b + c)) (4m + 4(−1)m )/5 2(4m − (−1)m )/5 Q← 2(4m − (−1)m )/5 (4m+1 + (−1)m )/5 else s0 ← s0 + (a − 1)(b − 1)/4 mod 2 r/2j0 ) good or bad iteration (a2 , b2) ← (b , j0 0 2 Q← 2j0 q m ← j0 s0 ← s0 + j0 (a22 − 1)/8 mod 2 k2 ← k − (m + j1 ) s2 , j2 , S ← HalfBinaryJacobi(a2 mod 22k2 +2 , b2 mod 22k2 +2 , k2 ) Return (s0 + s1 + s2 ) mod 2, j1 + j2 + m, S × Q × R

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Theorem 3. Let a, b, k be the inputs of Algorithm HalfBinaryJacobi, and s, j, R the corresponding outputs. If dc = 2−2j R ab , then: (b|a) = (−1)s (d|c)

and

ν(2j c) ≤ k < ν(2j d).

Proof (outline). We prove the theorem by induction on the parameter k. The key ingredient is that if we reduce a, b mod 22k1 +1 in step 4, then the GB sequence of a1 , b1 matches that of a, b, for the terms computed by the recursive call at step 5. This is a consequence of [13, Lemma 7] (which also holds for binary division with positive quotient). It follows that in all the binary divisions with inputs ai , bi in that recursive call, ai and bi /2ji match modulo 2ji +1 the corresponding values that would be obtained from the full inputs a, b (otherwise the corresponding binary quotient qi would be wrong). Since here we reduce a, b mod 22k1 +2 instead of mod 22k1 +1 , ai and bi /2ji now match modulo 2ji +2 — instead of modulo 2ji +1 — the values that would be obtained from the full inputs a, b, where 2ji +2 ≥ 8 since ji ≥ 1. At step 10, s0 depends only on j0 mod 2 and a mod 8, at step 17 it depends on m mod 2 and a mod 4, and at step 21 on a mod 4 and b mod 4. Since a and b at step 21 correspond to some ai and bi /2ji , it follows that a and b agree mod 8 with the values that would be computed from the full inputs, and thus the correction s0 is correct. This proves by induction that (b|a) = (−1)s (d|c). Now we prove that ν(2j c) ≤ k < ν(2j d). If there is no harmless iteration, it is a consequence of the proof of Theorem 1 in [13]. In case there is a harmless iteration, ﬁrst assume that m = ν(d) div 2 at step 15. The new values a2 , b2 at step 18 correspond to m successive ugly iterations, which yield j = j1 + m ≤ k. Thus ν(2j a2 ) ≤ k: we did not go too far, and since we are computing the same sequence of quotients as Algorithm QuadraticBinaryJacobi, the result follows. Now if k − j1 < ν(d) div 2, we would go too far if we performed ν(d) div 2 ugly iterations, since it would give j0 := ν(d) div 2 > k −j1 , thus j := j1 +j0 > k, and ν(2j a2 ) would exceed k. This is the reason why we “cut” the harmless iteration at m = k − j1 (step 15). The other invariants are unchanged. Finally we can present our O(M (n) log n) Algorithm FastBinaryJacobi, which computes the Jacobi symbol by calling Algorithm HalfBinaryJacobi. The general structure is similar to that described in [8] for several asymptotically fast GCD algorithms. Daireaux, Maume-Deschamps and Vall´ee [5] prove that, for the positive binary division, the average increase of the most signiﬁcant bits is 0.65 bits/iteration (which partly cancels an average decrease of two least signiﬁcant bits per iteration); compare this with only 0.05 bits/iteration on average for the centered division.4

4

We have computed more accurate values of these constants: 0.651993 and 0.048857 respectively.

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Algorithm 3.2. FastBinaryJacobi Input: a, b ∈ N with 0 = ν(a) < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: k ← max(ν(b), (b) div 3) 4: s , j, R ← HalfBinaryJacobi(a, b, k) 5: s ← (s + s ) mod 2 6: (a, b) ← 2−2j (R1,1 a + R1,2 b, R2,1 a + R2,2 b), 7: if a = 1 then return (−1)s else return 0

4

(b) is length of b in bits

j ← ν(b)

Experimental Results

We have implemented the diﬀerent algorithms in C (using 64-bit integers) and in GMP (using multiple-precision integers), as well as in Maple/Magma (for testing purposes). For max(a, b) < 226 the maximum number of iterations of Algorithm CubicBinaryJacobi is 64, with a = 15548029 and b = 66067306. The number of iterations seems to be O(n) for a, b < 2n : see Table 1. This is plausible because, from heuristic probabilistic arguments, we expect about half of the iterations to be good, and experiments conﬁrm this. For example, if we consider all admissible a, b < 220 , the cumulated number of iterations is 3.585×1012 for 238 calls, i.e., an average of 13.04 iterations per call (max 48); the cumulated number of good, bad and ugly iterations is 51.78%, 25.47%, and 22.75% respectively. For a, b < 260 , a random sample of 108 pairs (a, b) gave 42.72 iterations per call (max 89), with 50.54%, 25.14%, and 24.31% for good, bad and ugly respectively. These ratios seem to be converging to the heuristically expected 1/2 = 50%, 1/4 = 25%, and 1/4 = 25%. When we consider all admissible a, b < 220 , the maximum number of iterations of QuadraticBinaryJacobi is 37 when a = 933531, b = 869894, the cumulated number of iterations is 3.405 × 1012 (12.39 per call), the cumulated number of good, bad and harmless iterations is 54.51%, 26.82%, and 18.67% respectively. For a, b < 260 , a random sample of 108 pairs (a, b) gave 40.21 iterations per call (max 76), with 53.70%, 26.71%, and 19.59% for good, bad and harmless respectively. These ratios seem to be converging to the heuristically expected 8/15 = 53.33%, 4/15 = 26.67%, and 1/5 = 20%. We have also compared the time and average number of iterations for huge numbers, using the fast gcd algorithm in GMP, say gcd — which implements the algorithm from [8] — and an implementation of the algorithm from [13], say bgcd. For inputs of one million 64-bit words, gcd takes about 45.8s on a 2.83Ghz Core 2, while bgcd takes about 48.3s and 32,800,000 iterations: this is in accordance with the fact proven in [5] that each step of the binary gcd discards on average two least signiﬁcant bits, and adds on average about 0.05 most signiﬁcant bits. Our algorithm bjacobi (based on Algorithms 3.1–3.2) takes about 83.1s

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1e+06 mpz_jacobi FastBinaryJacobi 100000 10000 1000 100 10 1 0.1 0.01 0.001 0.0001 1

10

100

1000

10000

100000

Fig. 1. Comparison of GMP 4.3.1 mpz jacobi routine with our FastBinaryJacobi implementation in log-log scale. The x-axis is in 64-bit words, the y-axis in milliseconds on a 2.83Ghz Core 2. Table 1. Worst cases for CubicBinaryJacobi(b|a), max(a, b) < 2n n iterations example (a, b) 5 6 (7, 30) 10 19 (549, 802) 15 34 (23449, 19250) 20 48 (656227, 352966) 21 51 (1596811, 1493782)

n iterations example (a, b) 22 53 (2214985, 2781506) 23 55 (1383497, 8292658) 24 58 (2236963, 12862534) 25 62 (28662247, 30847950) 26 64 (15548029, 66067306)

and 47,500,000 iterations (for a version with steps 13–20 of Algorithm 3.1 omitted in the basecase routine), which agrees with the theoretical drift of 0.651993 bits per iteration. The break-even point between the O(n2 ) implementation of the Jacobi symbol in GMP 4.3.1 and our O(M (n) log n) implementation is about 535 words, that is about 34, 240 bits or about 10, 300 decimal digits (see Fig. 1).

5

Concluding Remarks

Weilert [15] says: “We are not able to use a GCD calculation in Z[i] similar to the binary GCD algorithm · · · because we do not get a corresponding quotient

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sequence in an obvious manner ”. In a sense we ﬁlled that gap for the computation of the Jacobi symbol, because we showed how it can be computed using a binary GCD algorithm without the need for a quotient sequence. We showed how to compute the Jacobi symbol with an asymptotically fast time bound, using such a binary GCD algorithm. Our implementation is faster than a good O(n2 ) implementation for numbers with bitsize n > 35000. Our subquadratic implementation is available from http://www.loria.fr/~zimmerma/ software/#jacobi. Binary division with a centered quotient does not seem to give a subquadratic algorithm; however we can use it with the “cubic” algorithm (which then becomes provably quadratic) since then we control the sign of a, b. For a better quadratic algorithm, we can choose the quotient q so that abq < 0, by replacing q by q − 2j+1 if necessary: experimentally, this gains on average 2.194231 bits per iteration, compared to 1.951143 for the centered quotient, and 1.348008 for the positive quotient. In comparison, Stein’s “binary” algorithm gains on average 1.416488 bits per iteration [4, §7][7, §4.5.2]. Acknowledgement. We thank Steven Galbraith who asked us about the existence of an O(M (n) log n) algorithm for the Jacobi symbol, Arnold Sch¨ onhage for his comments and a pointer to the work of his former student Andr´e Weilert, Damien Stehl´e who suggested adapting the binary gcd algorithm, and Marco Bodrato and Niels M¨oller for testing our implementation. We also thank the two anonymous reviewers, especially the one who actually implemented our new algorithm in Magma! We thank INRIA for its support of the ANC “´equipe associ´ee”. The ﬁrst author acknowledges the support of the Australian Research Council.

References 1. Bach, E.: A note on square roots in ﬁnite ﬁelds. IEEE Trans. on Information Theory 36(6), 1494–1498 (1990) 2. Bach, E., Shallit, J.O.: Algorithmic Number Theory: Eﬃcient Algorithms, vol. 1. MIT Press, Cambridge (1996) (Solution to problem 5.52) 3. Bachmann, P.: Niedere Zahlentheorie, Teubner, Leipzig, vol. 1 (1902); Reprinted by Chelsea, New York (1968) 4. Brent, R.P.: Twenty years’ analysis of the binary Euclidean algorithm. In: Davies, J., Roscoe, A.W., Woodcock, J. (eds.) Millennial Perspectives in Computer Science: Proceedings of the 1999 Oxford - Microsoft Symposium in honour of Professor Sir Antony Hoare, Palgrave, New York, pp. 41–53 (2000), http://wwwmaths.anu.edu.au/~brent/pub/pub183.html 5. Daireaux, B., Maume-Deschamps, V., Vall´ee, B.: The Lyapunov tortoise and the dyadic hare. In: Proceedings of the 2005 International Conference on Analysis of Algorithms, DMTCS Proc. AD, pp. 71–94 (2005), http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/81 6. Gauss, C.F.: Theorematis fundamentalis in doctrina de residuis quadraticis, demonstrationes et ampliatones novæ. Comm. Soc. Reg. Sci. Gottingensis Rec. 4 (presented February 10, 1817) (1818); Reprinted in Carl Friedrich Gauss Werke, Bd. 2: H¨ ohere Arithmetik, G¨ ottingen, pp. 47–64 (1876)

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7. Knuth, D.E.: The Art of Computer Programming. In: Seminumerical Algorithms, 3rd edn., vol. 2, Addison-Wesley, Reading (1997) 8. M¨ oller, N.: On Sch¨ onhage’s algorithm and subquadratic integer GCD computation. Mathematics of Computation 77(261), 589–607 (2008) 9. Sch¨ onhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971) 10. Sch¨ onhage, A.: Personal communication by email (December 2009) 11. Sch¨ onhage, A., Grotefeld, A.F.W., Vetter, E.: Fast Algorithms: A Multitape Turing Machine Implementation. BI-Wissenschaftsverlag, Mannheim (1994) 12. Shallit, J., Sorenson, J.: A binary algorithm for the Jacobi symbol. ACM SIGSAM Bulletin 27(1), 4–11 (1993), http://euclid.butler.edu/~sorenson/papers/binjac.ps 13. Stehl´e, D., Zimmermann, P.: A binary recursive gcd algorithm. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 411–425. Springer, Heidelberg (2004) 14. Vall´ee, B.: A unifying framework for the analysis of a class of Euclidean algorithms. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 343–354. Springer, Heidelberg (2000) 15. Weilert, A.: Fast Computation of the Biquadratic Residue Symbol. Journal of Number Theory 96, 133–151 (2002)

New Families of ECM Curves for Cunningham Numbers ´ Eric Brier1 and Christophe Clavier2,3

2

1 Ingenico S.A. 1, rue Claude Chappe, B.P. 346, 07530 Guilherand-Granges, France [email protected] Institut d’Ing´enierie Informatique de Limoges (3iL) 43, rue Sainte Anne F-87000 Limoges [email protected] 3 Universit´e de Limoges – XLIM D´epartement de Math´ematiques et Informatique 83, rue d’Isle F-87000 Limoges [email protected]

Abstract. In this paper we study structures related to torsion of elliptic curves deﬁned over number ﬁelds. The aim is to build families of elliptic curves more eﬃcient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational Z/4Z × Z/4Z torsion and positive rank over the ﬁeld Q(ζ8 ) and a family of elliptic curves with rational Z/6Z × Z/3Z torsion and positive rank over the ﬁeld Q(ζ3 ). These families have been used in ﬁnding new prime factors for the numbers 2972 + 1 and 21048 + 1. Along the way, we classify and give a parameterization of modular curves for some torsion subgroups.

1

Introduction

The Elliptic Curve Method (ECM in short) is a factoring algorithm, whose complexity depends on the size of the smallest prime factor instead of the size of the number to be factored. It can be seen as a variation of the p − 1 method. The idea is to build an elliptic curve over the ring Z/N Z with a point P on it and to compute the scalar multiplication M · P . Since N is not a prime, the elliptic curve is not deﬁned over a ﬁeld. However, computations are done as if we were working on a ﬁeld and if something fails, this means that a non-trivial factor of N has been found. The number M is chosen to be the product of powers of small primes and thus, a prime factor p is found as soon as the order of the elliptic curve reduced modulo p is smooth. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 96–109, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Many improvements of the ECM are described in the literature. We will focus on an improvement consisting in choosing the elliptic curve as the reduction modulo N of an elliptic curve deﬁned over the ﬁeld Q with a non-trivial torsion group and positive rank. The torsion group of an elliptic curve is the group of elements of ﬁnite order and the rank is the number of generators of the torsionfree part of the group. As soon as small prime factors have been removed from N , the torsion group is preserved in most cases by the modulo N reduction of the curve, which helps to make the order of the curve smooth. The positive rank is needed to set the starting point P of the algorithm. Possible torsion groups for elliptic curves deﬁned over Q are in ﬁnite number, with maximal order 16. For each possible torsion group, at least a family of elliptic curves with positive rank has been found. The idea we follow in this paper is to use a number ﬁeld K for which reduction modulo N can be made explicit and to build over K an elliptic curve with positive rank and a torsion subgroup as large as possible. Let us give an example : if the number to be factored is of the form N = u2 + 1, we can make use of the ﬁeld K = Q(i) with mapping i → u. The numbers of the Cunningham Project (i.e. numbers of the form am ±1) allow to use m-th roots of unity. It will be interesting to focus on cyclotomic ﬁelds or on their subﬁelds. It is important to note that all quadratic extensions of Q lie in cyclotomic ﬁelds. The paper is organized as follows. Section 2 introduces the necessary notions about modular curves and classiﬁes torsion subgroups that can be of any interest for ECM integer factoring. Section 3 is devoted to construction of parameterized elliptic curves with given torsion subgroup over some cyclotomic extensions of the ﬁeld of rationals. Section 4 focuses on the search for inﬁnite subfamilies of elliptic curves having nonzero rank, which is mandatory to ECM usage. Section 5 rephrases previous sections results in the context of ECM and gives some instances of new prime factors of Cunningham Project numbers discovered thanks to the work presented here. Finally, section 6 concludes and suggests some research areas to go further.

2

Elliptic Curve Torsion and Modular Curves

An elliptic curve E deﬁned over a number ﬁeld K turns out to be a commutative group. The Mordell-Weil theorem states that this group is ﬁnitely generated and can be written as: E(K) ∼ = T ⊗ Zr where the integer r is called rank and T is the so called torsion group, which consists in elements of ﬁnite order. Furthermore, T is isomorphic to Z/m1 Z × Z/m2 Z with the constraints that m2 divides m1 and the m2 -th roots of unity all lie in the ﬁeld K. Whereas it is conjectured that the rank is not constrained, the torsion group can take only ﬁnitely many diﬀerent shapes over the ﬁeld of rationals:

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Theorem 1 (Mazur). The torsion group T of an elliptic curve defined over the field Q is isomorphic to one of the following groups: Z/mZ with 1 ≤ m ≤ 10 or m = 12 Z/2mZ × Z/2Z with 1 ≤ m ≤ 4 This theorem is eﬀective in the sense that for each of these cases, it is possible to give equations of elliptic curves. These parameterizations come from modular curves. Over the ﬁeld C of complex numbers, there is a one-to-one correspondance between isomorphism classes of elliptic curves and the Riemann Surface X(1), which is the quotient H∗ /SL2 (Z), where H∗ is the compactiﬁed Poincar´e halfplane. For any subgroup Γ of SL2 (Z), the quotient surface H∗ /Γ is called a modular curve. Extending notations of [3], we deﬁne the following subgroups of SL2 (Z): a b ∈ SL2 (Z), a ≡ d ≡ 1 mod m, b ≡ c ≡ 0 mod m Γ (m) = c d Γ1 (m) = Γ0 (m) =

a b c d a b c d

∈ SL2 (Z), a ≡ d ≡ 1 mod m, c ≡ 0 mod m

∈ SL2 (Z), c ≡ 0 mod m

and the quotients: X(m) = H∗ /Γ (m) X1 (m) = H∗ /Γ1 (m) X0 (m) = H∗ /Γ0 (m) X1 (m1 , m2 ) = H∗ /(Γ1 (m1 ) ∩ Γ (m2 )) when m2 |m1 A point on the surface X(m) corresponds to an elliptic curve together with a basis for its [m]-torsion subgroup, up to isomorphism. A point on the surface X1 (m) corresponds, up to isomorphism, to an elliptic curve together with a [m]torsion point. A point on the surface X0 (m) corresponds, up to isomorphism, to an elliptic curve together with a cyclic torsion subgroup of order m. A point on the surface X(m1 , m2 ) corresponds to an elliptic curve with a [m1 ]-torsion point and an independent [m2 ]-torsion point. Though these notions make use of complex number and analytical tools, the modular curves can also be represented as algebraic curves. An algebraic model of X1 (m) can be found over Q and the correspondance with an elliptic curve and a [m]-torsion point on it is algebraic and deﬁned over Q. The curve X(m) involves the full [m]-torsion subgroup and, due to existence of Weil pairing, m-th roots of unity are involved. The rational models and correspondance for X(m) (resp. X1 (m1 , m2 )) are deﬁned over the cyclotomic ﬁeld Q(ζm ) (resp. Q(ζm2 )). The modular curves associated to torsion

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subgroups in Mazur’s theorem are genus 0 algebraic curves. This explains why it is possible to give parametric Weierstrass equations. The Elliptic Curve Method needs a non-bounded number of elliptic curves to compute with. Since algebraic curves of genus greater than 2 have only ﬁnitely many rational points over a given number ﬁeld, we will focus only on torsion structures for which the associated modular curve has genus 0 or 1. Computing the genus of an algebraic curve is not an easy task in the general case but the task is easy with a computer for modular curves of rather small level. A theorem from Shimura states that the genus of the modular curve X1 (p) for a prime p ≥ 5 is given by : g=

(p − 5)(p − 7) . 24

This implies that the only primes for which the genus of X1 (p) is 0 or 1 are {2, 3, 5, 7, 11}. When n|m, there is a surjective mapping X1 (m) → X1 (n), and thus the genus of X1 (m) is at least the genus of X1 (n). Computing the genus of X1 (m) for rather small values of m being easy, we can increase the power of these primes until the genus is strictly greater than 1 and we get that the only prime powers for which the genus of X1 (pe ) is 0 or 1 are {2, 4, 8, 3, 9, 5, 7, 11}. Now, combining this ﬁnite set, it is possible to check the following proposition with a ﬁnite amount of work: Proposition 1. The integers m such that X1 (m) is of genus 0 or 1 are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15} When m2 |m1 , there is a surjective mapping X1 (m1 , m2 ) → X1 (m1 ), and thus the genus of X1 (m1 , m2 ) is at least the genus of X1 (m1 ). This implies that if the genus of the modular curve X1 (m1 , m2 ) is 0 or 1, the number m1 is in the list given in proposition 1. Building on this, for any m1 in this list, we can check if the genus of X1 (m1 , m2 ) is 0 or 1 for all divisors m2 of m1 . The result is given in next proposition. Proposition 2. The torsion groups for which the associated modular curve is of genus 0 are: Z/2Z, Z/3Z, Z/4Z, Z/5Z, Z/6Z, Z/7Z Z/8Z, Z/9Z Z/10Z Z/12Z

Z/2Z × Z/2Z Z/3Z × Z/3Z Z/4Z × Z/2Z, Z/5Z × Z/5Z Z/6Z × Z/2Z, Z/8Z × Z/2Z

Z/4Z × Z/4Z Z/6Z × Z/3Z

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The torsion groups for which the associated modular curve is of genus 1 are: Z/6Z × Z/6Z Z/8Z × Z/4Z Z/9Z × Z/3Z Z/10Z × Z/2Z Z/11Z Z/12Z × Z/2Z Z/14Z Z/15Z

3

Parameterization of Elliptic Curves with Given Torsion Structure

When the base ﬁeld is Q, several papers (e.g. [8] and [4]) describe the construction of elliptic curves with prescribed torsion groups. We will study cases that need to work over extensions. 3.1

Construction of Z/3Z × Z/3Z

To study torsion points, one can use the division polynomials, whose roots are the abscises of torsion points. Since we wish all 3-torsion points to be rational, we start by imposing two rational roots x1 and x2 to the polynomial ϕ3 (x) = 3x4 + 6ax2 + 12bx − a2 The system ϕ3 (x1 ) = ϕ3 (x2 ) = 0 considered as equations in the variables a and b has roots if and only if −3x1 x2 is a square. A convenient parameterization is x1 = 6ξ x2 = −2ρ2 ξ and the corresponding parameters are a = −12ξ 2 ρ(ρ2 − 3ρ + 3) b = 2ξ 3 (ρ2 − 3)(ρ4 − 6ρ3 + 18ρ2 − 18ρ + 9) At this stage, we introduce two linear factors in ϕ3 . The remaining quadratic factor of ϕ3 has discriminant equal to −3(ρ − 1)2 (ρ − 3)2 . We need −3 to be a square, which is natural since the Weil pairing introduces cubic roots of unity. We thus have x-coordinates of point of order 3 rational. We now turn on to y-coordinates. Substitutions of x1 and x2 in x3 + ax + b yield y12 = 2ξ 3 (ρ − 3)2 (ρ2 + 3)2 and y12 = −6ξ 3 (ρ − 1)2 (ρ2 + 3)2 . To obtain squares, we set ξ = 2λ2 and for convenience ρ = 1 − τ . In conclusion, an elliptic curve in short Weierstrass form has rational 3-torsion over Q(ζ3 ) if and only if its parameters can be written as: a = 48λ4 (τ 3 − 1) b = 16λ6 (τ 6 − 20τ 3 − 8)

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101

Construction of Z/6Z × Z/3Z

Given as an input the results of previous section, we now have to ensure that x3 + ax + b has a linear factor to get a point of order 2. In a ﬁrst step, we set x = ξλ2 to get rid of the homogeneity parameter λ. We consider then x3 + ax + b as a equation in ξ and τ 3 , which is quadratic relatively to the unknown τ 3 . The discriminant of this quadratic equation is −(ξ − 12)3 . It is natural to set ξ = 12 − ν 2 . We now have: x3 + ax + b = λ6 (6ν 2 + ν 3 − 4τ 3 − 32)(6ν 2 − ν 3 − 4τ 3 − 32) Both factors diﬀer only in a sign change for ν. We will keep the ﬁrst factor, which is a cubic in ν and τ . Since the underlying modular curve has genus 0, this curve must have a singularity. We easily ﬁnd that the point (ν = −4, τ = 0) is singular and to reduce the degree of the curve, we set ν = μτ − 4. After replacement and factorization, we have a degree one equation in τ . To keep consistency in notations and to avoid denominators, we rename μ as 1/τ and modify the scaling factor λ. In conclusion, an elliptic curve in short Weierstrass form has rational 3-torsion and a point of order 2 over Q(ζ3 ) if and only if its parameters can be written as: a = −3λ4 (τ 12 − 8τ 9 + 240τ 6 − 464τ 3 + 16) b = −2λ6 (τ 18 − 12τ 15 − 480τ 12 + 3080τ 9 − 12072τ 6 + 4128τ 3 + 64) 3.3

Modular Curve for Z/6Z × Z/6Z

We know that the modular curve X(6) has genus 1. In this section, we will give a very simple model for this elliptic curve. Let us start with the equation for Z/6Z× Z/3Z torsion subgroup. The polynomial x3 + ax + b has by construction a linear and a quadratic factor. The discriminant of the quadratic factor is −9(8τ 3 − 1)3 . From this we derive the following model: X(6) : s2 = t3 + 1 3.4

Modular Curve for Z/9Z × Z/3Z

We start from parameterization of curves with full 3-torsion. One can note that the parameter is involved only to the third power, we thus note σ = τ 3 and will work in a ﬁrst stage only with σ. We introduce the polynomial χ9 whose roots are the sums of x-coordinates of points in cyclic subgroups of order 9 and whose degree is 12: χ9 (z) = z 12 + 792az 10 + 47520bz 9 + ... − 3543478272a6 We can de-homogenize this polynomial by setting λ = 1 and, since a and b are polynomials in σ, we get a polynomial equation in z and σ having a quadratic factor in σ. This factor has a root iﬀ z − 48 is six times a square. We set

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z = 6ζ 2 + 48 and get σ = (ζ 3 + 6ζ 2 + 12ζ + 72)/8. We can now factor the division polynomial ϕ9 (x) and obtain an equation of degree 3 in x and 6 in ζ. The solution x = 12 and ζ = −2 being a singularity, we set x = 12 + ξ(ζ + 2)2 and obtain the relation ζ = −2

ξ 3 + 3ξ 2 − 6ξ + 1 ξ 3 − 3ξ 2 + 1

We are guaranteed that a point of order 9 has rational x-coordinate, its happens that the y-coordinate is also rational. It is now time to remember that σ must be a cube. Elliptic curve with torsion group of type Z/9Z × Z/3Z have same parameters as for Z/3Z × Z/3Z, provided that τ3 =

8(ξ 2 − ξ + 1)3 (ξ 3 − 6ξ 2 + 3ξ + 1) (ξ 3 − 3ξ 2 + 1)3

Some algebraic manipulations turn the equation σ 3 = ξ 3 − 6ξ 2 + 3ξ + 1 into the elliptic model: X1 (9, 3) : s2 = t3 + 16 3.5

Construction of Z/4Z × Z/4Z

In short Weierstrass form, points of order 2 are points whose y-coordinate is 0. It follows that the general form of curve with Z/2Z torsion is: y 2 = (x − u)(x2 + ux + v) For the same reasons the general form of curve with Z/2Z × Z/2Z torsion is: y 2 = (x − u)(x − v)(x + u + v) On this elliptic curve, a point P = (x, y) can be written P = 2Q iﬀ the numbers x − u, x − v and x + u + v are squares, see [2, Theorem 4.2 page 85]. Thus, if we require that all 4-torsion are rational, all 2-torsion points must be doubles and we ask for 0, ±(u − v), ±(2u + v) and ±(u + 2v) being squares. One can note that −1 has to be a square, which is not a surprise: if 4-torsion is rational, the Weil pairing will produce fourth roots of unity, i.e. square roots of −1. We ﬁrst impose 2u + v and 2v + u to be squares. To do so, we invert the system: 2u + v = r2 u = (2r2 − s2 )/3 ⇐⇒ 2 v = (2s2 − r2 )/3 u + 2v = s Then, it remains to ensure that u − v is also a square. The factorization of u − v is (r − s)(r + s). It is convenient to write r = μ + ν and s = μ − ν. We get u − v = 4μν, which must be a square. We can set μ = τ 2 ν. Last, to get rid of denominators, we set ν = 3λ. In conclusion, an elliptic curve in short Weierstrass form has rational 4-torsion over Q(ζ4 ) if and only if its parameters can be written as: a = −27λ4 (τ 8 + 14τ 4 + 1) b = 54λ6 (τ 12 − 33τ 8 − 33τ 4 + 1)

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3.6

103

Modular Curve for Z/8Z × Z/4Z

To obtain a point of order 8, one of the points of order 4 must be expressed as the doubling of a rational point. We take for instance one of the points with x = 3τ 4 − 15. Diﬀerences with x-coordinates of 2-torsion points must be squares, these diﬀerences factor as: −18 (τ 2 + 1) 18 (τ 2 − 1) 9 (τ 4 − 1) We can easily impose the second expression to be a square by setting τ = (κ2 + 2)/(κ2 − 2) Then, the two other expressions are squares iﬀ κ4 + 4 is a square. In the equation σ 2 = κ4 + 4, we apply the change of variables σ = s2 /t2 − 2t and κ = −s/t and get the model: X1 (8, 4) : s2 = t3 − t 3.7

Construction of Z/5Z × Z/5Z

To reach full rational 5-torsion, we begin with two rational cyclic subgroups of order 5. Let χ5 denote the polynomial, whose roots are the sums of x-coordinates of points over the 6 cyclic subgroups of order 5: χ5 (z) = z 6 + 20az 4 + 160bz 3 − 80a2 z 2 − 128abz − 80b2 We note z1 and z2 two roots of χ5 and to take beneﬁt of symmetry use the transformation z1 = u + v and z2 = u − v. We consider the system χ5 (z1 ) = χ5 (z2 ) = 0 as equations in a and b and eliminate the unknown a, obtaining a quartic in b with parameters u and v. It is then convenient to set b = (u2 − v 2 )β to reduce degrees in u and v. This quartic presents a strong singularity when v = 0 and β = u/4, which leads us to set β = (u/4 + γv/8). The result is still a quartic in γ but the degree in v fell down to 2 and the discriminant of this quadratic equation in v is a square iﬀ 9 − 5γ 2 is ﬁve times a square. We use conic parameterization techniques to obtain: γ=

6(μ2 + μ − 1) 5(μ2 + 1)

Now v can be expressed as the product of u and a rational function of μ. We unroll substitutions to get the value of b and come back to equations χ5 (z1 ) = χ5 (z2 ) = 0. They have a common linear factor in a and we now have values for a and b. Knowing that χ5 has two rational roots, we can strengthen our wishes and factor the division polynomial ϕ5 . No surprise that we get two quadratic factors, whose discriminants are squares if and only if μ2 + 1 and 5(μ2 + 1) are squares.

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We remember that we are working over the ﬁeld of ﬁfth roots of unity, in which 5 is a square. We just have to set μ=

2τ τ2 − 1

Now that x-coordinates for 5-torsion points are rational, we choose the value of homogeneity parameter u to have y-coordinates rationals u = −6λ2 (τ 2 + 1)(τ 4 − 2τ 3 − 6τ 2 + 2τ + 1)(2τ 4 + τ 3 + 3τ 2 − τ + 2) In conclusion, an elliptic curve in short Weierstrass form has full rational 5-torsion over Q(ζ5 ) if and only if its parameters can be written as: a = −27λ4 (τ 20 + 228τ 15 + 494τ 10 − 228τ 5 + 1) b = 54λ6 (τ 30 − 522τ 25 − 10005τ 20 − 10005τ 10 + 522τ 5 + 1)

4 4.1

Construction of Elliptic Curve with Large Prescribed Torsion and Positive Rank Description of the Method

For an elliptic curve being useful for the Elliptic Curve Method, its rank has to be non-zero. This means that we still have to produce sub-families of curves with an extra rational point. When the modular curve is of genus 1, we did not ﬁnd any method because we are lacking of freedom on the parameters. This section is devoted to the method we use to produce sub-families with positive rank in the case of a parameterization by P1 (K). In this case, the parameters a and b are, up to the scaling factor λ, polynomials in K(τ ) and we can take x to be also a polynomial x = λ2 ξ(τ ). Then x3 + ax + b becomes itself λ6 times a polynomial. The polynomial ξ being ﬁxed, we can look for values of τ , which turns x3 + ax + b into a square. This approach is equivalent to looking for rational points on hyperelliptic curves of rather high genus and will yield only ﬁnitely many curves. Our method consists in choosing the polynomial ξ in such a way that x3 + ax + b contains as much as possible of square factors. We note a = λ4 α(τ ), b = λ6 β(τ ) and σ(τ ) = ξ(τ )3 + α(τ )ξ(τ ) + β(τ ). For readability, we will omit the parameter τ for polynomial and all derivatives will be taken relatively to τ . We wish to have square factors, i.e. relations of type σ ≡ 0 mod (τ − τ0 )2 . In most cases, this relation imposes to deﬁne ξ modulo (τ − τ0 )2 . Since increasing the degree of ξ will in the end increase the degree of σ, we try to obtain this relation with a constraint only on ξ modulo (τ − τ0 ). Let us compute derivatives: σ = (3ξ 2 + α)ξ + (ξα + β ) To avoid constraints modulo (τ − τ0 )2 , we must keep freedom on ξ , which leads to 3ξ 2 + α = 0. Combining this relation with ξ 3 + αξ + β, we get the criterion

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Δ = 4α3 + 27β 2 = 0 and the value for ξ = −3β/2α. Now, we have to check that ξα + β = 0. Under the previous conditions, this is equivalent to Δ = 0. The values τ0 that will be of interest will thus be multiple roots of the discriminant Δ. To have a maximum number of degrees of freedom, for each of these roots we try to impose conditions on ξ modulo (τ − τ0 )e and check whether we get σ ≡ 0 mod (τ − τ0 )2e . The last step is to combine multiple roots using the Chinese Remainder Theorem in K[τ ]. For each possible τ0 , we ﬁx ξ modulo some power (τ − τ0 )e , the exponent e being less than the maximum ”useful” exponent. We obtain candidates for ξ and for each candidate we factor σ. Since we wish to have σ being a square, we write σ = σ12 σ0 with σ0 square-free. If ξ does not correspond to torsion points and if the degree of σ0 is less than 5, we can parameterize by a curve of genus 0 or 1. If the auxiliary curve is of genus one (i.e. an elliptic curve) and if we can exhibit a point, we build an inﬁnite family of elliptic curves with given torsion and rank at least one. 4.2

Results for Z/4Z × Z/4Z

Taking the values for a and b given in section 3.5, we ﬁrst factor the discriminant Δ = −24 312 τ 4 (τ − 1)4 (τ + 1)4 (τ 2 + 1)4 The values of interest for τ are {0, 1, −1, ι, −ι}. We then check that each of them can be used up to the second power. The number of candidates we can generate for ξ is 35 . To simplify exploration of all these candidates, we compute once for all a polynomial Ξ that satisﬁes all modular conditions Ξ = 9τ 8 − 24τ 4 + 3 take its remainder modulo the polynomial τ e0 (τ − 1)e1 (τ + 1)e−1 (τ − ι)eι (τ − ι)e−ι We get values σ0 of degree 0 that are of no interest since they correspond to torsion points. We get no values of degree 1, 16 diﬀerent values of degree 2, 32 of degree 3 and 62 of degree 4. The simplest value of σ0 is 36 (τ 2 − 3), which corresponds to ξ = 9τ 6 − 15τ 4 − 9τ 2 + 3. To turn σ0 into a square, one can set τ=

ν2 + 3 and λ = 8ν 3 2ν

Unrolling substitutions, we have ⎧ a = −432ν 4 (ν 16 + 24ν 14 + 476ν 12 + 4200ν 10 + 18022ν 8 ⎪ ⎪ ⎪ ⎪ +37800ν 6 + 38556ν 4 + 17496ν 2 + 6561) ⎪ ⎪ ⎨ b = 3456ν 6 (ν 24 + 36ν 22 + 66ν 20 − 6732ν 18 − 101409ν 16 − 707256ν 14 ⎪ ⎪ ⎪ ⎪ −2772260ν 12 − 6365304ν 10 − 8214129ν 8 − 4907628ν 6 ⎪ ⎪ ⎩ +433026ν 4 + 2125764ν 2 + 531441)

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106

The point of inﬁnite order is given by ⎧ ⎨ x = 3 (3ν 12 + 34ν 10 + 117ν 8 + 316ν 6 + 1053ν 4 + 2754ν 2 + 2187) ⎩

y = 27 (ν 2 − 3)(ν 2 + 1)(ν 2 + 9)(ν 6 + 5ν 4 + 15ν 2 + 27)2

The choice of parameters giving such a torsion group when −1 is a square has also been studied to speed-up factorisation in [9]. 4.3

Results for Z/6Z × Z/3Z

Following the same steps, we start from formulae given in section 3.2 and factor the discriminant: Δ = −28 36 τ 3 (τ 3 + 1)6 (τ 3 − 8)3 The values of interest for τ0 are {−1, −ζ3, −ζ32 , 0, 2, 2ζ3 , 2ζ32 }. Only the ﬁrst 3 values can be used up to the second power, the four last ones being of interest only to the ﬁrst power. The solution for all modular constraints is Ξ = 2τ 9 − 9τ 6 − 42τ 3 − 4 The 432 possible candidates for ξ yield 32 cases where σ0 is of degree 4. Among them, one of the simplest corresponds to σ0 = −3τ (5τ 3 + 32) with ξ = −13τ 6 − 44τ 3 − 4. The elliptic curve ρ2 = −3τ (5τ 3 + 32) has nonzero rank over Q, a point of inﬁnite order being (−1, 9). The points of this auxiliary elliptic curve parameterize an inﬁnite family of elliptic curve having nonzero rank over Q and a torsion group containing Z/6Z × Z/3Z over Q(ζ3 ) 4.4

Results for Z/5Z × Z/5Z

Once again, we start by factoring: Δ = −28 312 τ 5 (τ 10 − 11τ 5 − 1)5 The eleven values of interest for τ0 can all be used up to the second power and the polynomial compatible with all constraints is Ξ=−

1 (252τ 20 − 5508τ 15 + 29019τ 10 + 7686τ 5 + 75) 25

Unfortunately, the 311 possible candidates for ξ all give σ0 polynomials of degree ﬁve or more, except for those corresponding to 5-torsion points. We also noticed that α is of degree 20 and β of degree 30. If we restrict ourselves to polynomials of degree 10 for ξ, the degree of σ will not exceed 30. In the case the leading coeﬃcient of ξ is −3, the degree of σ falls down to 28. One can see this as using the value τ0 = ∞. This is compatible with 10 modular constraints and we also tried the 24068 candidates built this way, with no success. Remark: To speed up computations and avoid to compute in a quartic extension of Q, we instead performed this computations in the ﬁeld F32621 , which contains ﬁfth roots of unity. For sure, if a solution had been found, we would have needed to perform actual computations in Q(ζ5 ).

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4.5

107

Half Way to Z/8Z × Z/4Z

As the modular curve X1 (8, 4) is of genus one, we lack freedom on curve parameters to ensure in addition a non zero rank. We will try to cover a part of the path from X1 (4, 4) to X1 (8, 4). We elaborate on the results of section 4.2 and will use the same parameterization, with a dedicated choice of values for the parameter ν. As in section 3.5, we use the characterization of points P that can be written P = [2]Q with Q a point with rational coordinates. Among the twelve points of order 4, there is one that needs the quantities ν(ν 2 + 3) and (ν 2 + 1)(ν 2 + 9) to be squares. Would both be squares, we would get a point of order 8. We limit ourselves to the ﬁrst condition only. At this stage, it is quite natural to consider the elliptic curve μ2 = ν(ν 2 + 3). The rank of this auxiliary curve over the ﬁeld Q is one and an inﬁnite subgroup is generated by the point P0 = (1, 2). Each of the points [k]P0 with k ∈ N yields a value of ν to be plugged into the formulæ of section 4.2. We thus get a inﬁnite family of elliptic curves with nonzero rank, Z/4Z×Z/4Z torsion over Q(ζ4 ), and better chances to get Z/8Z × Z/4Z torsion over the same number ﬁeld.

5

Application to Factoring

One can see an ECM implementation as a black box taking as inputs: A number N to be factored Elliptic curve paramaters a and b Coordinates of a point P on the curve modulo N and computing the scalar multiplication M · P on this curve for a smooth large integer M, expecting the result being at inﬁnity for some prime factor of N . In most implementations, projective coordinates are used and if M · P is at inﬁnity modulo a prime factor, this factor can be retrieved by a simple GCD between the number to be factored and the third coordinate. For full explanations on implementations and improvements of ECM, see [10], [1] and [7]. For the torsion groups Z/4Z×Z/4Z and Z/6Z×Z/3Z, we found curves having parameters and a point of inﬁnite order deﬁned over Q. These curves can be used for any number to be factored N . However, the beneﬁt of torsion is attained only when one knowns that suitable roots of unity exist in the ﬁnite ﬁelds deﬁned by prime factors of N . For order 16 torsion groups, numbers of the form a4n −b4n or a2n + b2n satisfy these conditions. The torsion group of order 18 can be used on numbers of the form a3n ± b3n . The suggested extension towards torsion group of order 32 can be used for numbers of the form a8n − b8n or a4n + b4n . To implement results of section 4.2, the parameter ν can be chosen at random or iteratively on integers. To implement results of section 4.3, things are slightly

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less simple, since an auxiliary elliptic curve has to be used. In this case one has to select an integer k randomly or in sequence, compute a scalar multiplication on the auxiliary elliptic curve to get the inputs of ECM. We adapted our ECM implementation in order to use these new families of elliptic curves. Making use of results on Z/4Z × Z/4Z torsion we found several factors of Cunningham numbers. Among them, one can mention the larger one: 5546025484206613872527377154544456740766039233|21048 + 1 We won’t give here full details of the factorization since they do not correspond to notations of these paper, this factor having been found in an early stage of development of this paper. We also implemented the variant with Z/6Z×Z/3Z torsion. Among the factors we found, the larger to mention is 1581214773543289355763694808184205062516817|2972 + 1 This factor has been discovered using the input parameters: ⎧ a = 29826081614523423723477944537088124780779 ⎪ ⎪ ⎨ b = 129980809632665349776106077981744185363149 ⎪ x = 479946793455925131408573042432160264988537 ⎪ ⎩ y = 341223966666174229961942234304018968605682

mod p mod p mod p mod p

The order of the curve modulo p factors as: #E(Fp ) = 2 × 32 × 29 × 241 × 691 × 5279 × 20353 × 252589 × 1489097 × 2258261 × 199312079

6

Conclusion

We exhibited two torsion groups, that can be used for ECM factoring, of orders 16 and 18. Classical implementations make use of the torsion group Z/8Z×Z/2Z that can be used for all numbers but of slightly smaller order. It would be really interesting to have a precise analysis of complexity improvements obtained by using torsion groups, as well as partial construction of torsion structure as in section 4.5. In the case of torsion group of order 25, we did not succeed in constructing elliptic curves having nonzero rank. This by no way means that no such curves exist. Solving this issue would result in speciﬁc implementations for numbers of the form a5n ± 1 with the larger available torsion group. Some torsion groups correspond to a modular curve of genus one. The obstruction in using them for ECM is the lack of freedom to build curve with nonzero rank: to build a curve with this torsion, one only have to select a multiple of a generator on this modular curve. Several approaches could improve the situation: being able to construct a large number of curves with nonzero rank

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by using rank computation software or being able to construct a point on the curve modulo N after the curve has been generated. Last, while inﬁnite families of curves are needed for ECM factoring of integers, individual curves providing large torsion groups over some number ﬁelds could be used during the sieving phase of the special number ﬁeld sieve (see [5] and [6]). Though further research is needed to hunt for interesting individual curves, we quote one preliminary result: the choice of ν = 1 in section 4.2 ensures a torsion subgroup of order 32 over the ﬁelds Q(ζ24 ) and Q(ζ40 ) and of order 64 over the ﬁeld Q(ζ120 ).

References 1. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1991) 2. Knapp, A.W.: Elliptic Curves. Princeton University Press, Princeton (1992) 3. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97. Springer, Heidelberg (1993) 4. Kubert, D.S.: Universal bounds on the torsion of elliptic curves. In: Proceedings of the London Mathematical Society, pp. 193–237 (1976) 5. Lenstra, A.K., Lenstra, H.W.: The Development of the Number Field Sieve. LNM, vol. 1554. Springer, Heidelberg (1993) 6. Lenstra, A.K., Lenstra, H.W., Manasse, M.S., Pollard, J.M.: The Factorization of the Ninth Fermat Number. In: Mathematics of Computation, vol. 61. American Mathematical Society, Providence (1993) 7. Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987) 8. Mazur, B.: Rational isogenies of prime degree. Invent. Math., 129–162 (1978) 9. Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987) 10. Zimmermann, P., Dodson, B.: Twenty Years of ECM. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 525–542. Springer, Heidelberg (2006)

Visualizing Elements of Sha[3] in Genus 2 Jacobians Nils Bruin and Sander R. Dahmen Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada [email protected], [email protected]

Abstract. Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H 1 (Gal(k/k), E) → H 1 (Gal(k/k), A). However, the abelian surface in Mazur’s construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the ShafarevichTate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

1

Introduction

Let E be an elliptic curve over a ﬁeld k with separable closure k. We write H 1 (k, E[3]) := H 1 (Gal(k/k), E[3](k)) for the ﬁrst galois cohomology group taking values in the 3-torsion of E (the notation H i (k, A) is used similarly for other group schemes A/k later in this paper). We are primarily concerned with the question which δ ∈ H 1 (k, E[3]) are visible in the jacobian of a genus 2 curve. Mazur deﬁnes visibility in the following way. Let 0 → E → A → B → 0 be a short exact sequence of abelian varieties over k. By taking galois cohomology, we obtain the exact sequence A(k)

/ B(k)

/ H 1 (k, E)

φ

/ H 1 (k, A) .

(1.1)

Elements of the kernel of φ are said to be visible in A. Mazur chose this term because a model of the principal homogeneous space corresponding to an element ξ ∈ H 1 (k, E) that is visible in A can be obtained as a ﬁber of A over a point in B(k) (this can readily be seen from (1.1)). By extension, we say that δ ∈ H 1 (k, E[n]) is visible in A if the image of δ under the natural homomorphism H 1 (k, E[n]) → H 1 (k, E) is visible in A. Let us restrict to the case that k is a number ﬁeld for the rest of this section. Inspired by some surprising experimental data [4], Mazur [5] proved, that for any element ξ in the Shafarevich-Tate group X(E/k) of order three, there exists an abelian variety A over k such that ξ is visible in A. The abelian variety that

Research of both authors supported by NSERC.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 110–125, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Mazur constructs is almost never principally polarizable over k and hence is almost never a jacobian of a genus 2 curve. In the present paper, we show that any element from X(E/k)[3] is in fact visible in the jacobian of a genus 2 curve. Moreover, we describe how to get an explicit model of such a genus 2 curve.

2

Torsors and Theta Groups

Throughout this section let n > 1 be an integer, let k be a perfect ﬁeld of characteristic not dividing n and let E denote an elliptic curve over k. In [2], many equivalent interpretations are given for the group H 1 (k, E[n]). For our purposes, we need two classes of objects. The ﬁrst is most closely related with descent in general and our question in particular. We consider E-torsors under E[n](k) and, following [2], call them n-coverings. Definition 1. An n-covering π : C → E of an elliptic curve E is an unramified covering over k that is galois and irreducible over k, with Autk (C/E) E[n](k). Two n-coverings π1 : C1 → E, π2 : C2 → E are called isomorphic if there exists a k-morphism φ : C1 → C2 such that π1 = π2 ◦ φ. Over k, all n-coverings are isomorphic to the trivial n-covering, the multiplicationby-n map [n] : E → E. Proposition 1 ([2, Proposition 1.14]). The k-isomorphism classes of n-coverings of E are classified by H 1 (k, E[n]). For δ ∈ H 1 (k, E[n]) we denote by Cδ the curve in the covering Cδ → E corresponding to δ. We remark that δ ∈ H 1 (k, E[n]) has trivial image in H 1 (k, E) if and only if Cδ has a k-rational point. We write O for the identity on E. The complete linear system |n·O| determines a morphism E → Pn−1 , where the translation action of E[n] extends to a linear action on Pn−1 . This gives a projective representation E[n] → PGLn . The lift of this representation to GLn gives rise to a group ΘE , which ﬁts in the following diagram. 1

/ Gm

1

/ Gm

αE

/ ΘE / GLn

βE

/ E[n]

/1

/ PGLn

/1

(2.1)

The group E[n](k) carries additional structure. It also has the Weil pairing eE , which is a non-degenerate alternating galois covariant pairing taking values in the n-th roots of unity eE : E[n](k) × E[n](k) → μn (k). The commutator of ΘE corresponds to the Weil pairing, meaning that for x, y ∈ ΘE we have xyx−1 y −1 = αE (eE (βE (x), βE (y))).

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Definition 2. A theta group for E[n] is a central extension of group schemes β

α

1 → Gm → Θ → E[n] → 1 such that the Weil-pairing on E[n] corresponds to the commutator, i.e. for x, y ∈ Θ we have xyx−1 y −1 = α(eE (β(x), β(y))). Two theta groups 1 → Gm → Θi → E[n] → 1,

i = 1, 2

are called isomorphic if there exists a group scheme isomorphism φ : Θ1 → Θ2 over k making the following diagram commutative. 1

/ Gm

/ Θ1

/ Gm

/ Θ2

/ E[n]

/1

/ E[n]

/1

φ

1

Over k, all theta-groups are isomorphic to ΘE as central extensions; see [2, Lemma 1.30]. Proposition 2. ([2, Proposition 1.31]). Let E[n] be the n-torsion subscheme of an elliptic curve E over a field k, equipped with its Weil pairing. The isomorphism classes of theta-groups for E[n] over k are classified by H 1 (k, E[n]). The theta group associated to δ ∈ H 1 (k, E[n]) may allow for a matrix representation Θ → GLn that ﬁts in a diagram like (2.1). This is measured by the obstruction map Ob introduced in [6] and [2]. This map can be obtained by taking non-abelian galois cohomology of the deﬁning sequence of ΘE : Ob

· · · −→ H 1 (k, ΘE ) −→ H 1 (k, E[n]) −→ H 2 (k, Gm ) = Br(k) −→ · · · . Note that, except in some trivial cases, Ob is not a group homomorphism. The map Ob also has an interpretation in terms of n-coverings. Let C → E be an n-covering associated to δ ∈ H 1 (k, E[n]). We have that Ob(δ) = 0 if and only if C admits a model C → Pn−1 with Autk (C/E) = E[n](k) acting linearly, in which case C is k-isomorphic to E as a curve and the covering C → E is simply a translation composed with multiplication-by-n. Remark 1. Note that if k is a number ﬁeld, then any element in Br(k) that restricts to the trivial element in Br(kv ) in all completions kv of k, is trivial itself. It follows that Ob is trivial on the n-Selmer group S (n) (E/k).

Visualizing Elements of Sha[3] in Genus 2 Jacobians

3

113

Visibility in Surfaces

Let E1 be an elliptic curve over a perfect ﬁeld k of characteristic distinct from 3. In what follows, we will consider δ ∈ H 1 (k, E1 [3]) with Ob(δ) = 0. A possible way of constructing an abelian surface A such that δ is visible in A starts by taking a suitable elliptic curve E2 /k together with a k-group scheme isomorphism λ : E1 [3] → E2 [3]. Let Δ ⊂ E1 × E2 be the graph of λ so that Δ(k) = {(P, λ(P )) : P ∈ E1 [3](k)}. Let A := (E1 × E2 )/Δ and write φ : E1 × E2 → A for the corresponding isogeny. Since Δ ⊂ E1 [3] × E2 [3], we have another isogeny φ : A → E1 × E2 such that φ

φ ◦ φ = 3. We write p∗ for the composition E1 → (E1 × E2 ) → A and p∗ for the φ

composition A → (E1 × E2 ) → E1 and q ∗ , q∗ for the corresponding morphisms concerning E2 . It is straightforward to verify that p∗ , q ∗ are embeddings, that φ = p∗ − q ∗ (where the projections are understood), and that φ = p∗ × q∗ . We combine the galois cohomology of the short exact sequences p∗

q∗

q∗

p∗

0 → E1 → A → E2 → 0, 0 → E2 → A → E1 → 0, and 3

0 → Ei [3] → Ei → Ei → 0 for i = 1, 2 to obtain the big (symmetric) commutative diagram with exact rows and columns E2 (k) 3

E2 (k) α

E1 (k)

3

/ E1 (k)

/ H 1 (k, Δ)

p∗

/ E1 (k)

/ H 1 (k, E2 )

p∗

A(k)

q∗

/ A(k) q∗

E2 (k) / H 1 (k, E1 ) / H 1 (k, A)

where we note that H 1 (k, Δ) H 1 (k, E1 [3]) H 1 (k, E2 [3]). We see that δ is visible in A precisely if δ ∈ H 1 (k, E1 [3]) = H 1 (k, Δ) lies in the image of α, i.e., if the curve Cλ(δ) corresponding to λ(δ) ∈ H 1 (k, E2 [3]) has a rational point. We summarize these observations, which are due to Mazur. Lemma 1. Let E1 be an elliptic curve over a perfect field k of characteristic distinct from 3 and let δ ∈ H 1 (k, E[3]) with Ob(δ) = 0. Suppose that there exists an elliptic curve E2 /k and a k-group scheme isomorphism λ : E1 [3] → E2 [3] such that the curve Cλ(δ) corresponding to λ(δ) has a k-rational point. Then δ is visible in the abelian surface (E1 × E2 )/Δ where Δ denotes the graph of λ.

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Mazur also observed, in the case of a number ﬁeld k, that if δ ∈ S (3) (E/k), then Cδ admits a plane cubic model. Furthermore, there is a pencil of cubics through the 9 ﬂexes of Cδ , and each non-singular member corresponds to a 3-covering Ct → Et , where Et [3] E[3] and Ct → Et represents δ. It is therefore easy to ﬁnd a t such that Ct has a rational point; simply pick a rational point and solve for t. To reﬁne the construction, one can ask Question 1. Can one make δ ∈ H 1 (k, E[3]) visible in the jacobian of a genus 2 curve? Note that E1 ×E2 is principally polarized via the product polarization. This gives rise to a Weil pairing on (E1 × E2 )[3], corresponding to the product pairing. If A is a jacobian, then A must be principally polarized over k. One way this could happen is if the isogeny φ : E1 × E2 → A gives rise to a principal polarization. This would be the case if the kernel Δ is a maximal isotropic subgroup of E1 [3]× E2 [3] with respect to the product pairing. That means that λ : E1 [3] → E2 [3] must be an anti-isometry, i.e. for all P, Q ∈ E1 [3] we must have eE2 (λ(P ), λ(Q)) = eE1 (P, Q)−1 . Note that the original cubic C is a member of the pencil that Mazur constructs, so in his construction λ is actually an isometry, i.e. it preserves the Weil-pairing. Below we consider a pencil of cubics that leads to an anti-isometry λ.

4

Anti-isometric Pencils

Let k be a perfect ﬁeld of characteristic distinct from 2, 3. Following [7], we associate to a ternary cubic form F ∈ k[x, y, z] three more ternary cubic forms. Namely, the Hessian of F ∂F 2 ∂F 2 ∂F 2 ∂x∂y ∂x∂z 1 ∂x∂x 2 2 2 ∂F ∂F ∂F , H(F ) := − ∂y∂x ∂y∂z 2 ∂F 2 ∂y∂y 2 ∂F ∂F 2 ∂z∂x ∂z∂y ∂z∂z

the Caylean of F

∂F (0, z, −y) ∂x 1 ∂F P (F ) := − ∂x (−z, 0, x) xyz ∂F ∂x (y, −x, 0)

∂F ∂y (0, z, −y) ∂F ∂y (−z, 0, x) ∂F ∂y (y, −x, 0)

∂F ∂z (0, z, −y) ∂F ∂z (−z, 0, x) ∂F ∂z (y, −x, 0)

and a ternary cubic form denoted Q(F ), for which we refer to [7, Section 11.2]. For most cases one can take Q(F ) to be H(P (F )) or P (H(F )), but there are some exceptional cases where P (F ), Q(F ) span an appropriate pencil and P (F ), H(P (F )) do not. The left action of GL3 on k 3 induces a right action of GL3 on ternary cubic forms (or, more generally, on k[x, y, z]). For a ternary cubic form F and an M ∈ GL3 we denote this action simply by F ◦M . The signiﬁcance

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of the three associated ternary cubic forms lies in the fact that H(F ) depends covariantly on F (of weight 2) and P (F ) and Q(F ) depend contravariantly on F (of weights 4 and 6 respectively). This means that for every ternary cubic form F and every M ∈ GL3 we have, with d := det M that H(F ◦ M ) = d2 H(F ) ◦ M P (F ◦ M ) = d4 P (F ) ◦ M −T Q(F ◦ M ) = d6 Q(F ) ◦ M −T , where M −T denotes the inverse transpose of M . Now consider a smooth cubic curve C in P2 given by the zero locus of a ternary cubic form F . Then C has exactly 9 diﬀerent ﬂex points Φ, which all lie on the (not necessarily smooth) curve given by H(F ) = 0. The smoothness of C guarantees that F and H(F ) will be linearly independent over k. Hence Φ can be described as the intersection F = H(F ) = 0. We call Φ the flex scheme of C. At least one of P (F ) and Q(F ) turns out to be nonsingular (still assuming that C is nonsingular) and the intersection P (F ) = Q(F ) = 0 equals the ﬂex points Φ∗ of the nonsingular cubics among P (F ) and Q(F ) (if, say, P (F ) is nonsingular, then Φ∗ can of course also be written as P (F ) = H(P (F )) = 0). We can consider the pencil of cubics through Φ, explicitly given by C(s:t) : sF (x, y, z) + tH(F )(x, y, z) = 0.

(4.1)

Classical invariant theory tells us the following. This pencil has exactly 4 singular members and all other members have ﬂex scheme equal to Φ. Conversely, any nonsingular cubic with ﬂex scheme Φ occurs in this pencil. Furthermore, both P (sF + tH(F )) and Q(sF + tH(F )) are linear combinations of P (F ) and Q(F ). This shows that the ﬂex scheme Φ∗ is independent of the choice of C through Φ and only depends on Φ. We call Φ∗ the dual ﬂex scheme of Φ and we will justify this name below. As a simple, but important example we take F := x3 + y 3 + z 3 . Then we compute H(F ) = −108xyz,

P (F ) = −54xyz,

Q(F ) = 324(x3 + y 3 + z 3 ).

Now deﬁne Φ0 to be the ﬂex scheme of F = 0, i.e. Φ0 := {[x : y : z] ∈ P2 : x3 + y 3 + z 3 = xyz = 0}.

(4.2)

Then we see that the ﬂex scheme given by P (F ) = Q(F ) = 0 (which is the ﬂex scheme of Q(F ) = 0) equals Φ0 , i.e. Φ∗0 = Φ0 . The pencil of cubics through Φ0 (note that 108 = 0 in k), which is given by s(x3 + y 3 + z 3 ) = txyz,

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is a model over k for the universal elliptic curve over the (genus zero) modular curve X(3); see [5, p. 225]. Geometrically all ﬂex schemes are linear transformations of each other. In particular, for any ﬂex scheme Φ there exists an M ∈ GL3 (k) such that Φ = M Φ0 . This shows that the pencil (4.1) associates to a general ﬂex scheme Φ is a twist of the universal elliptic curve over X(3). The contravariance of P and Q implies that the assignment Φ → Φ∗ has the contravariance property that for any ﬂex scheme Φ and M ∈ GL3 (M Φ)∗ = M −T Φ∗ .

(4.3)

We also note that this implies that the assignment Φ → Φ∗∗ := (Φ∗ )∗ is covariant in the sense that for any ﬂex scheme Φ and M ∈ GL3 we have (M Φ)∗∗ = M Φ∗∗ . Writing Φ = M Φ0 and using (Φ0 )∗∗ = Φ∗0 = Φ0 we now get Φ∗∗ = (M Φ0 )∗∗ = M Φ∗∗ 0 = M Φ0 = Φ. This justiﬁes calling Φ∗ the dual ﬂex scheme of Φ. Remark 2. In the discussion above it was convenient to consider just one projective plane P2 . A more canonical way would be to consider a projective plane P2 with coordinates x, y, z (for a point) and the dual projective plane, denoted (P2 )∗ , where the point with coordinates u, v, w describes the line ux+vy +wz = 0. Now let C be a smooth cubic curve in P2 given by the zero locus of the ternary cubic form F (x, y, z) with ﬂex scheme Φ. The 9 tangent lines through Φ determine 9 points in (P2 )∗ . Generically, these 9 points in (P2 )∗ will not be the ﬂex points of a smooth cubic curve, hence generically there will a unique cubic curve going through these points. This curve in (P2 )∗ is exactly given by the zero locus of the Caylean, i.e. P (F )(u, v, w) = 0; see also [10, pp.151,190–191]. Moreover, if the characteristic of k is zero, then it turns out that this cubic curve is nonsingular if and only if the j-invariant of C is nonzero. To any ﬂex scheme Φ we associate a group Θ(Φ) ⊂ GL3 as follows. Choose a nonsingular cubic curve C through Φ and let E be its jacobian. After identifying E and C as curves over k, we get an action of E[3] on C, which extends to a linear action on P2 . This determines an embedding χ : E[3] → PGL3 . Obviously, the image χ(E[3]) only depends on Φ. We deﬁne Θ(Φ) to be the inverse image of χ(E[3]) in GL3 . Actually Θ(Φ) can be deﬁned just in terms of Φ, without choosing C, since it turns out that χ(E[3]) consists exactly of the linear transformations that preserve Φ. (One way of quickly ﬁnding these linear transformations explicitly is by using the fact that, for any two distinct points of Φ, the line through these two points intersects Φ in a unique third point.) The construction gives rise to the theta group 1 → Gm → Θ(Φ) → E[3] → 1. Note that the isomorphism class of this theta group may still depend on the choice of identiﬁcation of C with E. This corresponds to the choice of an isomorphism between Θ(Φ)/Gm and E[3]. If Φ is deﬁned over k, then E[3] and

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Θ(Φ) are also deﬁned over k and the element in H 1 (k, E[3]) corresponding to this theta group is the same as the element corresponding to the 3-covering C → C/E[3] E for any nonsingular cubic curve C through Φ. The construction also shows that for any M ∈ GL3 we have Θ(M Φ) = M Θ(Φ)M −1 .

(4.4)

Proposition 3. Let Φ1 ⊂ P2 be a flex scheme and let Φ2 := Φ∗1 be the dual flex scheme. For i = 1, 2 let Ci be a smooth plane cubic with flex scheme Φi , denote its jacobian by Ei and consider an induced theta group 1

/ Gm

αi

/ Θ(Φi )

βi

/ Ei [3]

/1.

(4.5)

Then the outer automorphism (−T ) : GL3 → GL3 given by M → M −T , yields an isomorphism Θ(Φ1 ) → Θ(Φ2 ). There exists an anti-isometry λ : E1 [3] → E2 [3] making the following diagram commutative. 1

/ Gm

α1

x→x−1

1

/ Gm

/ Θ(Φ1 )

β1

(−T )

α2

/ Θ(Φ2 )

β2

/ E1 [3]

/1

(4.6)

λ

/ E2 [3]

/1

In particular, let δi ∈ H 1 (k, Ei [3]) correspond to the theta group (4.5). Then under the isomorphism H 1 (k, E1 [3]) H 1 (k, E2 [3]) induced by λ, the cocycle δ1 maps to δ2 . Proof. Once the isomorphism Θ(Φ1 ) → Θ(Φ2 ) given by M → M −T is established, the existence of an isomorphism λ : E1 [3] → E2 [3] making the diagram (4.6) commutative, follows immediately. That λ must be an anti-isometry can readily be seen as follows. Let P, Q ∈ E1 [3] and choose x, y ∈ Θ(Φ1 ) such that P = β1 (x) and Q = β1 (y). Then α2 (eE2 (λ(P ), λ(Q))) = α2 (eE2 (β2 (x−T ), β2 (y −T ))) = x−T y −T xT y T = (xyx−1 y −1 )−T = α1 (eE1 (β1 (x), β1 (y)))−T = α1 (eE1 (P, Q)−1 ). The last statement of the proposition is also immediate, so we are left with ∼ establishing (−T ) : Θ(Φ1 ) → Θ(Φ2 ). It suﬃces to show that for a ﬂex scheme 2 −T Φ ⊂ P we have Θ(Φ) = Θ(Φ∗ ). Write Φ = M Φ0 for some M ∈ GL3 with Φ0 given by (4.2). Then a straightforward calculation shows that Θ(Φ0 )−T = Θ(Φ0 ). We also know that Φ∗0 = Φ0 , so we get Θ(Φ0 )−T = Θ(Φ∗0 ). Together with (4.3) and (4.4) we ﬁnally obtain,

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Θ(Φ)−T = Θ(M Φ0 )−T = M −T Θ(Φ0 )−T M T = M −T Θ(Φ∗0 )(M −T )−1 = Θ(M −T Φ∗0 ) = Θ((M Φ0 )∗ ) = Θ(Φ∗ ).

Remark 3. The construction above of the dual ﬂex scheme Φ∗ of a ﬂex scheme Φ involved choosing a smooth cubic going through Φ. Without using theta groups, it was not obvious from this construction that the degree 9 ´etale algebra k(Φ) is isomorphic to k(Φ∗ ). However, there exists a nice explicit geometric construction of the dual ﬂex scheme that remedies these shortcomings of the earlier construction. Given a ﬂex scheme Φ, we proceed as follows. We label its 9 points over k with P1 , . . . , P9 . There are 4 sets of 3 lines, (corresponding to the 4 singular members of the pencil of cubics through φ) containing these points. We label the line that contains Pi , Pj , Pk with l{i,j,k} . One can label the points such that the subscripts are {1, 2, 3} {1, 4, 7} {1, 5, 9} {1, 6, 8} {4, 5, 6} , {2, 5, 8} , {2, 6, 7} , {2, 4, 9} , {7, 8, 9} {3, 6, 9} {3, 4, 8} {3, 5, 7} Naturally, two diﬀerent lines l{i1 ,j1 ,k1 } , l{i2 ,j2 ,k2 } meet in a unique point. If for example i1 = i2 , then the intersection point is Pi1 . If the two sets {i1 , j1 , k1 } {i2 , j2 , k2 } are disjoint, then the two lines meet in a point outside Φ. We name this point L{i3 ,j3 ,k3 } , where {i1 , j1 , k1 , i2 , j2 , k2 , i3 , j3 , k3 } = {1, . . . , 9}. As it turns out, the four points that have i in their label all lie on a line pi . It is also straightforward to check that the pi together with the L{i,j,k} form a conﬁguration in (P2 )∗ that is completely dual to the Pi with the l{i,j,k} . The pi form the k points of a ﬂex scheme in (P2 )∗ , which is justiﬁably a ﬂex scheme Φ∗ dual to Φ, and its construction immediately implies the contravariance property (M Φ)∗ = M −T Φ∗ . We can easily verify that the two constructions of Φ∗ coincide for one ﬂex scheme, for instance Φ0 . The general result then follows because any ﬂex scheme can be expressed as M Φ0 for some M ∈ GL3 (k). Since the action of Gal(k/k) on {P1 , . . . , P9 } must act via collinearity-preserving permutations, we see that if σ(Pi ) = Pσ(i) , then σ(pi ) = pσ(i) . Hence, we see that the k-points of Φ and its dual have the same Galois action and hence k(Φ) is isomorphic as a k-algebra to k(Φ∗ ).

5

Recovering the Genus 2 Curve

Let k be a ﬁeld and let E1 , E2 be two elliptic curves over k with an anti-isometry λ : E1 [3] → E2 [3] and denote by Δ the graph of λ as before. Recall that E1 × E2 is

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principally polarized via the product polarization and that the induced polarization on A := (E1 × E2 )/Δ is also principal in this case. It is a classical fact that if A is not geometrically isomorphic to a product of elliptic curves, then A (together with its principal polarization) is isomorphic to the jacobian of a genus 2 curve C. Let us assume from now on that E1 and E2 are non-isogenous. In [8] it is shown that in this case A is always isomorphic over k to the jacobian of a genus 2 curve C/k. This is enough to get our main theoretical result. Theorem 4. Let E be an elliptic curve over a number field k and let ξ ∈ X(E/k)[3]. Then ξ is visible in the jacobian of a genus 2 curve C/k. Proof. Let δ ∈ S (3) (E/k) be a cocycle representing ξ. By Proposition 2, there is a 3-covering Cδ → E corresponding to δ. According to Remark 1, we have that Ob(δ) = 0 and hence that Cδ ⊂ P2 . Let Φ ⊂ P2 be its ﬂex scheme. The construction in Section 4 gives us a pencil of cubics through Φ∗ , so we can easily pick a non-singular one with a rational point. It follows from Proposition 3 that such a curve is of the form Cλ(δ) for some elliptic curve E2 and some anti-isometry λ : E[3] → E2 [3]. This places us in the situation of Lemma 1, so δ is visible in an abelian surface A = (E × E2 )/Δ. We have ensured that λ is an anti-isometry, which implies that the surface is principally polarized. As long as we make sure that E, E2 are non-isogenous (and this is easy given the freedom we have in choosing Cλ (δ)) it follows that A is a jacobian.

Remark 4. We could of course state a more general result about visibility of elements δ ∈ H 1 (k, E[3]) with Ob(δ) = 0 for an elliptic curves E over a perfect ﬁeld k of characteristic distinct from 2 or 3. Note however that if k is too small, there might not be enough non-isogenous elliptic curves available. The exclusion of ﬁelds of characteristic 3 is a serious one, the exclusion of non-perfect ﬁelds less so. Most of what we are saying could be generalized to the non-perfect case, basically because for an elliptic curve over any ﬁeld of characteristic distinct from 3, the multiplication by 3 map is separable. The exclusion of ﬁelds of characteristic 2 stems from the fact that the necessary invariant theory in this case is not readily available. We continue with the construction of the genus 2 curve C. Deﬁne the divisor Θ := 01 × E2 + E1 × 02 on E1 × E2 , which gives a principal polarization on E1 × E2 . Next, consider the set D of eﬀective divisors on E1 × E2 over k which are linearly equivalent to 3Θ and invariant under Δ. Also consider the set C of eﬀective divisors C on A over k whose pull-back to E1 × E2 are linearly equivalent to 3Θ and which satisfy (C · C) = 2. Frey and Kani show that there exist unique curves D ∈ D and C ∈ C deﬁned over k which are invariant under multiplication by −1. Furthermore, because E1 and E2 are not isogenous, D and C are irreducible smooth curves of genus 10 and 2 respectively and the natural map D → C is unramiﬁed of degree 9. If k is a perfect ﬁeld of characteristic distinct from 2 or 3, the curves D and C can be explicitly constructed as follows. Embed E1 in P2 , given by, say

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F (x, y, z) = 0, for a ternary cubic F/k (such an F is readily obtained if E1 is given by a Weierstrass model). Express E2 as G := sP (F ) + tQ(F ) = 0 for some s, t ∈ k. This way, we obtain an embedding of E1 × E2 in P2 × P2 given by F (x, y, z) = G(u, v, w) = 0. Moreover, by appealing to Proposition 3 we obtain that the curve on this surface given by xu + yv + zw = 0 must be the curve D. The genus 2 curve C is the image of D in (E1 × E2 )/Δ. E1 × E2O OOO OOO OOO O' [3]×[3] (E1 × E2 )/Δ o ooo o o oo wooo E1 × E2 The map [3] × [3] is much more accessible, though. We claim that the subgroup of E1 [3] × E2 [3] under which D is invariant is equal to Δ. Hence, we can ﬁnd a (singular) model of C as a curve on E1 × E2 by computing ([3] × [3])(D). This can easily be done via interpolation, as explained in the next section by means of an example. As for our claim above, suppose that D is invariant under some σ ∈ E1 [3] × E2 [3] with σ ∈ / Δ. Without loss of generality we may assume that σ = (P, 0E2 ) ∈ E1 [3] × E2 [3] with P = 0E1 . Denote by M ∈ PGL3 (k) the linear action corresponding to translation by P . Now for all ([x : y : z], [u : v : w]) on D we have (x, y, z)(u, v, t)T = (x, y, z)M T (u, v, w)T = 0. This yields (u, v, t) = (x, y, z) × (x, y, z)M T , where × denotes the standard cross product. This association actually deﬁnes a birational transformation φ : P2 → P2 (a Cremona transformation with singular points corresponding to the eigenspaces of M ). Note that φ is deﬁned on all the [x : y : z] on E1 , so the image of E1 under φ is an irreducible curve birational to E1 . Together with the assumption that E1 and E2 are not isogenous, we get that this image intersects E2 in only ﬁnitely many points, so D is not invariant under σ.

6

Examples

Following the ﬁrst example in [4, Table 1], consider the elliptic curve 681b1 (in Cremona’s notation), given by the minimal Weierstrass equation E1 : y 2 + xy = x3 + x2 − 1154x − 15345. It turns out that the plane cubic curve C1 : x3 + 5x2 y + 5x2 z + 2xy 2 + xyz + xz 2 + y 3 − 5y 2 z + 2yz 2 + 6z 3 = 0

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deﬁnes an element ξ (up to inverse) of order three in X(E1 /Q). The contravariants, denoted P0 , Q0 , are given by P0 = −478x3 + 2525x2 y + 916x2 z − 1127xy 2 + 29xyz −160xz 2 + 753y 3 − 1228y 2z + 260yz 2 + 301z 3, Q0 = −122314x3 + 618551x2y + 191092x2z − 271157xy 2 − 7825xyz −28120xz 2 + 184011y 3 − 264916y 2z + 55892yz 2 + 73663z 3. Now the curve C2 : 55033P0 − 235Q0 = 0 has a rational point [x : y : z] = [10 : 8 : 7] and its jacobian is the elliptic curve 681c1, given by the minimal Weierstrass equation E2 : y 2 + y = x3 − x2 + 2. To construct the corresponding genus two curve C such that ξ becomes visible in its jacobian we could take the curve in C1 × C2 ⊂ P2 × P2 with coordinates ([x : y : z], [u : v : w]) given by the equation xu + yv + zw = 0, and take its image under C1 × C2 → E1 × E2 , since this is a twist of [3] × [3] : E1 × E2 → E1 × E2 anyway. We will follow Section 5 more closely. Obviously, E1 is given by F = 0 if we deﬁne F := y 2 z + xyz − (x3 + x2 z − 1154xz 2 − 15345z 3). The contravariants of the ternary cubic F are given by P = −2308x3 + 3462x2 y − 5x2 z − 275056xy 2 + 5xyz +6xz 2 + 136951y 3 + 13853y 2z − 3yz 2, Q = −725020x3 + 1087530x2y + 27721x2z − 65861608xy 2 − 27721xyz −30xz 2 + 32749549y 3 + 3217559y 2z + 15yz 2 + 24z 3 . Write j(s, t) for the j-invariant of the curve given by sP +tQ = 0. The j-invariant of E2 equals −4096/2043 and the equation j(s, t) = −4096/2043 has exactly one solution in P1 (Q), namely [s : t] = [55033 : −235] (compare with the deﬁnition of C2 ). This gives us a new model for E2 , namely E2 : 55033P − 235Q = 0. We consider the surface E1 × E2 embedded in P2 × P2 as F (x, y, z) = 0,

55033P (u, v, w) − 235Q(u, v, w) = 0.

The curve D on this surface is given by xu + yv + zw = 0.

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The image of D under multiplication by 3 on E1 × E2 is the genus two curve C. Using the deﬁning properties of C from Section 5 (such as the invariance under multiplication by −1), we get that as a curve on E1 × E2 it must be of the form axu + byv + czw + dxw + ezu = 0 for some a, b, c, d, e ∈ Q. We simply generate 4 points on C (over a number ﬁeld), compute the image under multiplication by 3 of these points and solve for a, b, c, d, e. If the dimension of the solution space is greater than 1, we must of course add points (or take 4 better ones) so that the solution space becomes 1-dimensional. This gives us our equation for C. By a linear change of the u, v, w coordinates we can change the model for E2 back to the original minimal Weierstrass model. Thus, the model for E1 × E2 embedded in P2 × P2 is E1 : y 2 z + xyz = x3 + x2 z − 1154xz 2 − 15345z 3, E2 : v 2 w + vw2 = u3 − u2 w + 2w3 and C is the curve on this surface given by 4xu − 155zu + xv + 2yv − 40xw + yw + 1314zw = 0. Hyperelliptic models for C are Y 2 + (X + 1)Y = 3X 5 + 5X 4 + X 3 − 8X 2 − 5X + 2 or Y 2 = (3X − 1)(X + 1)(4X 3 + 4X 2 − 9). Next, consider the elliptic curve 2006e1, given by the minimal Weierstrass equation E1 : y 2 + xy = x3 + x2 − 58293654x − 171333232940. It turns out that the plane cubic curve C1 : 20x3 +44x2 y +21x2z −77xy 2 +71xyz +44xz 2 +31y 3 +3y 2 z +150yz 2 +z 3 = 0 deﬁnes an element ξ (up to inverse) of order three in X(E1 /Q). In the sixth example in [4, Table 1] the elliptic curve E2 which ‘explains’ X(E1 /Q) is 2006d1. However, for this choice of E2 , there only exists an isometry between E1 [3] and E2 [3] and not an anti-isometry. The corresponding abelian surface (E1 × E2 )/Δ visualizing ξ will not be the jacobian of a genus 2 curve. If instead we take for E2 the elliptic curve 6018c1, then we do have an anti-isometry between E1 [3] and E2 [3]. Following the same route as in the ﬁrst example, we ﬁnd that ξ is visible in the jacobian of the genus 2 curve C with hyperelliptic models Y 2 + (X 2 + X)Y = − 9675X 6 − 94041X 5 − 914X 4 + 1301674X 3 − 352310X 2 − 2071181X − 945269

or

Y = 43(2X + 13)(18X − 81X + 89)(25X 3 + 193X 2 + 224X + 76). 2

2

Visualizing Elements of Sha[3] in Genus 2 Jacobians

7

123

Applications to 3-Descent

In this section we survey some of the ways in which explicit visibility might aid computations of Mordell-Weil groups and related quantities of elliptic curves. We recall that given an abelian variety A over a number ﬁeld k, the group X(A/k) ⊂ H 1 (k, A) consists of the cocycle classes that are everywhere locally trivial. It measures the diﬀerence between the Mordell-Weil group A(k) and the Selmer 1 group S(A/k) ⊂ lim ←−n H (k, A[n]) which is an everywhere local approximation to A(k), in the sense that the following sequence is exact. 0 → A(k) → S(A/k) → X(A/k) → 0 An n-descent usually means an explicit computational process to compute S (n) (A/k) = S(A/k)/nS(A/k) ⊂ H 1 (k, A[n]). It provides a bound on rkA(k) and conversely, if A(k) is known, then we can use 0 → A(k)/nA(k) → S (n) (A/k) → X(A/k)[n] → 0 to compute #X(A/k)[n] and thus obtain information on #X(A/k). In principle, one can use visibility to reﬁne this information. We will argue using an example. Stein and Watkins [12] found the following elliptic curve E : y 2 + xy = x3 − x2 + 94x + 9. Using a 2-descent and some point searching (with for instance Magma [1]) it is straightforward to verify that E(Q) Z × Z and that #X(E/Q)[2] = 1. Using a 3-descent (see [2, 3, 11], implemented in Magma), with unproved S-unit data we ﬁnd that C1 : x3 + 2x2 z + 2xy 2 + xyz − xz 2 − y 3 + 3y 2 z − 6yz 2 + z 3 = 0, C2 : x3 − 2xy 2 + 3xyz + 2y 3 + y 2 z + yz 2 + 3z 3 = 0 are 3-coverings of E that have points everywhere locally and we can verify by looking at preimages of representatives of E(Q)/3E(Q) that C1 , C2 have no rational points. The same process allows us to ﬁnd more than 18 such spaces, verifying unconditionally that #X(E/Q)[3] ≥ 9. The conditional 3-descent computation suggests that C1 , C2 represent cocycles generating S (3) (E/Q)/E(Q), so one expects that #X(E/Q)[3] = 9 and indeed BSD predicts that #X(E/Q) = 9. Visibility could help with proving that #X(E/Q)[3∞ ] = 9. The construction in this paper yields an abelian surface A = Jac(C), together with a map φ∗ : X(E/Q) × X(E /Q) → X(A/Q) where we know that ker(φ∗ ) is contained in the 3-torsion, because multiplicationby-three factors through φ. If we can make sure that ker(φ∗ ) contains the classes

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represented by C1 , C2 (this implies that E (Q) is of rank at least 2), it may well be that #X(A/Q)[3] = 1. If we can compute S (3) (A/Q), we can check this and the result would follow. Thus, visibility allows us to substitute a 9-descent on an elliptic curve with a 3-descent on the Jacobian of a genus 2 curve. Both are theoretically computable, but in neither case does it seem practical at this point. Since A has a 3-isogeny to E × E , the 3-torsion algebra (generically of degree 80), splits in two algebras of degrees 72 and 8 respectively. However, doing class group computations for degree 72 algebras over Q still seems well out of range. It is conceivable that some appropriate galois-stable set S of divisors on C exists with #S < 72. The group Sp4 (F3 ) has an index 27 subgroup, for instance, predicting a transitive action on 27 objects somewhere. If for some ﬁxed divisor D0 we have that A[3] = [D − D0 ] : D ∈ S, it may be possible to adapt ideas about fake Selmer groups [9] for application to A and only require class group information for algebras of degree #S. At this point it is unclear if this approach has any advantages to a direct 9descent on E and whether either method can be made practical for the example given in this section. Acknowledgments. The authors would like to thank the referees, who provided various helpful comments which found their way into this article.

References [1] Bosma, W., Cannon, J., Playoust, C.: The Magma computer algebra system is described in the Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997) [2] Cremona, J.E., Fisher, T.A., O’Neil, C., Simon, D., Stoll, M.: Explicit n-descent on elliptic curves. I. Algebra. J. Reine Angew. Math. 615, 121–155 (2008) [3] Cremona, J.E., Fisher, T.A., Stoll, M.: Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, arXiv: 0908.1741 (2009), http://arxiv.org/abs/0908.1741 [4] Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich-Tate group. Experiment. Math. 9(1), 13–28 (2000) [5] Mazur, B.: Visualizing elements of order three in the Shafarevich-Tate group. Asian J. Math. 3(1), 221–232 (1999); Sir Michael Atiyah: a great mathematician of the twentieth century [6] O’Neil, C.: The period-index obstruction for elliptic curves. J. Number Theory 95(2), 329–339 (2002) [7] Fisher, T.: The Hessian of a genus one curve, arXiv: math/0610403 (2006), http://arxiv.org/abs/math/0610403 [8] Frey, G., Kani, E.: Curves of genus 2 covering elliptic curves and an arithmetical application. In: Arithmetic Algebraic Geometry (Texel, 1989), pp. 153–176 (1991) [9] Poonen, B., Schaefer, E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997)

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[10] Salmon, G.: A treatise on the higher plane curves, 3rd edn. Hodges, Foster, and Figgis, Grafton Street, Dublin (1879) [11] Schaefer, E.F., Stoll, M.: How to do a p-descent on an elliptic curve. Trans. Amer. Math. Soc. 356(3), 1209–1231 (2004) [12] Stein, W.A., Watkins, M.: A database of elliptic curves—first report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002), http://wstein.org/Tables/ecdb/

On Weil Polynomials of K3 Surfaces Andreas-Stephan Elsenhans1, and J¨ org Jahnel2 1

2

Universit¨ at Bayreuth, Mathematisches Institut, Universit¨ atsstraße 30, D-95447 Bayreuth, Germany [email protected] Fachbereich 6, Mathematik, Universit¨ at Siegen, Walter-Flex-Straße 3, D-57072 Siegen, Germany [email protected]

Abstract. For K3 surfaces, we derive some conditions the characteristic polynomial of the Frobenius on the ´etale cohomology must satisfy. These conditions may be used to speed up the computation of Picard numbers and the decision of the sign in the functional equation∗∗ . Our investigations are based on the Artin-Tate formula.

1

Introduction

√ An algebraic integer such that all its conjugates have absolute value r is called an r-Weil number. Correspondingly, a possibly √ reducible monic polynomial Φ ∈ Z[T ] such that all roots have absolute value r is called an r-Weil polynomial. Let q be a prime power and r = q k . Then, for every smooth projective variety V over Fq , the eigenvalues of the Frobenius endomorphism Frob on the ´etale cohomology H´ekt (VFq, Ql ) are r-Weil numbers [3, Lemme 1.7]. Conversely, every q k -Weil number is an eigenvalue of Frob on H´ekt (VFq, Ql ) for a suitable smooth projective variety V over Fq . Actually, this fact is a direct consequence of the results of T. Honda [9]. In this note, we will study the Weil numbers of K3 surfaces. As the second Betti number of a K3 surface is b2 (V ) = 22 and q is always a root of the characteristic polynomial, the possible Weil numbers are of degree at most 20. We will show that not all q 2 -Weil polynomials Φ ∈ Z[T ] satisfying deg Φ = 22 and Φ(q) = 0 occur as characteristic polynomials of Frob on the ´etale cohomology of K3 surfaces. Concerning K3 surfaces of ﬁxed degree, even more restrictions result. Our investigations are based on the Artin-Tate formula which we will recall in section 3.

The first author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) through a funded research project. The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Computing at the G¨ ottingen Mathematisches Institut. Both authors are grateful to Prof. Y. Tschinkel for the permission to use these machines as well as to the system administrators for their support.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 126–141, 2010. c Springer-Verlag Berlin Heidelberg 2010

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An application. The characteristic polynomial of Frob may be computed by counting points over extensions of the ground ﬁeld. Indeed, for V a K3 surface over Fq , the Lefschetz trace formula [13, Ch. VI, §12] yields tr(Frobe ) = #V (Fqe ) − q 2e − 1. When we denote the eigenvalues of Frob by r1 , . . . , r22 , we have e tr(Frobe ) = r1e + · · · + r22 =: σe (r1 , . . . , r22 ). Newton’s identity [20] sk (r1 , . . . , r22 ) =

k−1 1 (−1)k+j+1 σk−j (r1 , . . . , r22 )sj (r1 , . . . , r22 ) k j=0

shows that the knowledge of σe (r1 , . . . , r22 ), for e = 1, . . . , k, is suﬃcient in order to determine the coeﬃcient (−1)k sk of T 22−k of the characteristic polynomial Φ of Frob. Further, there is the functional equation q deg Φ Φ(T ) = ±T deg Φ Φ(q 2/T )

(1)

which, as deg Φ = 22, relates the coeﬃcient of T k with that of T 22−k . Nevertheless, this method is time-consuming. The size of the ﬁelds to be considered grows exponentially. One would like to avoid point counting over large ﬁelds and, nevertheless, determine Φ suﬃciently well in order to decide things such as the sign in (1). Algorithms of this type were presented in [6]. For example, Algorithm 22 of [6] veriﬁes that the geometric Picard rank is 2, having counted points over Fp , . . . , Fp9 for p a prime number. The main result of the present article leads to a more substantial approach to this problem. In fact, we will show that certain hypothetical characteristic polynomials are impossible, in general. This leads to an improvement of [6, Algorithm 22]. Sections 7 and 8 will be devoted to examples showing how this improvement works in practice. Remark 1. A continuation of this application, which we have in mind, is the computation of the geometric Picard rank for K3 surfaces over Q. Here, the general strategy is to use reduction modulo p. One applies the inequality rk Pic(VQ ) ≤ rk Pic(VFp ) which is true for every smooth variety V over Q and every prime p of good reduction. Then, the number of eigenvalues of Frob which are roots of unity is an upper bound for the Picard number. More details are given in [6] and [7].

2

The Galois Group of a Weil Polynomial

For a randomly chosen irreducible polynomial over Q, one expects the Galois group to be the full symmetric group. In this sense, the irreducible factors of a Weil polynomial are not very random. When we consider the operation of Frob on a cohomology group of even degree, cyclotomic factors do arise. They correspond to the algebraic part of the cohomology, i.e., to the image of the Picard group and its analogues in higher codimension. The corresponding Galois group is always abelian.

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Concerning the remaining factors, still, there are restrictions on the Galois group. Note that, for each root of an irreducible r-Weil polynomial not of degree 1, the complex conjugate is a root, too. This means, the roots come in pairs. The product of each pair is equal to r. The Galois group therefore acts on the pairs. For a suitable integer n, it is a subgroup of the semi-direct product (Z/2Z)n Sn ⊂ S2n . Here, each factor (Z/2Z) acts on one pair by complex conjugation. The complex conjugation itself belongs to the center of the group. An experimental result. One could ask for further restrictions on the Galois group. For that, we computed the characteristic polynomial of Frob for a few thousand randomly chosen K3 surfaces. In each case, the factorization of that polynomial had precisely one irreducible factor which was not cyclotomic. This coincides with Zarhin’s results [18] for ordinary K3 surfaces. Furthermore, in the vast majority of the examples, the Galois group of the last factor was actually equal to the semi-direct product (Z/2Z)n Sn ⊂ S2n . For example, this was true for 875 out of 1 000 K3 surfaces of degree 2 over F3 and 923 out of 1 000 K3 surfaces of degree 2 over F7 . The resolvent algebra. Let Φ ∈ Q[T ] be a polynomial such that its set of roots is of the particular form {r1 , r1 , . . . , rn , rn } such that r1 r1 = . . . = rn rn =: r ∈ Q. Then, the sums r1 + r1 , . . . , rn + rn are the roots of a polynomial R ∈ Q[T ] of half the degree. We will call R the resolvent polynomial and A := Q[T ]/R the resolvent algebra of Φ. Remarks 2. a) When Φ √ is an r-Weil polynomial of even degree, the assumption is satisﬁed if and only if r is a root of even multiplicity (or no root) of Φ. In this √ case, (− r) has even multiplicity, too. In fact, this means exactly that Φ fulﬁlls the functional equation (1) with the plus sign. b) On the other hand, when one wants to verify that a given polynomial satisfying the functional equation is, in fact, a Weil polynomial, the resolvent is helpful. √ Observe that the roots of the initial polynomial are all of absolute value √ √r if and only if the roots of the resolvent are all real and in the interval [−2 r, 2 r]. That property may easily be checked using Sturm’s chain theorem. This is a fast and exact replacement of [6, Algorithm 23].

3

The Artin-Tate Formula

Let us recall the Artin-Tate conjecture in the special case of a K3 surface. Conjecture 3 (Artin-Tate). Let V be a K3 surface over a ﬁnite ﬁeld Fq . Denote by ρ the rank and by Δ the discriminant of the Picard group of V, deﬁned over Fq . Then, lim Φ(T )ρ T →q (T −q) . |Δ| = 21−ρ q #Br(V )

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Here, Φ denotes the characteristic polynomial of Frob on H´e2t (VFq, Ql ). Finally, Br(V ) is the Brauer group of V . Remarks 4. i) The characteristic polynomial Φ is independent of the choice of the auxiliary prime l as long as l = p for q = pe [3, Th´eor`eme 1.6]. ii) For a general non-singular, projective surface, the exponent of q in the numerator is b2 (V ) − h02 (V ) − ρ. Here, h02 (V ) denotes the Hodge number. iii) The Artin-Tate conjecture is proven for most K3 surfaces. Most notably, the Tate conjecture implies the Artin-Tate conjecture [11, Theorem 6.1]. iv) The Tate conjecture claims that all zeroes of Φ of the form qζ for ζ a root of unity belong to the algebraic part of H´e2t (VFq, Ql ). I.e., it asserts that the transcendental part never generates a zero of this form. The evidence for this is overwhelming as far as K3 surfaces are concerned. The Tate conjecture is proven for elliptic K3 surfaces [1] and ordinary K3 surfaces [15]. In characteristic diﬀerent from 2 and 3, even more particular cases were successfully treated [16]. v) It is expected that Br(V ) is always a ﬁnite group. This is actually equivalent to the Tate conjecture. In this case, #Br(V ) is automatically a perfect square. We may therefore compute the square class of Δ making use of the ArtinTate conjecture. An unconditional version of the Artin-Tate formula Notation 5. i) For n a positive integer, we will denote by μn the sheaf of n-th roots of unity with respect to the fppf topology. When l is a prime number, d d we put Hfppf (VFq, Tl μ) := lim Hfppf (VFq, μle ). ←e− ii) For l a prime number and M an abelian group, the notation Ml-pow shall be used for the l-power torsion subgroup of M . Similarly, we will write Ml-div ⊆ Ml-pow for the subgroup of inﬁnitely l-divisible elements. iii) We will denote by M Frob and MFrob the invariants, respectively coinvariants, under the operation of Frob on the abelian group M . The coinvariants may have torsion even when M is torsion-free. Write MFrob for the torsion-free quotient. Proposition 6. Let V be a K3 surface over a ﬁnite ﬁeld Fq and l be any prime. Write Φ for the characteristic polynomial of Frob on the ´etale cohomology of VFq and ρ for the multiplicity of q as a zero of Φ. i) Then, the Brauer group Br(V ) is a torsion group. The quotient Br0 (V, l) := Br(V )l-pow / Br(V )l-div is a ﬁnite group of square order. 2 ii) Further, Hfppf (VFq, Tl μ)Frob is a free Zl -module of rank ρ. iii) Denote by Δl the discriminant of the bilinear form 2 2 (VFq, Tl μ)Frob × Hfppf (VFq, Tl μ)Frob −→ Zl Hfppf

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deﬁned by Poincar´e duality. Then, νl (Δl ) = νl

Φ(T ) lim ρ T →q (T −q) q21−ρ #Br0 (V,l)

.

Proof. i) Finiteness of Br0 (V, l) follows immediately from [8, (8.9)]. Further, there is a non-degenerate alternating pairing Br0 (V, l) × Br0 (V, l) → Ql /Zl constructed in [19, Lemma 3.4.1]. This ensures that the group order is a perfect square. ii) and iii) We denote the zeroes of Φ by r1 , . . . , r22 . 2 (VFq, Tl μ) = H´e2t (VFq, Zl (1)) is the same as First case. l = p. Here, H := Hfppf l-adic ´etale cohomology. It is a free Zl -module of rank 22. In the present case, the operation of Frob on H is known to be semi-simple [4, Corollary 1.10]. The eigenvalues are r1 /q, . . . , r22 /q. Assertion ii) follows immediately from this. Further, we have νl (Δl ) = νl (#coker(H Frob → Hom(H Frob , Zl ))), the map being induced by Poincar´e duality. Identifying Hom(H, Zl ) with H, the module Hom(H Frob , Zl ) goes over into HFrob . Here, as shown in [19, Proposition 1.4.2], (HFrob )tors ∼ ho= Br0 (V, l). Further, the order of the cokernel of the canonical momorphism H Frob → HFrob is equal to the l-primary part of (1 − rj /q). rj =q Altogether, this implies the claim. Second case. l = p. Here, some modiﬁcations are necessary which are described in [11]. More concretely, the short exact sequence 2 n 0 → Pic(VFq)⊗Z Zp → Hfppf (VFq, Tp μ) → lim ←− Br(VFq)p → 0 2 immediately shows that H := Hfppf (VFq, Tp μ) is a torsion-free Zp -module. Otherwise, its structure is rather diﬀerent from the previous case. The rank of H is, in general, less than 22. Eigenvalues of Frob are only those rj /q which are units in Qp [11, 1.4]. But this is enough to show ii). Generally, there are unipotent connected quasi-algebraic groups U d and ´etale group schemes Dnd for d = 2, 3 and n 0 which provide short exact sequences d 0 → U d (Fq ) → Hfppf (VFq, μpn) → Dnd (Fq ) → 0. For varying n, the vector groups 3 U (Fq ) are connected by identities. Further, Dn3 = 0. Hence, if dim U 3 = s then 3 #Hfppf (VFq, Tp μ)Frob = q s the operation of Frob being semi-simple. Actually, one has s = 0 except when V is supersingular. Poincar´e duality is available [12, Theorem 5.2 and Corollary 2.7.c)] only at the level of torsion coeﬃcients. Thereby, U 2 (Fq ) and U 3 (Fq ) are dual to each other. 2 1 2 One has ← lim − U (Fq )2= 0 and R lim ←∼ − U (Fq2) = 0 as the connecting homomorphisms are zero. Hence, Hfppf (VFq, Tp μ) = lim Dn (Fq ). Further, it turns out that the ho←− momorphism HFrob → Hom(H Frob , Zp ) does not need to be bijective. It has a cokernel exactly of order q s (cf. [11, Lemma 5.2]). Summarizing, we ﬁnd that Δp has the same p-adic valuation as qs · (1 − rj /q) . νp (rj /q)=0 rj =q

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For iii), it remains to show p-adic units, the product of the following. Up tos−1 the remaining factors, i.e. (1 − rj /q), equals q . This is worked out in [11, νp (rj /q)=0 sec. 7]. 2 Remark 7. The Tate conjecture implies Hfppf (VFq, Tl μ)Frob ∼ = Pic(VFq )⊗ZZl . Further, it is equivalent to Br(V )l-div = 0. Thus, Proposition 6 goes over into the Artin-Tate formula in its usual form. However, the Tate conjecture is unknown in general, even for K3 surfaces. For this reason, we prefer to apply the version of the Artin-Tate formula which holds unconditionally.

4

The Rank-1 Condition

Let V be a K3 surface of degree d over a ﬁnite ﬁeld Fq . Assume that q is a simple zero of the characteristic polynomial of Frob. Then, the Tate conjecture is true for V and the arithmetic Picard rank is equal to 1. The discriminant of Pic(V ) is equal to d. A comparison with the analytic discriminant computed via the ArtinTate formula leads to a non-trivial condition for hypothetical Weil polynomials. Remarks 8. a) This is a condition for rank-1 surfaces of a given degree d. It is not a condition for K3 surfaces, in general. b) The degree of a K3 surface may be any even integer greater than zero. On the other hand, when the arithmetic Picard rank is 1, the number (−q) is necessarily among the Frobenius eigenvalues. Hence, the Artin-Tate formula can generate only even numbers. c) The Artin-Tate conjecture implies the inequality #Br(V )|Δ| ≤ 222−ρ q. Thus, the left hand side is O(q). Observe the following striking consequence. Over the ﬁeld Fq , there is no K3 surface of a square-free degree d > 221 q and arithmetic Picard rank 1. Remark 9. The rank-1 condition may be extended to other situations where a subgroup of the Picard group is known. For this, one has to compare the predicted ranks and discriminants with the known ones.

5

The Field Extension Condition

Notation 10. For q a positive integer, let Φ be a q 2 -Weil polynomial. Then, we will write q c − rjc (c) EΦ := q (c−1)(21−ρ) . q − rj rj =q

Here, rj runs over all the zeroes of Φ. Further, ρ is the multiplicity of the zero q. Observation 11 (Field extension for the characteristic polynomial). Let V be any smooth, projective variety over Fq and j (T − rj ) the characteristic polynomial on H´e2t (VFq, Ql ). Then, the corresponding polynomial for VFqd of Frob d is j (T − rj ).

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Theorem 12. Let V be a K3 surface over Fq . Further, let c be a positive in(c) teger. Then, for Φ the characteristic polynomial of Frob, the expression EΦ is a perfect square in Q. (c) Proof. If there is an rj = q such that rjc = q c then EΦ = 0. Otherwise, for every 2 Frobqc 2 prime l, Hfppf(VFq, Tl μ) is a sublattice of ﬁnite index in Hfppf (VFq, Tl μ)Frobq . In particular, the discriminants diﬀer by a factor being a perfect square. Dividing the Artin-Tate formulas for VFqc and VFq through each other yields that (c) (c) νl (EΦ ) is even for every l. Finally, it is easy to see that EΦ > 0. (c)

Remark 13. Assume the Tate conjecture. Then, EΦ is non-zero if and only if rk Pic(VFq) = rk Pic(VFqc). (c)

Definition 14. We will call the condition on EΦ to be a perfect square, the ﬁeld extension condition for the ﬁeld extension Fqc /Fq . (c)

Explicit computation of the expression EΦ . Our goal is now to describe (c) the square class of EΦ more explicitly. It will turn out that, for an arbitrary Weil (c) polynomial, EΦ may be a non-square. In other words, Theorem 12 provides a non-trivial condition. Remark 15. A priori, there are inﬁnitely many conditions, one for each value of c. The main result of this section is that there is in fact only one condition. Further, this condition may be checked easily. Lemma 16. Let f ∈ Q[T ] be a q 2 -Weil polynomial. Suppose f (q) = 0 and f (−q) = 0. Then, for r1 , . . . , r2l the zeroes of f , 2l q c − rjc (Q∗ )2 ∪ {0} for c odd, ∈ f (−q)(Q∗ )2 ∪ {0} for c even. q − rj j=1 Further, the left hand side is actually in f (−q)(Q∗ )2 for c = 2. Proof. First observe that, for c = 2, the numerators q 2 − rj2 are all non-zero according to the assumption. Hence, the additional assertion is clear once we showed the main one. For that, let us start with the contribution of one pair of complex conjugate roots. Put rj = q(u + iv). Then, the corresponding factor is (q c − rjc )(q c − r cj ) (q c − q c (u + iv)c )(q c − q c (u − iv)c ) = (q − rj )(q − r j ) (q − q(u + iv))(q − q(u − iv) = q 2(c−1)

c−1

(1 − ζck (u + iv))(1 − ζck (u − iv)) .

k=1

Using (u + iv)(u − iv) = 1, we get q 2(c−1)

c−1 k=1

(1 − 2ζck u + ζc2k ) .

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Next, for k = c/2, let us multiply the factors for k and c − k. This yields (1 − 2ζck u + ζc2k )(1 − 2ζcc−k u + ζc2c−2k ) = 2 + 4u2 − 8u Re(ζck ) + 2 Re(ζc2k ) . As Re(ζc2k ) = 2 Re(ζck )2 − 1, the latter term is the same as 4u2 − 8u Re(ζck ) + 4 Re(ζck )2 = (2u − 2 Re(ζck ))2 . Multiplying over all k such that 1 ≤ k < c/2, we ﬁnd a square in Q(u). Consequently, up to the factor for k = c/2, if present, the contribution of the pair {rj , rj } is a square in the resolvent algebra A of f . Multiplying over all l pairs means to form a norm for the extension A/Q. As the norm of a square is a square, the result is a perfect square in Q. For c odd, this completes the argument. For c even, the factors for k = c/2 are still missing. These are the ones for ζck = −1. We ﬁnd the product l

(1 + rj /q)(1 + r j /q) = q −2l f (−q) .

j=1

The assertion follows. Proposition 17. Let Φ be a q -Weil polynomial of even degree. Then, (Q∗ )2 ∪ {0} for c odd, (c) EΦ ∈ qΦ(−q)(Q∗ )2 ∪ {0} for c even. 2

For c = 2, we actually have EΦ ∈ qΦ(−q)(Q∗ )2 . Proof. First case: c is odd. Then, the denominator q (c−1)(21−ρ) is a perfect square. The zeroes (−q) con(c) tribute factors q c−1 which are squares, too. Finally, the contribution to EΦ of the zeroes not being real is a perfect square according to Lemma 16. Second case: c is even. (c) If (−q) is a zero of Φ then EΦ = 0. This coincides with the claim as Φ(−q) = 0. Otherwise, write Φ(T ) = (T − q)ρ f (T ) where f (q) = 0 and f (−q) = 0. By assumption, ρ is even. Hence, q (c−1)(21−ρ) is in the square class of q. Further, the zeroes of Φ diﬀering from q are exactly the zeroes of f . Their contribution is in f (−q)(Q∗ )2 for c = 2 and in f (−q)(Q∗ )2 ∪ {0}, in general. As ρ is even, f (−q)(Q∗ )2 is the same class as Φ(−q)(Q∗ )2 . The assertion follows. (c)

Corollary 18. Let f ∈ Z[T ] be a q 2 -Weil polynomial. i) Then, all ﬁeld extension conditions for Fqc /Fq are satisﬁed if only if the condition for the quadratic extension Fq2 /Fq does hold. ii) For extensions of odd degree, the ﬁeld extension condition is always satisﬁed. iii) If Fq and Fq2 lead to diﬀerent Picard ranks then all the ﬁeld extension conditions are satisﬁed. Remark 19. One might want to study the ﬁeld extension conditions for Fqac /Fqa , i.e., for an extended ground ﬁeld. Our calculations show that this does not lead to new conditions.

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Simplification of the field extension test. Denote by φn the n-th cyclotomic polynomial. Correspondingly, there is the monic polynomial ψn given by ψn (T ) := q ϕ(n) φn (T /q). This is a q 2 -Weil polynomial. Lemma 20. Let n > 1 be an integer. Then, ⎧ ⎨ (Q∗ )2 if n is not a power of 2 , ψn (−q) ∈ 2(Q∗ )2 for n = 2m , m ≥ 2 , ⎩ {0} for n = 2 . Proof. It is well known (see, e.g., [14, sec. 3]) that φn (−1) = 1 unless n is e−1 a power of 2. Further, the formula φ2e (t) = t2 + 1 shows φ2 (−1) = 0 and φ2e (−1) = 2 for e > 1. Observe, ﬁnally, that ϕ(n) is always even for n > 2. Remark 21. The result used here is a very special case of the value of a cyclotomic polynomial at a root of unity. Theorem 22. Let Φ ∈ Z[T ] be a q 2 -Weil polynomial of even degree. Factorize Φ as Φ(T ) = (T − q)r (T + q)s ψn1 (T ) · . . . · ψnk (T )Φ1 (T ) such that Φ1 has no root being a root of unity multiplied by q. Denote by M the number of the powers of 2 among the n1 , . . . , nk . Then, i) if c is odd then EΦ ∈ (Q∗ )2 ∪ {0}. (c)

(c)

ii) If c is even and s > 0 then EΦ = 0 for every c.

iii) Finally, if c is even and s = 0 then EΦ ∈ 2MqΦ1 (−q)(Q∗ )2 ∪ {0}. Furthermore, for c = 2, one actually has (c)

EΦ ∈ 2MqΦ1 (−q)(Q∗ )2 . (2)

Proof. i) and ii) are immediate consequences from Proposition 17. For iii), observe the assumption implies that r is even. In particular, (−2q)r is a perfect square. The assertion now follows from Proposition 17 together with Corollary 20. Remark 23. Suppose Φ ∈ Z[T ] is a q 2 -Weil polynomial of degree 22. In order to show that Φ may not be the characteristic polynomial of the Frobenius for a K3 surface over Fq , it suﬃces to verify that s = 0 and 2MqΦ1 (−q) is a non-square. Example 24. As an example, we look at K3 surfaces of Picard rank 18 such that the Picard group is deﬁned over an extension of odd degree. Then, (−q) is not an eigenvalue of the Frobenius. The transcendental part of the characteristic polynomial is given by (T 4 + aT 3 + bT 2 + aq 2 T + q 4 ). Hence, the ﬁeld extension condition usually requires that (2q 2 − 2aq + b)q is a perfect square. If, however, the cyclotomic factors contain an odd number of type ψ2n then 2(2q 2 − 2aq + b)q is required to be a square.

On Weil Polynomials of K3 Surfaces

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135

The Special Case of a Degree-2 Surface – Twisting

When a K3 surface has a non-trivial automorphism, one can hope to get more conditions by inspecting the corresponding twist. This is the case for degree-2 surfaces. The Twist. Let the K3 surface V be given by the equation w2 = f6 (x, y, z) . Then, for n a non-square in Fq , consider the twist V of V given by nw2 = f6 (x, y, z) . Fact 25. Assume that q, r2 , . . . , r22 are the eigenvalues of Frob for V . Then, the eigenvalues for V are q, −r2 , . . . , −r22 . Proof. For e even, VFqe and V Fqe are isomorphic. When e is odd, we have #V (Fqe ) + #V (Fqe ) = 2·#P2 (Fqe ) = 2q 2e + 2q e + 2 . It is easy to check that the Lefschetz trace formula, applied to the eigenvalues q, −r2 , . . . , −r22 , implies exactly this relation. Proposition 26. Let V be a K3 surface of degree 2 over Fq . Denote by Φ the the corresponding polynomial characteristic polynomial of Frob for V and by Φ for the twist V . does not have a zero at (−q). i) Then, Φ has a simple zero at q if and only if Φ I.e., the rank-1 condition can be applied to the one precisely when the ﬁeld extension condition is non-empty for the other one. ii) The two conditions are equivalent to each other. Proof. i) immediately follows from Fact 25. ii) By assumption, we can write Φ(T ) = (T − q)(T + q)2n−1 f (T ). Here both, f (q) and f (−q) are non-zero. Fact 25 shows, the corresponding polyno ) = (T − q)2n f (−T ). Using these two formulas, one mial for the twist is Φ(T can make the conditions explicit. The rank-1 condition for Φ simply means (2q)2n−1 f (q) = 2 in Q∗/(Q∗ )2 which is equivalent to saying that qf (q) is a per fect square. This is precisely the ﬁeld extension condition for Φ.

7

Examples

Let us show in detail the data for a few examples. Our goal is to illustrate how the Artin-Tate conditions work in practice. Example 27 (A K3 surface of degree 2 over F7 ). Consider the surface V over F7 , given by w2 = y 6 + 3z 6 + 5xz 5 + 5x2 y 4 + x2 z 4 + 3x3 y 3 + x3 z 3 + 5x4 y 2 + x4 z 2 + 5x5 y + 2x6 . Over F7 , . . . , F79 , there are exactly 66, 2 378, 118 113, 5 768 710, 282 535 041,

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13 841 275 877, 678 223 852 225, 33 232 944 372 654, and 1 628 413 551 007 224 points. We claim that rk Pic(VF7 ) = 2. Assuming the characteristic polynomial of the Frobenius has more than two zeroes of the form 7 times a root of unity, [6, Algorithm 22] leaves us with three candidates Φ1 , Φ2 , Φ3 . Φi (t) = t22 − 16 t21 + 140 t20 − 1 029 t19 + 5 831 t18 − 36 015 t17 + 268 912 t16 − 1 882 384 t15 + 11 529 602 t14 − 46 118 408 t13 + ai t12 + bi t11 + ci t10 + (−1)ji [−110 730 297 608 t9 + 1 356 446 145 698 t8 − 10 851 569 165 584 t7 + 75 960 984 159 088 t6 − 498 493 958 544 015 t5 + 3 954 718 737 782 519 t4 − 34 196 685 556 119 429 t3 + 227 977 903 707 462 860 t2 − 1 276 676 260 761 792 016 t + 3 909 821 048 582 988 049]

for j1 = 0,

(a1 , b1 , c1 ) = (161 414 428, −1 129 900 996, 7 909 306 972) ,

j2 = 1,

(a2 , b2 , c2 ) = ( 80 707 214,

0, −3 954 653 486) ,

j3 = 1,

(a3 , b3 , c3 ) = (121 060 821,

0, −5 931 980 229) .

Each of the three polynomials leads to an upper bound of 4 for the rank of the geometric Picard group. All three have roots of absolute value 7, only. Applying the Artin-Tate formula, we ﬁnd the following. Table 1. Hypothetical ranks and discriminants polynomial Φ1 Φ2 Φ3

ﬁeld F7 F49 F7 F49 F7 F49

arithmetic Picard rank 2 2 1 2 1 2

#Br(V )|Δ| 58 4524 4 1996 6 2997

The polynomial Φ1 is excluded by the ﬁeld extension condition as the two values in the rightmost column deﬁne diﬀerent square classes. On the other hand, the rank-1 condition excludes Φ2 and Φ3 since we have a degree-2 example. Thus, relative to the Tate conjecture, geometric Picard rank 2 is proven. Example 28 (continuation). On the same surface, point counting over F710 leads to a number of 79 792 267 067 823 523. For the characteristic polynomial of the Frobenius, we ﬁnd the two candidates Φ4 , Φ5 , Φi (t) = t22 − 16 t21 + 140 t20 − 1 029 t19 + 5 831 t18 − 36 015 t17 + 268 912 t16 − 1 882 384 t15 + 11 529 602 t14 − 46 118 408 t13 + 40 353 607 t12 + ai t11 + (−1)ji [ −1 977 326 743 t10 + 110 730 297 608 t9 − 1 356 446 145 698 t8 + 10 851 569 165 584 t7 − 75 960 984 159 088 t6 + 498 493 958 544 015 t5 − 3 954 718 737 782 51 9t4 + 34 196 685 556 119 429 t3 − 227 977 903 707 462 860 t2 + 1 276 676 260 761 792 016 t − 3 909 821 048 582 988 049]

On Weil Polynomials of K3 Surfaces

137

for j4 = 0, a4 = 0, j5 = 1, and a5 = 564 950 498. Φ4 corresponds to the minus sign in the functional equation, Φ5 to the case of the plus sign. Both candidates, according to the Tate conjecture, imply geometric Picard rank 2. To decide which sign is the right one, one would ﬁrst check the absolute values of the roots. Unfortunately, both polynomials only have roots of absolute value 7. The Artin-Tate formula provides the picture given in the table below. Table 2. Hypothetical ranks and discriminants polynomial Φ4 Φ5

ﬁeld F7 F49 F7 F49

arithmetic Picard rank 1 2 2 2

#Br(V )|Δ| 2 997 55 4125

Thus, Φ5 is excluded by the ﬁeld extension condition. The minus sign in the functional equation is correct. Example 29 (A K3 surface of degree 8 over F3 ). Consider the complete intersection V of the three quadrics in P5F3 , given by q1 , q2 , and q3 , q1 := −xy + xz + xu + xv + xw − y 2 − yz − yv + yw + z 2 + zu + zw − u2 − uw + v 2 + w2 , q2 := −x2 + xy + xz − xv + xw − y 2 + yz − yu − yv + yw − zu − zw + uw − v 2 + vw , q3 := xu − yz . V is smooth and, therefore, a K3 surface. As q3 is of rank 4, V carries an elliptic ﬁbration. There are precisely 14, 98, 794, 6 710, 59 129, 532 460, 4 784 990, 43 049 510, and 387 374 024 points over F3 , . . . , F39 . From these data, let us check whether one can prove rk Pic(VF3 ) = 2. Assume that the characteristic polynomial of the Frobenius has more than two zeroes of the form 3 times a root of unity. Then, [6, Algorithm 22] leaves us with ﬁve polynomials Ψ1 , . . . , Ψ5 , Ψi (t) = t22 − 4 t21 + 27 t18 + 81 t17 − 243 t16 + 6 561 t13 + a1 t12 + b1 t11 + c1 t10 + (−1)ji [531 441 t9 − 14 348 907 t6 + 43 046 721 t5 + 129 140 163 t4 − 13 947 137 604 t + 31 381 059 609]

for

j1 = 0,

(ai , bi , ci ) = (−59 049, 236 196, −531 441) ,

j2 = 0,

(a2 , b2 , c2 ) = (

j3 = 0,

(a3 , b3 , c3 ) = ( 19 683, −236 196, 177 147) ,

j4 = 1,

(a4 , b4 , c4 ) = (−59 049,

0, 531 441) ,

j5 = 1,

(a5 , b5 , c5 ) = (−39 366,

0, 354 294) .

0, −118 098,

0) ,

Applying the Artin-Tate formula to these polynomials, we obtain the following data.

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A.-S. Elsenhans and J. Jahnel Table 3. Hypothetical ranks and discriminants polynomial Ψ1 Ψ2 Ψ3 Ψ4 Ψ5

ﬁeld F3 F9 F3 F9 F3 F9 F3 F9 F3 F9

arithmetic Picard rank 2 4 2 2 2 2 3 4 1 2

#Br(V )|Δ| 24 1116 27 81 28 112 144 1152 2 65

Observe that an elliptic surface of Picard rank 2 automatically has a discriminant of the form (−n2 ) for n an integer. We may therefore exclude everything except for Ψ4 . Note that Ψ2 is, in addition, incompatible with the ﬁeld extension condition. Thus, using the numbers of points over the ﬁelds up to F39 , we only obtain that, either the geometric Picard rank is equal to 2, or Ψ4 is the characteristic polynomial of the Frobenius in which case it is 4. Example 30 (continuation). The number of points over F310 is 34 871 648 631. This additional information reproduces Ψ1 and Ψ4 as possible characteristic polynomials of Frob. Consequently, the minus sign holds in the functional equation and the geometric Picard rank of V is equal to 4.

8

Statistics

We tested the Artin-Tate conditions on samples of K3 surfaces of degrees 2, 4, 6, and 8. The possibilities of computing are limited by the fact that point counting over large ﬁnite ﬁelds is slow. In degree 2, decoupling [6, Algorithm 17] (see also [5]) leads to a substantial speed-up. In higher degrees, one may focus on elliptic K3 surfaces and exploit the fact that point counting on the elliptic ﬁbers is fast. The numbers and particularities of the examples treated are listed in Table 4. Table 4. Numbers of examples computed d d d d

= = = =

2 4 6 8

p=2 1000 rand 1000 rand 1000 rand 1000 rand

p=3 1000 rand 1000 ell 1000 ell 1000 ell

p=5 1000 dec

p=7 1000 dec

dec = decoupled, ell = elliptic, rand = random

The remaining parameters of the surfaces were chosen by a random number generator. We stored the equations and the numbers of points over Fp , . . . , Fp10 in a ﬁle.

On Weil Polynomials of K3 Surfaces

139

Results I. Point counting until Fp9 . First, we tried to show that the geometric Picard-rank was equal to 2 only using the numbers of rational points over Fp , . . . , Fp9 . I.e., we applied [6, Algorithm 22]. This algorithm produces a list of hypothetical Weil polynomials for each surface. If one is able to exclude all of them then, relative to the Tate conjecture, rank 2 is proven. To exclude a particular polynomial, we ﬁrst checked whether the roots are of absolute value p. When a surface was known to be elliptic over Fp , we checked in addition that the predicted Picard rank over Fp was at least equal to 2. Then, we applied the Artin-Tate conditions to the polynomials. We checked the ﬁeld extension condition and the rank-1 condition. For surfaces known to be elliptic over Fp , we observed the fact that arithmetic Picard rank 2 forces the discriminant to be minus a perfect square. The results are summarized in Table 5. Table 5. Distribution of the remaining hypothetical characteristic polynomials d = 2, p = 2 d = 2, p = 3 d = 2, p = 5 d = 2, p = 7 d = 4, p = 2 d = 4, p = 3 d = 6, p = 2 d = 6, p = 3 d = 8, p = 2 d = 8, p = 3

Number of polynomials without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions

0 84 149 116 214 85 158 92 214 40 81 22 53 39 83 16 50 25 29 12 20

1 479 598 480 573 581 651 534 611 532 638 669 785 549 645 713 797 657 723 720 803

2 312 218 285 193 209 169 232 154 303 249 242 161 312 257 217 148 268 239 236 175

3 89 28 88 20 96 20 98 21 87 27 57 1 70 14 47 5 38 5 27 2

4 21 7 24 0 25 2 37 0 29 5 9 0 22 1 7 0 8 4 4 0

5 12 0 4 0 4 0 7 0 8 0 1 0 6 0 0 0 4 0 1 0

6 3 0 3 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0

Results II. Point counting until Fp10 . Using data up to Fp10 , one obtains two hypothetical Weil polynomials for each of the surfaces. The two polynomials correspond to the possible signs in the functional equation (1). One has to exclude one of them. For this, we ﬁrst checked the absolute values of the roots. For surfaces known to be elliptic over Fp , we then tested whether the predicted arithmetic Picard rank is at least 2. Then, we applied the Artin-Tate conditions. We checked the ﬁeld extensions and the rank-1 condition. For elliptic surfaces, supposed to be of arithmetic Picard rank 2, we tested, in addition, whether the predicted discriminant was minus a square. Table 6 shows the number of surfaces with known signs. In the case that the sign is not known, we computed the numbers of points predicted over further extensions of Fp . Comparing these numbers for both hypothetical polynomials indicates whether further point counting would lead to a decision of the sign. We count how often which ﬁelds had to be considered in order to decide the sign.

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A.-S. Elsenhans and J. Jahnel Table 6. Sign decision in the functional equation p d Known signs without A-T Known signs using A-T Remaining unknown signs Data up to Fp11 insuﬃcient Data up to Fp12 insuﬃcient Data up to Fp13 insuﬃcient Data up to Fp14 insuﬃcient Data up to Fp15 insuﬃcient Data up to Fp16 insuﬃcient Data up to Fp17 insuﬃcient Data up to Fp18 insuﬃcient Data up to Fp19 insuﬃcient Data up to Fp20 insuﬃcient

2 2 768 863 137 84 41 22 13 7 4 4 4 2 0

3 2 843 940 60 23 11 5 2 0 0 0 0 0 0

5 2 864 940 60 15 2 1 0 0 0 0 0 0 0

7 2 869 961 39 12 1 0 0 0 0 0 0 0 0

2 4 761 863 137 69 39 24 12 8 3 2 0 0 0

3 4 876 943 57 19 3 2 0 0 0 0 0 0 0

2 6 790 868 132 77 42 20 13 7 2 2 1 1 0

3 6 888 933 67 25 11 2 1 0 0 0 0 0 0

2 8 822 867 133 72 47 24 8 5 4 0 0 0 0

3 8 897 944 56 21 7 2 0 0 0 0 0 0 0

Using these data, we repeated our attempt to prove that the geometric Picard rank is equal to 2. More precisely, we checked whether only two roots of the characteristic polynomial are of the form p times a root of unity. The numbers of surfaces for which we succeeded are listed in Table 7. Table 7. Numbers of rank-2 cases using Fp10 -data p = 2, d = 2 p = 3, d = 2 p = 5, d = 2 p = 7, d = 2 p = 2, d = 4 p = 3, d = 4 p = 2, d = 6 p = 3, d = 6 p = 2, d = 8 p = 3, d = 8

without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T

conditions conditions conditions conditions conditions conditions conditions conditions conditions conditions

rank 2 proven 271 278 397 409 353 360 460 464 132 138 79 79 145 152 74 74 65 65 23 23

rank 2 possible 330 301 460 428 425 382 511 476 197 163 114 81 183 163 101 81 93 74 47 25

Conclusion. The Artin-Tate conditions usually halve the number of cases with unknown signs. Furthermore, they double the number of cases where geometric Picard rank 2 may be proven only using data up to Fp9 . Comparing Table 5 with Table 7, we see, however, that still only about one half of the cases with Picard rank 2 may be detected when counting until Fp9 . Remark 31. Let us ﬁnally mention that the Artin-Tate conditions came to us as a big surprise. It is astonishing that the Artin-Tate formula may be incompatible with itself under ﬁeld extensions. Thus, it seems not entirely unlikely that there are even more constraints and one can still do better.

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References 1. Artin, M., Swinnerton-Dyer, S.P.: The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math. 20, 249–266 (1973) 2. Beauville, A.: Surfaces alg´ebriques complexes, Ast´erisque 54, Soci´et´e Math´ematique de France, Paris (1978) 3. Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974) 4. Deligne, P.: Rel`evement des surfaces K3 en caract´eristique nulle. In: Prepared for publication by Luc Illusie, Algebraic surfaces (Orsay 1976–78). LNM, vol. 868, pp. 58–79. Springer, Berlin (1981) 5. Elsenhans, A.-S., Jahnel, J.: The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds. In: Algorithmic Number Theory (ANTS 7), pp. 317–332. Springer, Berlin (2006) 6. Elsenhans, A.S., Jahnel, J.: K3 surfaces of Picard rank one and degree two. In: Algorithmic Number Theory (ANTS 8), pp. 212–225. Springer, Berlin (2008) 7. Elsenhans, A.S., Jahnel, J.: On the computation of the Picard group for K3 surfaces (2009) (preprint) 8. Grothendieck, A.: Le groupe de Brauer, III: Exemples et compl´ements. In: Grothendieck, A. (ed.) Dix expos´es sur la Cohomologie des sch´emas, pp. 88–188. North-Holland, Amsterdam (1968) 9. Honda, T.: Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan 20, 83–95 (1968) 10. van Luijk, R.: K3 surfaces with Picard number one and infinitely many rational points. Algebra & Number Theory 1, 1–15 (2007) 11. Milne, J.S.: On a conjecture of Artin and Tate. Ann. of Math. 102, 517–533 (1975) ´ 12. Milne, J.S.: Duality in the flat cohomology of a surface. Ann. Sci. Ecole Norm. Sup., 4e s´erie 9, 171–201 (1976) ´ 13. Milne, J.S.: Etale Cohomology. Princeton University Press, Princeton (1980) 14. Motose, K.: On values of cyclotomic polynomials. VIII. Bull. Fac. Sci. Technol. Hirosaki Univ. 9, 15–27 (2006) 15. Nygaard, N.O.: The Tate conjecture for ordinary K3 surfaces over finite fields. Invent. Math. 74, 213–237 (1983) 16. Nygaard, N.O., Ogus, A.: Tate’s conjecture for K3 surfaces of finite height. Ann. of Math. 122, 461–507 (1985) 17. Tate, J.: Conjectures on algebraic cycles in l-adic cohomology. In: Motives, Proc. Sympos. Pure Math., vol. 55(1), pp. 71–83. Amer. Math. Soc., Providence (1994) 18. Zarhin, Y.I.: Transcendental cycles on ordinary K3 surfaces over finite fields. Duke Math. J. 72, 65–83 (1993) 19. Zarhin, Y.I.: The Brauer group of an abelian variety over a finite field. Izv. Akad. Nauk SSSR Ser. Mat. 46, 211–243 (1982) (Russian) 20. Zeilberger, D.: A combinatorial proof of Newtons’s identities. Discrete Math. 49, 319 (1984)

Class Invariants by the CRT Method Andreas Enge1 and Andrew V. Sutherland2 2

1 INRIA Bordeaux–Sud-Ouest, France Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract. We adapt the CRT approach for computing Hilbert class polynomials to handle a wide range of class invariants. For suitable discriminants D, this improves its performance by a large constant factor, more than 200 in the most favourable circumstances. This has enabled record-breaking constructions of elliptic curves via the CM method, including examples with |D| > 1015 .

1

Introduction

Every ordinary elliptic curve E over a ﬁnite ﬁeld Fq has complex multiplication by an imaginary quadratic order O, by which we mean that the endomorphism ring End(E) is isomorphic to O. The Deuring lifting theorem implies that E ˆ is the reduction of an elliptic curve E/C that also has complex multiplication by O. Let K denote the fraction ﬁeld of O. The j-invariant of Eˆ is an algebraic integer whose minimal polynomial over K is the Hilbert class polynomial HD , where D is the discriminant of O. Notably, the polynomial HD actually lies in Z[X], and its splitting ﬁeld is the ring class field KO for the order O. Conversely, an elliptic curve E/Fq with complex multiplication by O exists whenever q satisﬁes the norm equation 4q = t2 − v 2 D, with t, v ∈ Z and t ≡ 0 modulo the characteristic of Fq . In this case HD splits completely over Fq , and its roots are precisely the j-invariants of the elliptic curves E/Fq that have complex multiplication by O. Such a curve has q + 1 ± t points, where t is determined, up to a sign, by the norm equation. With a judicious selection of D and q one may obtain a curve with prescribed order. This is known as the CM method. The main challenge for the CM method is to obtain the polynomial HD , which has degree equal to the class number h(D), and total size O(|D|1+ ). There are three approaches to computing HD , all of which, under reasonable assumptions, can achieve a running time of O(|D|1+ ). These include the complex analytic method [12], a p-adic algorithm [9, 7], and an approach based on the Chinese Remainder Theorem (CRT) [2]. The ﬁrst is the most widely used, and it is quite eﬃcient; the range of discriminants to which it may be applied is limited not by its running time, but by the space required. The polynomial HD is already likely to exceed available memory when |D| > 109 , hence one seeks to apply the CM method to alternative class polynomials that have smaller coeﬃcients than HD . This makes computations with |D| > 1010 feasible. Recently, a modiﬁed version of the CRT approach was proposed that greatly reduces the space required for the CM method [30]. Under the Generalised Riemann Hypothesis (GRH), this algorithm is able to compute HD mod P using G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 142–156, 2010. c Springer-Verlag Berlin Heidelberg 2010

Class Invariants by the CRT Method

143

O(|D|1/2+ log P ) space and O(|D|1+ ) time. (Here and in the following, all complexity estimates refer to bit operations.) The reduced space complexity allows it to handle much larger discriminants, including examples with |D| > 1013 . An apparent limitation of the CRT approach is that it depends on some speciﬁc features of the j-function. As noted in [2], this potentially precludes it from computing class polynomials other than HD . The purpose of the present article is to show how these obstructions may be overcome, allowing us to apply the CRT method to many functions other than j, including two inﬁnite families. Subject to suitable constraints on D, we may then compute a class polynomial with smaller coeﬃcients than HD (by a factor of up to 72), and, in certain cases, with smaller degree (by a factor of 2). Remarkably, the actual running time with the CRT method is typically better than the size diﬀerence would suggest. Fewer CRT moduli are needed, and we may choose a subset for which the computation is substantially faster than on average. We start §2 with a brief overview of the CRT method, and then describe a new technique to improve its performance, which also turns out to be crucial for certain class invariants. After discussing families of invariants in §3, we consider CRT-based approaches applicable to the diﬀerent families and give a general algorithm in §4. Computational results and performance data appear in §5.

2 2.1

Hilbert Class Polynomials via the CRT The Algorithm of Belding, Br¨ oker, Enge, Lauter and Sutherland

The basic idea of the CRT-based algorithm for Hilbert class polynomials is to compute HD modulo many small primes p, and then lift its coeﬃcients by Chinese remaindering to integers, or to their reductions modulo a large (typically prime) integer P , via the explicit CRT [4, Thm. 3.1]. The latter approach sufﬁces for most applications, and while it does not substantially reduce the running time (the same number of small primes is required), it can be accomplished using only O(|D|1/2+ log P ) space with the method of [30, §6]. For future reference, we summarise the algorithm to compute HD mod p for a prime p that splits completely in the ring class ﬁeld KO . Let h = h(D). Algorithm 1 (Computing HD mod p) 1. Find the j-invariant j1 of an elliptic curve E/Fp with End(E) ∼ = O. 2. Enumerate the other roots j2 , . . . , jh of HD mod p. 3. Compute HD (X) mod p = (X − j1 ) · · · (X − jh ). The ﬁrst step is achieved by varying j1 (systematically or randomly) over the elements of Fp until it corresponds to a suitable curve; details and many practical improvements are given in [2, 30]. The third step is a standard building block of computer algebra. Our interest lies in Step 2.

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Enumerating the Roots of HD mod p

2.2

The key idea in [2] leading to a quasi-linear complexity is to apply the Galois action of Cl(O) Gal(KO /K). The group Cl(O) acts on the roots of HD , and when p splits completely in KO there is a corresponding action on the set EllO (Fp ) = {j1 , . . . , jh } containing the roots of HD mod p. For an ideal class [a] in Cl(O) and a j-invariant ji ∈ EllO (Fp ), let us write [a]ji for the image of ji under the Galois action of [a]. We then have EllO (Fp ) = {[a]j1 : [a] ∈ Cl(O)}. As in [30, §5], we use a polycyclic presentation deﬁned by a sequence of ideals l1 , . . . , lm with prime norms 1 , . . . , m whose classes generate Cl(O). The relative order rk is the least positive integer for which [lrkk ] ∈ [l1 ], . . . , [lk−1 ]. We may then uniquely write [a] = [le11 ] · · · [lemm ], with 0 ≤ ek < rk . To maximise performance, we use a presentation in which 1 < · · · < m , with each k as small as possible subject to rk > 1. Note that the relative order rk divides the order nk of [lk ] in Cl(O), but for k > 1 we can (and often do) have rk < nk . For each ji ∈ EllO (Fp ) and each O-ideal l of prime norm , the j-invariant [l]ji corresponds to an -isogenous curve, which we may obtain as a root of Φ (ji , X), where Φ ∈ Z[J, J ] is the classical modularpolynomial [31, §69]. The polynomial Φ has the pair of functions j(z), j(z) as roots, and parameterises isogenies of degree . Fixing an isomorphism End(E) ∼ = O, we let π ∈ O denote the Frobenius endomorphism. When the order Z[π] is maximal at , the univariate polynomial Φ (ji , X) ∈ Fp [X] has exactly two roots [l]ji and [¯l]ji when splits in O, and a single root [l]ji if is ramiﬁed [25, Prop. 23]. To simplify matters, we assume here that Z[π] is maximal at each k , but this is not necessary, see [30, §4]. We may enumerate EllO (Fp ) = {[a]j1 : [a] ∈ [l1 ], . . . , [lm ]} via [30, Alg. 1.3]: Algorithm 2 (Enumerating EllO (Fp ) — Step 2 of Algorithm 1) 1. Let j2 be an arbitrary root of Φm (j1 , X) in Fp . 2. For i from 3 to rm , let ji be the root of Φm (ji−1 , X)/(X − ji−2 ) in Fp . 3. If m > 1, then for i from 1 to rm : Recursively enumerate the set {[a]ji : [a] ∈ [l1 ], . . . , [lm−1 ]}. In general there are two distinct choices for j2 , but either will do. Once j2 is chosen, j3 , . . . , jrm are determined. The sequence (j1 , . . . , jrm ) corresponds to a path of m -isogenies; we call this path an m -thread. The choice of j2 in Step 1 may change the order in which EllO (Fp ) is enumerated. Three of the sixteen possibilities when m = 2, r1 = 4, and r2 = 3 are shown below; we assume [l32 ] = [l1 ], and label each vertex [le2 ]j1 by the exponent e. 0 l2 1 l2 2

l1 l1 l1

3 4 5

l1 l1 l1

6 7 8

l1 l1 l1

9

0 l2

10

1 l2

11

2

¯l1 ¯l1 l1

9 10 5

¯l1 ¯l1 l1

6 7 8

¯l1 ¯l1 l1

3

0 ¯l2

4

11 ¯l2

11

10

l1 ¯l1 l1

3 8 1

l1 ¯l1 l1

6 5 4

l1 ¯l1 l1

9 2 7

Bold edges indicate where a choice was made. Regardless of these choices, Algorithm 2 correctly enumerates EllO (Fp ) in every case [30, Prop. 5].

Class Invariants by the CRT Method

2.3

145

Finding Roots with Greatest Common Divisors (gcds)

The potentially haphazard manner in which Algorithm 2 enumerates EllO (Fp ) is not a problem when computing HD , but it can complicate matters when we wish to compute other class polynomials. We could distinguish the actions of l and ¯l using an Elkies kernel polynomial [10], as suggested in [7, §5], however this slows down the algorithm signiﬁcantly. An alternative approach using polynomial gcds turns out to be much more eﬃcient, and actually speeds up Algorithm 2, making it already a useful improvement when computing HD . We need not distinguish the actions of l and ¯l at this stage, but we wish to ensure that our enumeration of EllO (Fp ) makes a consistent choice of direction each time it starts an -thread. The ﬁrst -thread may be oriented arbitrarily, but for each subsequent -thread (j1 , j2 , . . . , jr ), we apply Lemma 1 below. This allows us to “square the corner” by choosing j2 as the unique common root of Φ (X, j1 ) and Φ (X, j2 ), where (j1 , . . . , jr ) is a previously computed -thread and j1 is -isogenous to j1 . The edge (j1 , j1 ) lies in an -thread that has already been computed, for some > . j1 l j1

l j 2 l

l j2

l j 3

l

· · · l jr

j1 l j1

l j 2

l j 3

l

l

l

j2

l

j3

l

· · · l jr

l

l · · · l jr

Having computed j2 , we could compute j3 , . . . , jr as before, but it is usually better to continue using gcds, as depicted above. Asymptotically, both rootﬁnding and gcd computations are dominated by the O(2 M(log p)) time it takes to instantiate Φ (X, ji ) mod p, but in practice is small, and we eﬀectively gain a factor of O(log p) by using gcds when ≈ . This can substantially reduce the running time of Algorithm 2, as may be seen in Table 1 of §5. With the gcd approach described total number of root-ﬁnding m above,the m operations can be reduced from k=1 rk to k=1 rk . When m is large, this is a big improvement, but it is no help when m = 1, as necessarily occurs when h(D) is prime. However, even in this case we can apply gcds by looking for an auxiliary ideal l1 , with prime norm 1 , for which [l1 ] = [le1 ]. When r1 is large, such an l1 is easy to ﬁnd, and we may choose the best combination of 1 and e available. This idea generalises to k -threads, where we seek [lk ] ∈ [l1 ] . . . , [lk ]\[l1 ] . . . , [lk−1 ]. Lemma 1. Let j1 , j2 ∈ EllO (Fp ), and let 1 , 2 = p be distinct primes with 421 22 < |D|. Then gcd Φ1 (j1 , X), Φ2 (j2 , X) has degree at most 1. Proof. It follows from [25, Prop. 23] that Φ1 (X, j1 ) and Φ2 (X, j2 ) have at most two common roots in the algebraic closure Fp , which in fact lie in EllO (Fp ). If there are exactly two, then both 1 = l1 l1 and 2 = l2 l2 split in O, and one of l21 l22 or l21¯l22 is principal with a non-rational generator. We thus have a norm equation 421 22 = a2 − b2 D with a, b ∈ Z and b = 0, and the lemma follows.

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Class Invariants

Due to the large size of HD , much eﬀort has been spent seeking smaller generators of KO . For a modular function f and O = Z[τ ], with τ in the upper half plane, we call f (τ ) a class invariant if f (τ ) ∈ KO . The class polynomial for f is HD [f ](X) = (X − [a]f (τ )) . [a]∈Cl(O)

The contemporary tool for determining class invariants is Shimura’s reciprocity law; see [28, Th. 4] for a fairly general result. Class invariants arising from many diﬀerent modular functions have been described in the literature; we brieﬂy summarise some of the most useful ones. Let η be Dedekind’s function, and let ζn = exp(2πi/n). Weber considered z+1 √ η(2z) η z2 −1 η 2 , f1 (z) = , f2 (z) = 2 , f = ζ48 η(z) η(z) η(z) √ 3 powers of which yield class invariants when D j, which 2 = −1, and also γ2 = is a class invariant whenever 3 D. The Weber functions can be generalised [15, 16, 21, 20, 23], and we have the simple and double η-quotients z z z η η N p1 η p2 wN (z) = ; wp1 ,p2 = with N = p1 p2 , z η(z) η(z) η p1 p2

where p1 and p2 are primes. Subject to constraints on D, including that no prime dividing N is inert in O, suitable yield class invariants, powers of these functions see [15, 16]. For s = 24/ gcd 24, (p1 − 1)(p2 − 1) , the canonical power wsp1 ,p2 0 is invariant under the Fricke involution W |N : z → −N z for Γ (N ), equivalently, the Atkin-Lehner involution of level N , by [17, Thm. 2]. The theory of [28] applies to any functions for Γ0 (N ), in particular to those of primelevel N invariant under the Fricke involution, which yield class invariants D when N = −1. Atkin developed a method to compute such functions AN , which are conjectured to have a pole of minimal order at the unique cusp [10, 26]. These are used in the SEA algorithm, and can be found in Magma or Pari/GP. The functions above all yield algebraic integers, so HD [f ] ∈ OK [X]. Except for weN or when gcd(N, D) = 1, in which cases additional restrictions may apply, one actually has HD [f ] ∈ Z[X], cf. [16, Cor. 3.1]. The (logarithmic) height of HD [f ] = ai X i is log max |ai |, which determines the precision needed to compute the ai . We let cD (f ) denote the ratio of the heights of HD [j] and HD [f ]. With c(f ) = lim|D|→∞ cD (f ), we have: c(γ2 ) = 3; c(f) = 72 (when D 2 = 1); c(weN ) =

24(N + 1) ; e(N − 1)

c(wsp1 ,p2 ) =

12ψ(p1 p2 ) ; s(p1 − 1)(p2 − 1)

c(AN ) =

N +1 , 2|vN |

where e divides the exponent s deﬁned above, vN is the order of the pole of AN at the cusp, and ψ(p1 p2 ) is (p1 + 1)(p2 + 1) when p1 = p2 , and p1 (p1 + 1) when

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p1 = p2 . Morain observed in [27] that c(A71 ) = 36, which is so far the best value known when D 2 = −1. We conjecture that in fact for all primes N > 11 with +1 , and that for N ≡ −1 mod 60 we N ≡ 11 mod 60 we have c(AN ) = 30 NN−11 have c(AN ) = 30. This implies that given an arbitrary discriminant D, we can always choose N so that AN yields class invariants with cD (AN ) ≥ 30 + o(1). When the prime divisors of N are all ramiﬁed in K, both wp1 ,p2 and AN yield class polynomials that are squares in Z[X], see [11, §1.6] and [18]. Taking the square root of such a class polynomial reduces both its degree and its height by a factor of 2. For a composite fundamental discriminant D (the most common case), this applies to HD [AN ] for any prime N | D. In the best case, D is divisible by 71, and we obtain a class polynomial that is 144 times smaller than HD . 3.1

Modular Polynomials

Each function f (z) considered above is related to j(z) by a modular polynomial Ψf ∈ Z[F, J] satisfying Ψf (f (z), j(z)) = 0. For primes not dividing the level N , we let Φ,f denote the minimal polynomial satisfying Φ ,f (f (z), f (z)) = 0; it is a factor of ResJ ResJ (Φ (J, J ), Ψf (F, J)), Ψf (F , J ) , and as such, an element of Z[F, F ]. Thus Φ,f generalises the classical modular polynomial Φ = Φ,j . The polynomial Φ,f has degree d(+1) in F and F , where d divides degJ Ψf , see [6, §6.8], and 2d divides degJ Ψf when f is invariant under the Fricke involution. In general, d is maximal, and d = 1 is achievable only in the relatively few cases where X0 (N ), respectively X0+ (N ), is of genus 0 and, moreover, f is a hauptmodul, that is, it generates the function ﬁeld of the curve. Happily, this includes many cases of practical interest. The polynomial Ψf characterises the analytic function f in an algebraic way; when d = 1, the polynomials Φ and Φ,f algebraically characterise -isogenies between elliptic curves given by their j-invariants, or by class invariants derived from f , respectively. These are key ingredients for the CRT method.

4

CRT Algorithms for Class Invariants

To adapt Algorithm 1 to class invariants arising from a modular function f (z) other than j(z), we only need to consider Algorithm 2. Our objective is to enumerate the roots of HD [f ] mod p for suitable primes p, which we are free to choose. This may be done in one of two ways. The most direct approach computes an “f -invariant” f1 , corresponding to j1 , then enumerates f2 , . . . , fh using the modular polynomials Φ,f . Alternatively, we may enumerate j1 , . . . , jh as before, and from these derive f1 , . . . , fh . The latter approach is not as eﬃcient, but it applies to a wider range of functions, including two inﬁnite families. Several problems arise. First, an elliptic curve E/Fp with CM by O unambiguously deﬁnes a j-invariant j1 = j(E), but not the corresponding f1 . The f1 we seek is a root of ψf (X) = Ψf (X, j1 ) mod p, but ψf may have other roots, which may or may not be class invariants. The same problem occurs for the

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p-adic lifting algorithm and can be solved generically [6, §6]; we describe some more eﬃcient solutions, which are in part speciﬁc to certain types of functions. When ψf has multiple roots that are class invariants, these may be roots of distinct class polynomials. We are generally happy to compute any one of these, but it is imperative that we compute the reduction of “the same” class polynomial HD [f ] modulo each prime p. The lemma below helps to address these issues for at least two inﬁnite families of functions: the double η-quotients wp1 ,p2 and the Atkin functions AN . 0 Lemma 2. Let f be a modular function Γ (N ), invariant under the Fricke −1for involution W |N , such that f (z) and f z have rational q-expansions. Let the imaginary quadratic order O have conductor coprime to N and contain an √ √ B02 −D 0+ D ideal n = N, B0 +2 D . Let A0 = 4N and τ0 = −B2A , and assume that 0 gcd(A0 , N ) = 1. Then f (τ0 ) is a class invariant, and if f (τ ) is any of its conjugates under the action of Gal(KO /K) we have and Ψf f (τ ), [n]j(τ ) = 0. Ψf f (τ ), j(τ ) = 0 Proof. By deﬁnition, Ψf f (z), j(z) = 0. Applying the Fricke involution yields = Ψf f (z), j Nz . The 0 = Ψf ((W |N f )(z), (W |N j)(z)) = Ψf f (z), j −N z value f (τ0 ) is a class invariant by [28, Th. 4]. By the same result, we may assume −B+√D that τ is the basis quotient of an ideal a = A, with gcd(A, N ) = 1 2 √ τ −B+ D . It is the basis quotient of an = AN, and B ≡ B0 mod 2N. Then N 2 τ , and replacing z above by τ completes the proof. follows that [n]j(τ ) = j N

If we arrange the roots of HD into a graph of n-isogeny cycles corresponding to the action of n, the lemma yields a dual graph deﬁned on the roots of HD [f ], in which vertices f (τ ) correspond to edges j(τ ), [n]j(τ ) . In computational terms, f (τ ) is a root of gcd Ψf X, j(τ ) , Ψf X, [n]j(τ ) . Generically, we expect this gcd to have no other roots modulo primes p that split completely in KO . For a ﬁnite number of such primes, there may be additional roots. We have observed this for p dividing the conductor of the order generated by f (τ ) in the maximal order of KO . Such primes may either be excluded from our CRT computations, or addressed by one of the techniques described in §4.3. 4.1

Direct Enumeration

When the polynomials Φ,f have degree + 1 we can apply Algorithm 2 with essentially no modiﬁcation; the only new consideration is that must not divide the level N , but we can exclude such when choosing a polycyclic presentation for Cl(O). When the degree is greater than + 1 the situation is more complex, moreover the most eﬃcient algorithms for computing modular polynomials do not apply [8, 13], making it diﬃcult to obtain Φ,f unless is very small. Thus in practice we do not use Φ,f in this case; instead we apply the methods of §4.3 or §4.4. For the remainder of this subsection and the next we assume that we do have polynomials Φ,f of degree + 1 with which to enumerate f1 , . . . , fh , and

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consider how to determine a starting point f1 , given the j-invariant j1 = j(E) of an elliptic curve E/Fp with CM by O. When ψf (X) = Ψf (X, j1 ) mod p has only one root, our choice of f1 is immediately determined. This is usually not the case, but we may be able to ensure it by restricting our choice of p. As an example, for f = γ2 with 3 D, if we require that p ≡ 2 mod 3, then f1 is the unique cube root of j1 in Fp . If we additionally have D ≡ 1 mod 8 and p ≡ 3 mod 4, then the equation γ2 = (f24 − 16)/f8 uniquely determines the square of the Weber f function, by [8, Lem. 7.3]. To treat f itself we need an additional trick described in §4.2. The next simplest case occurs when only one of the roots of ψf is a class invariant. This necessarily happens when f is invariant under the Fricke involution and all the primes dividing N are ramiﬁed in O. In the context of Lemma 2, each root of HD [f ] then corresponds to an isolated edge j(τ ), [n]j(τ ) in the n-isogeny graph on the roots of HD , and we compute f1 as the unique root of ¯, and each f (τ ) occurs twice gcd Ψf (X, j1 ), Ψf (X, [n]j1 ) . In this situation n = n as a root of HD [f ]. By using a polycyclic presentation for Cl(O)/[n] rather than Cl(O), we enumerate each double root of HD [f ] mod p just once. Even when ψf has multiple roots that are class invariants, it may happen that they are all roots of the same class polynomial. This applies to the Atkin functions f = AN . When N is a split prime, there are two N -isogenous pairs (j1 , [n]j1 ) and ([¯ n]j1 , j1 ) in EllO (Fp ), and under Lemma 2 these correspond to roots f1 and [¯ n]f1 of ψf . Both are roots of HD [f ], and we may choose either. The situation is slightly more complicated for the double η-quotients wp1 ,p2 , with N = p1 p2 composite. If p1 = p1 ¯ p1 and p2 = p2 ¯p2 both split and p1 = p2 , then there are four distinct N -isogenies corresponding to four roots of ψf . Two of these roots are related by the action of [n] = [p1 p2 ]; they belong to the same class polynomial, which we choose as HD [f ] mod p. The other two are related by [p1 ¯ p2 ] and are roots of a diﬀerent class polynomial. We make an arbitrary choice for f1 , explicitly compute [n]f1 , and then check whether it occurs among the other three roots; if not, we correct the initial choice. The techniques of §4.3 may be used to eﬃciently determine the action of [n]. Listed below are some of the modular functions f for which the roots of HD [f ] mod p may be directly enumerated, with suﬃcient constraints on D and p. In each case p splits completely in KO and D < −4N 2 has conductor u. (1) γ2 , with 3 D and p ≡ 2 mod 3; (2) f2 , with D ≡ 1 mod 8, 3 D, and p ≡ 11 mod 12; (3) wsN , for N ∈ {3, 5, 7, 13} and s = 24/ gcd(24, N − 1), with N | D and N u; (4) w25 , with 3 D, 5 | D, and 5 u; (5) AN , for N ∈ {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}, with and N u.

D N

= −1

(6) wsp1 ,p2 , for (p1 , p2 ) ∈ {(2, 3), (2, 5), (2, 7), (2, 13), (3, 5), (3, 7), (3, 13), (5, 7)} and s = 24/ gcd 24, (p1 − 1)(p2 − 1) , with pD1 , pD2 = −1 and p1 , p2 u. (7) w63,3 with D 3 = 1 and 3 u.

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The Trace Trick

In §4.1 we were able to treat the square of the Weber f function but not f itself. To remedy this, we generalise a method suggested to us by Reinier Br¨oker. We consider the situation where there are two modular functions f and f that are roots of Ψf (X, j(z)), both of which yield class invariants for O, and we wish to apply the direct enumeration approach. We assume that p is chosen so that ψf (X) = Ψf (X, j1 ) mod p has exactly two roots, and depending on which root we take as f1 , we may compute the reduction of either HD [f ](X) or HD [f ](X) modulo p. In the case of Weber f, we have f = −f , and HD [f ] diﬀers from HD [f ] only in the sign of every other coeﬃcient. Consider a ﬁxed coeﬃcient ai of HD [f ](X) = ai X i ; most of the time, the trace t = −ah−1 = f1 + · · · + fh will do (if f = −f , we need to use ai with i ≡ h mod 2). The two roots f1 and f1 lead to two possibilities t and t modulo p. However, the elementary symmetric functions T1 = t + t and T2 = tt are unambiguous modulo p. Computing these modulo many primes p yields T1 and T2 as integers (via the CRT), from which t and t are obtained as roots of the quadratic equation X 2 − T1 X + T2 . If these are diﬀerent, we arbitrarily pick one of them, which, going back, determines the set of conjugates {f1 , . . . , fh } or {f1 , . . . , fh } to take modulo each of the primes p t − t . In the unlikely event that they are the same (the suspicion t = t being conﬁrmed after, say, looking at the second prime), we need to switch to a diﬀerent coeﬃcient ai . If f and f diﬀer by a simple transformation (such as f = −f ), the second set of conjugates and the value t are obtained essentially for free. As a special case, when h is odd and the class invariants are units (as with Weber f), we can simply ﬁx t = a0 = 1, and need not compute T1 = 0 and T2 = −1. The key point is that the number of primes p we use to determine t is much less than the number of primes we use to compute HD [f ]. Asymptotically, the logarithmic height of the trace is smaller than the height bound we use for HD [f ] by a factor quasi-linear in log |D|, under the GRH. In practical terms, determining t typically requires less than one tenth of the primes used to compute HD [f ], and these computations can be combined. The approach described above generalises immediately to more than two roots, but this case does not occur for the functions we examine. Unfortunately it can be used only in conjunction with the direct enumeration approach of §4.1; otherwise we would have to consistently distinguish not only between f1 and f1 , but also between fi and fi for i = 2, . . . , h. 4.3

Enumeration via the Fricke Involution

For functions f to which Lemma 2 applies, we can readily obtain the roots of HD [f ] mod p without using the polynomials Φ,f . We instead enumerate the roots of HD mod p (using the polynomials Φ ), and arrange them into a graph G of n-isogeny cycles, where n is the ideal of norm N appearing in Lemma 2. We then obtain roots of HD [f ] mod p by computing gcd Ψf (X, ji ), Ψf (X, [n]ji ) for each edge (ji , [n]ji ) in G.

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The graph G is composed of h/n cycles of length n, where n is the order of [n] in Cl(O). We assume that the O-ideals of norm N are all non-principal and inequivalent (by requiring |D| > 4N 2 if needed). When every prime dividing N is ramiﬁed in O we have n = 2; as noted in §4.1, every root of HD [f ] then occurs with multiplicity 2, and we may compute the square-root of HD [f ] by taking each root just once. Otherwise we have n > 2. Let [l1 ], . . . , [lm ] be a polycyclic presentation for Cl(O) with relative orders √ r1 , . . . , rm , as in §2.2. For k from 1 to m let us ﬁx lk = k , −Bk2+ D with Bk ≥ 0. To each vector e = (e1 , . . . , em ) with 0 ≤ ek < rk , we associate a unique root je enumerated by Algorithm 2, corresponding to the path taken from j1 to je , where ek counts steps taken along an k -thread. For o = (0, . . . , 0) we have jo = j1 , and in general je = [lσ1 1 e1 · · · lσmm em ]jo , with σk = ±1. Using the method of §2.3 to consistently orient the k -threads ensures that each σk depends only on the orientation of the ﬁrst k -thread. To compute the graph G we must determine the signs σk . For those [lk ] of order 2, we let σk = 1. We additionally ﬁx σk = 1 for the least k = k0 (if any) for which [lk ] has order greater than 2, since we need not distinguish the actions of n ¯. It suﬃces to show how to determine σk , given that we know σ1 , . . . , σk−1 . and n We may assume [lk0 ] and [lk ] both have order greater than 2, with k0 < k ≤ m. Let l be an auxiliary ideal of prime norm such that [l] = [ab] = [le11 · · · lekk ], with 0 ≤ ei < ri , where b = lekk , and [a] and [b] have order greater than 2. Our ˇ assumptions guarantee that such an l exists, by the Cebotarev density theorem, and under the GRH, is relatively small [1]. The fact that [a] and [b] have order ¯ is distinct from [l] and its inverse. It follows that greater than 2 ensures that [ab] σk = 1 if and only if Φ (jo , je ) = 0, where e = (e1 , . . . , ek , 0, . . . , 0). Having determined the σk , we compute the unique vector v = (v1 , . . . , vm ) for which [n] = [lσ1 1 v1 · · · lσmm vm ]. We then have [n]jo = jv , yielding the edge (jo , jv ) of G. In general, we obtain the vector corresponding to [n]je by computing e + v xk−1 and using relations [lrkk ] = [lx1 1 · · · lk−1 ] to reduce the result, cf. [30, §5]. This method may be used with any function f satisfying Lemma 2, and in particular it applies to two inﬁnite families of functions: D (8) AN , for N > 2 prime, with N = −1 and N u. s (9) wp1 ,p2 , for p1 , p2 primes not both 2, with pD1 , pD2 = −1 and p1 , p2 u. As above, u denotes the conductor of D < −4N 2 . As noted earlier, for certain primes p we may have diﬃculty computing the edges of G when gcd Ψf (X, ji ), Ψf (X, [n]ji ) has more than one root in Fp . While we need not use such primes, it is often easy to determine the correct root. Here we give two heuristic techniques for doing so. The ﬁrst applies when N is prime, as with the Atkin functions. In this case problems can arise when HD [f ] has repeated roots modulo p. By Kummer’s criterion, this can happen only when p divides the discriminant of HD [f ], and even then, a repeated root x1 is only actually a problem when it corresponds to two alternating edges in G, say (j1 , j2 ) and (j3 , j4 ), with the edge (j2 , j3 ) between them.

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In this scenario we will get two roots x1 and x2 of gcd Ψf (X, j2 ), Ψf (X, j3 ) . But if we already know that x1 corresponds to (j1 , j2 ), we can unambiguously choose x2 . In each of the N -isogeny cycles of G, it is enough to ﬁnd a single edge that yields a unique root. If no such edge exists, then every edge must yield the same two roots x1 and x2 , and we count each with multiplicity n/2. The second technique applies when the roots of HD [f ] are units, as with the double η-quotients [16, Thm. 3.3]. The product of the roots is then ±1. Assuming that the number of edges in G for which multiple roots arise is small (it is usually zero, and rarely more than one or two), we simply test all the possible choices of roots and see which yield ±1. If only one combination works, then the correct choices are determined. This is not guaranteed to happen, but in practice it almost always does. 4.4

A General Algorithm

We now brieﬂy consider the case of an arbitrary modular function f of level N , and sketch a general algorithm to compute HD [f ] with the CRT method. Let us assume that f (τ ) is a class invariant, and let D be the discriminant and u the conductor of the order O = [1, τ ]. The roots of Ψf (X, j(τ )) ∈ KO [X] lie in the ray class ﬁeld of conductor uN over K, and some number n of these, including f (τ ), actually lie in the ring class ﬁeld KO . We may determine n using the method described in [6, §6.4], which computes the action of (O/N O)∗ /O∗ on the roots of Ψf (X, j(τ )). We note that the complexity of this task is essentially ﬁxed as a function of |D|. Having determined n, we use Algorithm 2 to enumerate the roots j1 , . . . , jh of HD mod p as usual, but if for any ji we ﬁnd that Ψf (X, ji ) mod p does not have (1) (n) exactly n roots fi , . . . , fi , we exclude the prime p from our computations. The number of such p is ﬁnite and may be bounded in terms of the discriminants of the polynomials Ψf (X, α) as α ranges over the roots of HD [f ]. We then h n (r) compute the polynomial H(X) = i=1 r=1 X − fi of degree nh in Fp [X]. After doing this for suﬃciently many primes p, we can lift the coeﬃcients by Chinese remaindering to the integers. The resulting H is a product of n distinct class polynomials, all of which may be obtained by factoring H in Z[X]. Under suitable heuristic assumptions (including the GRH), the total time to compute HD [f ] is quasi-linear in |D|, including the time to factor H. This approach is practically eﬃcient only when n is small, but then it can be quite useful. A notable example is the modular function g for which Ψg (X, J) = (X 12 − 6X 6 − 27)3 − JX 18 . This function was originally proposed by Atkin, and is closely related to certain class invariants of Ramanujan [3, Thm. 4.1]. The function g yields class invariants when D ≡ 13 mod 24. In terms of our generic algorithm, we have n = 2, and for p ≡ 2 mod 3 we get exactly two roots of Ψg (X, ji ) mod p, which diﬀer only in sign. Thus H(X) = HD [g 2 ](X 2 ) = HD [g](X)HD [g](−X), and from this we easily obtain HD [g 2 ], and also HD [g] if desired.

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5

153

Computational Results

This section provides performance data for the techniques developed above. We used AMD Phenom II 945 CPUs clocked at 3.0 GHz for our tests; the software was implemented using the gmp [22] and zn poly [24] libraries, and compiled with gcc [19]. To compute the class polynomial HD [f ], we require a bound on the size of its coeﬃcients. Unfortunately, provably accurate bounds for functions f other than j are generally unavailable. As a heuristic, we take the bound B on the coeﬃcients of HD given by [30, Lem. 8], divide log2 B by the asymptotic height factor c(f ), and add a “safety margin” of 256 bits. We note that with the CM method, the correctness of the ﬁnal result can be eﬃciently and unconditionally conﬁrmed [5], so we are generally happy to work with a heuristic bound. 5.1

Class Polynomial Computations Using the CRT Method

Our ﬁrst set of tests measures the improvement relative to previous computations with the CRT method. We used discriminants related to the construction of a large set of pairing-friendly elliptic curves, see [30, §8] for details. We reconstructed many of these curves, ﬁrst using the Hilbert class polynomial HD , and then using an alternative class polynomial HD [f ]. In each case we used the explicit CRT to compute HD or HD [f ] modulo a large prime q (170 to 256 bits). Table 1 gives results for four discriminants with |D| ≈ 1010 , three of which appear in [30, Table 2]. Each column lists times for three class polynomial computations. First, we give the total time Ttot to compute HD mod q, including the time Tenum spent enumerating EllD (Fp ), for all the small primes p, using Algorithm 2 as it appears in §2.2. We then list the times Tenum and Ttot obtained when Algorithm 2 is modiﬁed to use gcd computations whenever it is advantageous to do so, as explained in §2.3. The gcd approach typically speeds up Algorithm 2 by a factor of 2 or more. For the third computation we selected a function f that yields class invariants for D, and computed HD [f ] mod q. This polynomial can be used in place of HD in the CM method (one extracts a root x0 of HD [f ] mod q, and then extracts a root of Ψf (x0 , J) mod q). For each function f we give a “size factor”, which approximates the ratio of the total size of HD to HD [f ] (over Z). In the ﬁrst three examples this is just the height factor c(f ), but in Example 4 it is 4c(f ) because the prime 59 is ramiﬁed and we actually work with the square root of HD [A59 ], as noted in §4.1, reducing both the height and degree by a factor of 2. We then list the speedup Ttot /Ttot [f ] attributable to computing HD [f ] rather than HD . Remarkably, in each case this speedup is about twice what one would expect from the height factor. This is explained by a particular feature of the CRT method: The cost of computing HD mod p for small primes p varies significantly, and, as explained in [30, §3], one can accelerate the CRT method with a careful choice of primes. When fewer small primes are needed, we choose those for which Step 1 of Algorithm 1 can be performed most quickly. The last line in Table 1 lists the total speedup Ttot /Ttot [f ] achieved.

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A. Enge and A.V. Sutherland Table 1. Example class polynomial computations (times in CPU seconds) Example 1

Example 2

Example 3

Example 4

|D| h(D) log2 B r (r11 , . . . , kk )

13569850003 20203 2272564 (720203 )

11039933587 11280 1359134 (171128 , 1910 )

12901800539 54706 5469776 (327038 , 52 )

12042704347 9788 1207412 (292447 , 312 , 432 )

Tenum (roots) Ttot

6440 19900

10200 23700

10800 52200

21700 42400

Tenum (gcds) Ttot

2510 15900

2140 15500

3440 44700

4780 25300

A71 36 213

A47 24 305

A71 36 629

A59 120* 191

75 93

51 78

71 83

132 222

Function f Size factor [f ] Ttot Speedup (Ttot /Ttot [f ]) [f ]) Speedup (Ttot /Ttot

5.2

Comparison to the Complex Analytic Method

Our second set of tests compares the CRT approach to the complex analytic method. For each of the ﬁve discriminants listed in Table 2 we computed class polynomials HD [f ] for the double η-quotient w3,13 and the Weber f function, using both the CRT approach described here, and the implementation [14] of the complex analytic method as described in [12]. With the CRT we computed HD [f ] both over Z and modulo a 256-bit prime q; for the complex analytic method these times are essentially the same. Table 2. CRT vs. complex analytic (times in CPU seconds) complex analytic

CRT mod q

CRT

|D|

h(D)

w3,13

f

w3,13

f

w3,13

f

6961631 23512271 98016239 357116231 2093236031

5000 10000 20000 40000 100000

15 106 819 6210 91000

5.4 33 262 1900 27900

2.2 10 52 248 2200

1.0 4.1 22 101 870

2.1 9.8 47 213 1800

1.0 4.0 22 94 770

We also tested a “worst case” scenario for the CRT approach: the discriminant D = −85702502803, for which the smallest non-inert prime is 1 = 109. Choosing the function most suitable to each method, the complex analytic method computes HD [w109,127 ] in 8310 seconds, while the CRT method computes HD [A131 ]

Class Invariants by the CRT Method

155

in 7150 seconds. The CRT approach beneﬁts from the attractive height factor of the Atkin functions, c(A131 ) = 33 versus c(w109,127 ) ≈ 12.4, and the use of gcds in Algorithm 2. Without these improvements, the time to compute HD with the CRT method is 1460000 seconds. The techniques presented here yield more than a 200-fold speedup in this example. 5.3

A Record-Breaking CM Construction

To test the scalability of the CRT approach, we constructed an elliptic curve using |D| = 1000000013079299 > 1015 , with h(D) = 10034174 > 107 . This yielded a curve y 2 = x3 − 3x + c of prime order n over the prime ﬁeld Fq , where c = 12229445650235697471539531853482081746072487194452039355467804333684298579047; q = 28948022309329048855892746252171981646113288548904805961094058424256743169033; n = 28948022309329048855892746252171981646453570915825744424557433031688511408013.

This curve was obtained by computing the square root of HD [A71 ] modulo q, a polynomial of degree h(D)/2 = 5017087. The height bound of 21533832 bits was achieved with 438709 small primes p, the largest of which was 53 bits in size. The class polynomial computation took slightly less than a week using 32 cores, approximately 200 days of CPU time. Extracting a root over Fq took 25 hours of CPU time using NTL [29]. We estimate that the size of HD [A71 ] is over 13 terabytes, and that the size of the Hilbert class polynomial HD is nearly 2 petabytes. The size of HD [A71 ] mod q, however, is under 200 megabytes, and less than 800 megabytes of memory (per core) were needed to compute it.

References [1] Bach, E.: Explicit bounds for primality testing and related problems. Mathematics of Computation 55(191), 355–380 (1990) [2] Belding, J., Br¨ oker, R., Enge, A., Lauter, K.: Computing Hilbert class polynomials. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 282–295. Springer, Heidelberg (2008) [3] Berndt, B.C., Chan, H.H.: Ramanujan and the modular j-invariant. Canadian Mathematical Bulletin 42(4), 427–440 (1999) [4] Bernstein, D.J.: Modular exponentiation via the explicit Chinese Remainder Theorem. Mathematics of Computation 76, 443–454 (2007) [5] Bisson, G., Sutherland, A.V.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory (2009) (to appear), http://arxiv.org/abs/0902.4670 [6] Br¨ oker, R.: Constructing elliptic curves of prescribed order. Universiteit Leiden, Proefschrift (2006) [7] Br¨ oker, R.: A p-adic algorithm to compute the Hilbert class polynomial. Mathematics of Computation 77, 2417–2435 (2008) [8] Br¨ oker, R., Lauter, K., Sutherland, A.V.: Modular polynomials via isogeny volcanoes (2009) (preprint), http://arxiv.org/abs/1001.0402

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[9] Couveignes, J.-M., Henocq, T.: Action of modular correspondences around CM points. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 234– 243. Springer, Heidelberg (2002) [10] Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory, pp. 21–76. AMS, Providence (1998) [11] Enge, A.: Courbes alg´ebriques et cryptologie. In: Habilitation ` a diriger des recherches, vol. 7. Universit´e Denis Diderot, Paris (2007) [12] Enge, A.: The complexity of class polynomial computation via floating point approximations. Mathematics of Computation 78(266), 1089–1107 (2009) [13] Enge, A.: Computing modular polynomials in quasi-linear time. Mathematics of Computation 78(267), 1809–1824 (2009) [14] Enge, A.: cm, 0.2 edition (2010), http://cm.multiprecision.org/ [15] Enge, A., Morain, F.: Generalised Weber functions. I. Technical Report 385608, HAL-INRIA (2009), http://hal.inria.fr/inria-00385608 [16] Enge, A., Schertz, R.: Constructing elliptic curves over finite fields using double eta-quotients. Journal de Th´eorie des Nombres de Bordeaux 16, 555–568 (2004) [17] Enge, A., Schertz, R.: Modular curves of composite level. Acta Arithmetica 118(2), 129–141 (2005) [18] Enge, A., Schertz, R.: Singular values of multiple eta-quotients for ramified primes (in preparation 2010) [19] Free Software Foundation. GNU Compiler Collection, 4.2.4 edition (2008), http://gcc.gnu.org/ [20] Gee, A.: Class fields by Shimura reciprocity. Universiteit Leiden, Proefschrift (2001) [21] Gee, A., Stevenhagen, P.: Generating class fields using Shimura reciprocity. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 441–453. Springer, Heidelberg (1998) [22] Granlund, T., et al.: gmp, 4.3.1 edition (2009). http://gmplib.org/. [23] Hajir, F., Villegas, F.R.: Explicit elliptic units, I. Duke Mathematical Journal 90(3), 495–521 (1997) [24] Harvey, D.: zn poly: a library for polynomial arithmetic, 0.9 edn. (2008), http://cims.nyu.edu/~ harvey/zn_poly [25] Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California at Berkeley (1996) [26] Morain, F.: Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques. Journal de Th´eorie des Nombres de Bordeaux 7(1), 111–138 (1995) [27] Morain, F.: Advances in the CM method for elliptic curves. In: Slides of Fields Cryptography Retrospective Meeting, May 11-15 (2009), http://www.lix.polytechnique.fr/~ morain/Exposes/fields09.pdf [28] Schertz, R.: Weber’s class invariants revisited. Journal de Th´eorie des Nombres de Bordeaux 14(1), 325–343 (2002) [29] Shoup, V.: NTL: A library for doing number theory, 5.5 edn. (2008), http://www.shoup.net/ntl/ [30] Sutherland, A.V.: Computing Hilbert class polynomials with the Chinese Remainder Theorem. Mathematics of Computation (to appear 2010), http://arxiv.org/abs/0903.2785 [31] Weber, H.: Lehrbuch der Algebra, 3rd edn., vol. III. Chelsea, New York (1961)

Short Bases of Lattices over Number Fields Claus Fieker1 and Damien Stehl´e1,2 1

Magma Computer Algebra Group, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia 2 CNRS and Macquarie University [email protected], [email protected]

Abstract. Lattices over number ﬁelds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, “short” (pseudo)-basis is important in many applications. In this article, we provide the ﬁrst algorithm that computes such a “short” (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to ﬁnd a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a ﬁeld invariant. As it runs in polynomial time, this provides an eﬀective variant of Minkowski’s second theorem for lattices over number ﬁelds.

1

Introduction

Let K be a number ﬁeld and OK be its maximal order. An OK -module M is a ﬁnitely generated set of elements which is closed under addition and multiplication by elements in OK . Frequently, we have M ⊆ K m for some m. In the case of K being Q, we have OK = Z, thus OK -modules are just the classical Z-lattices. Since Z is a principal ideal domain, every (torsion free) module is free, thus there exists a basis b1 , . . . , bn ∈ M for some n ≤ m such that M = ⊕i≤n Zbi . Any two bases (bi )i and (ci )i have the same cardinality and are linked by some unimodular matrix T ∈ GL(n, Z). The choice of a good basis is crucial for almost all computational problems attached to M . Generally one tries to ﬁnd a basis whose vectors have short Euclidean norms, using, for example, the LLL algorithm [15]. Replacing Z by the maximal order OK makes the classiﬁcation more complicated since OK may no longer be a principal ideal domain. However, since OK is still a Dedekind domain, the modules M ⊆ K m have a well known structure ([7, Cor. 1.2.25], [23, Th. 81:3]): there exist linearly independent elements b1 , . . . , bn ∈ K m and (non-zero fractional) ideals b1 , . . . , bn such that M = ⊕i≤n bi bi , i.e., every b ∈ M has a unique representation as b = i≤n xi bi with xi ∈ bi for all i ≤ n. Such a representation is commonly called a pseudo-basis. It should be noted that bi may not belong to M , and in fact bi ∈ M if and only if 1 ∈ bi . Similarly to the case of Z-lattices, diﬀerent pseudo-bases share the same cardinality, and it is known how to move from a pseudo-basis to another. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 157–173, 2010. c Springer-Verlag Berlin Heidelberg 2010

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As for Z-lattices, the choice of the pseudo-basis is of utmost importance. However, a key diﬀerence is that no analogue of LLL is known, as repeatedly noted in [7]. There have been attempts [10,22,11] but the algorithms are either limited to certain ﬁelds or give no guaranteed bounds on the output size. While every OK -module is also a Z-lattice and can thus be analyzed with all the tools available over Z, for many applications the additional structure as an OK -module is important. This structure is typically lost when applying techniques over Z. Originally, OK -modules mainly came from the study of ﬁnite extensions of K but now they occur in a wider range of problems from group theory (matrix groups and representations [9]) to applications in geometry (automorphism algebras of Abelian varieties). OK -modules also occur in lattice-based cryptography [17,19,24,25,26], and in that context the module rank n is usually polylogarithmic in the degree of the number ﬁeld. Cryptography based on OK modules is increasingly popular, as on one side they lead to compact representations and to fast operations, and on the other side they enjoy a worst-case to average-case reduction for variants of the shortest vector problem, which allows the cryptographic security to be based on worst-case hardness assumptions. As diverse as the applications are the requirements: only one (or more) short module element(s) may be needed, or a short (pseudo)-basis may be required, some applications rely on canonical representations, while any representation may suﬃce for others. We note that canonical representations tend to have components that are much larger than short representations as obtained by lattice reduction or our techniques. To ﬁnd one short element it suﬃces to consider the underlying Z-module (of dimension nd with d = [K : Q]). For Z-lattices contained in Qm , a canonical representation is the Hermite Normal Form (HNF). It has been generalized (BPC-HNF) to OK -modules contained in K m by Bosma and Pohst [4] and Cohen [7, Chap. 1.4] (see also [12]). Our results. In the present work, we describe an algorithm that computes a pseudo-basis made of short vectors. Given an arbitrary pseudo-basis [(ai )i , (ai )i ] of a module M ⊆ K m , it returns a pseudo-basis [(bi )i , (bi )i ] such that: 2

∀i ≤ n : bi ∈ M, N (bi ) ∈ [2−O(d ) , 1] and bi ≤ 2O(dn) λi (M ), where the O(·)’s depend only on the ﬁeld K and the choice of a given LLLreduced integral basis, the euclidean norm · is a module extension of the T2 -norm over K, and the λi (M )’s correspond to the module minima. We refer to Corollary 1 for a precise statement. Overall, this provides a module equivalent to LLL-reduced bases of Z-lattices in the sense that the vectors cannot be arbitrarily longer than the minima. Since it runs in polynomial time, it can also be interpreted as an eﬀective approximate variant of the adaptation to OK -modules of Minkowski’s second theorem (given in Theorem 2). We also study the representation of one-dimensional OK -modules, i.e., modules that are isomorphic to ideals of OK . We show how to modify Belabas’ 2-element representation algorithm [2, Alg. 6.15] so that the output is provably small. Combining the latter and our module pseudo-reduction algorithm leads to compact representations of OK -modules.

Short Bases of Lattices over Number Fields

159

The most natural approach to obtain reduced pseudo-bases consists in trying to generalize LLL, but as mentioned earlier all previous attempts have only partially succeeded. In contrast, we start by viewing the OK -module as a highdimensional Z-lattice. We ﬁnd short module elements by applying LLL to a basis of the latter lattice and interpreting the output as module elements. At this point, we have a pseudo-basis (the input) and a full-rank set of short module vectors (produced by LLL). If we had a Z-lattice instead of an OK -module, we would then use a technique common in the lattice-based cryptography community (see, e.g., [20, Le. 7.1]), consisting in using the HNF to convert a full rank set of short lattice vectors to a short basis. We adapt this technique to number ﬁelds, using the BPC-HNF and introducing a size-reduction algorithm for pseudo-bases. Let us compare (pseudo-)LLL-reduced and BPC-HNF pseudo-bases. A theoretical advantage of the LLL approach is that it is not restricted to K m but also works in a continuous extension (similarly to LLL-reduction being welldeﬁned for real lattices). It should also be signiﬁcantly more eﬃcient to work with pseudo-bases made of short vectors because smaller integers and polynomials of smaller degrees are involved. On the other side, (pseudo-)LLL-reduced pseudo-bases are far from being unique, and seem more expensive to obtain. Road-map. In Section 2, we give some reminders and elementary results on lattices, number ﬁelds and modules. In Section 3, we modify Belabas’ 2-element representation algorithm for ideals of OK , as described above. We then give our module reduction algorithm in Section 4. Finally, in Section 5 we describe our implementation and give some examples. Implementation. The algorithms have been implemented in the Magma computer algebra system [3,18] and are available on request. They will be part of upcoming releases.

2

Preliminaries

We assume the reader is familiar with the geometry of numbers and algebraic number theory. We refer to [16,20], [5,21] and [7, Chap. 1] for introductions to the computational aspects of lattices, elementary algebraic number theory and to modules over Dedekind domains, respectively. 2.1

Lattices

In this work, we will call any ﬁnitely generated free Z-module L a lattice. A usual lattice corresponds to the case where L is a discrete additive subgroup of Rn for some n. Any lattice can be written L = ⊕i≤d Zbi . If the bi ’s are Z-free, they are called a basis of L. A given lattice may have inﬁnitely many bases but their cardinality d is constant and called rank. Any two bases are related by a unimodular transformation, i.e., one is obtained from the other by multiplying by a matrix in Zd×d of determinant ±1. If L ⊆ Qn is of rank d, then there exists a basis B = (bi )i ∈ Qn×d of L such that μj = min{i : Bi,j = 0} (strictly) increases with j, and for all j > k we

160

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have Bμj ,j > Bμj ,k ≥ 0. If d = n, this means that B is a row-wise diagonally strictly dominant lower triangular matrix and that its entries are non-negative. This basis is unique and called the Hermite Normal Form (HNF) of L. It can be computed in polynomial time from any basis [13]. In order to quantify the smallness of an element of a lattice L, we associate to L a positive deﬁnite bilinear form q : LR × LR → R. We use it to map a basis (bi )i to its Gram matrix Gq (b1 , . . . , bd ) := (q(bi , bj ))i,j . We denote q(b, b) by bq , and may omit the subscript if it is clear from the context. The determinant of L, deﬁned as detq (L) = det(Gq (b1 , . . . , bd ))1/2 , does not depend on the particular choice of the basis of L. Note that if L ⊆ Rn and q is the euclidean inner product, then det(L) is the d-dimensional volume of the parallelepiped { i yi bi : yi ∈ [0, 1]}. We deﬁne the lattice minima as follows: ∀i ≤ d, λi,q (L) = min{r : ∃c1 , . . . , ci ∈ L free, maxk≤i ck q ≤ r}. √ d Minkowski’s second theorem states that i≤d λi,q (L) ≤ d detq (L). Frequently one tries to represent a lattice L by a basis that approximates the minima. In this article, we assume that we have an algorithm LatRed that takes as input an arbitrary basis of L and returns a reduced basis satisfying bi ≤ γλi (L), for all i ≤ d. For example, if we use the LLL algorithm [15], then we can take γ = 2d/2 . We proceed as follows: compute the Gram matrix G of the input basis; use the Gram matrix LLL algorithm (see, e.g., [5, p. 88]), to ﬁnd U unimodular such that U t GU is reduced; apply U to the input lattice basis. If the arithmetic over L is eﬃcient, and if q can be eﬃciently computed or approximated with high accuracy, then this provides an eﬃcient algorithm. Apart from being well-deﬁned for more general lattices (not only for lattices on a rational vector space), a signiﬁcant advantage of the LLL-reduction over the HNF is that it provides small lattice elements. However, it seems more expensive to obtain and the uniqueness of the representation is lost. Taking√ the HKZ-reduction instead of the LLLreduction allows one to take γ = 1/2 d + 3 (see [14]), but the complexity of the best algorithm for computing it [1] is exponential in d. ∗ ∗ ∗ Let (bi )i≤d be a lattice j, we deﬁne μi,j = q(b i , bj )/q(bj , bj ), basis. For any i > ∗ ∗ where bi = argminbi + ji |xj |, which gives the bound. To complete the proof, note that the reducedness of the ri ’s gives minj rj∗ ≥ √ 2−d/2 minj rj , and that rj ≥ d for all j. 2.3

OK -Modules

Let b1 , . . . , bn ∈ KRm with n = rankK (bi )i , and b1 , . . . , bn be fractional ideals of OK . The OK -module M [(bi )i , (bi )i ] spanned by the pseudo-basis [(bi )i , (bi )i ] is bi bi . The bi ’s are called the coeﬃcient ideals. As each bi is a Z-lattice, so (j) (j) is M . More precisely, if bi = j≤d Zβi , then M = i,j Zβi bi . Two pseudobases [(bi )i , (bi )i ] and [(ci )i , (ci )i ] represent the same OK -module M if and only if there exists a non-singular U ∈ K n×n with ([23, §81 C]): 1. (c1 , . . . , cn ) = (b1 , . . . , bn )U ; 2. For all i, j, we have Ui,j ∈ bi c−1 j ; −1 3. For all i, j, we have Ui,j ∈ ci b−1 . j , where U = U Cohen [6] generalized the HNF to modules in K m . The algorithm of [4] may also be interpreted as such a generalization. We refer to [12, Chap. 4] for a detailed exposure and comparison.

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163

Theorem 1. Let M ⊆ K m be an OK -module of rank n. There exists a pseudobasis [(bi )i , (bi )i ] of M such that μj = min{i : Bi,j = 0} (strictly) increases with j, for all j we have Bμj ,j = 1 and for all j > k the entry Bμj ,k ∈ K is size-reduced modulo the HNF of bj b−1 k . This unique pseudo-basis is called the HNF of M . It can be computed in polynomial time from any pseudo-basis of M . Similarly to the HNF for lattices, the above HNF can only handle OK -modules M ⊆ K m (as opposed to KRm ) and does not necessarily contain small elements of M . We now deﬁne the concept of small-ness for elements of KRm . For any two vectors b = (b1 , . . . , bm )t , b = (b1 , . . . , bm )t ∈ KRm , we deﬁne T2⊗m (b, b ) = T2⊗m (b, b) by b. Notice that for any (r, b) ∈ i≤m T2 (bi , bi ), and we denote KR × KRm , we have rb ≤ r · b. With this deﬁnition at hand, we can deﬁne the minima of M : ∀i ≤ n, λi (M ) = min{r : ∃c1 , . . . , ci ∈ M, rankK (ck )k = i and max ck ≤ r}. Let [(bi )i , (bi )i ] be a pseudo-basis of an OK -module M ⊆ KRm . Assume that bi = (j) j≤d Zβi . We deﬁne det(M ) as the square root of the determinant of the nd×nd (j )

symmetric positive deﬁnite matrix T2⊗m (βi bi , βi bi )i,j;i ,j . This is a module invariant. When M is a non-zero fractional ideal of OK , this matches detT2 (M ). It should be noted that det(M ) is not immediately related to the (Steinitz) class of M nor to the maximal exterior power of M . The following is a direct consequence of Minkowski’s second theorem over Z-lattices. Theorem 2. Let M ⊆ KRm be an OK -module of rank n. Then i≤n λi (M ) ≤ √ n dn det(M )1/d . (j)

Proof. The module M can be seen as a lattice L of dimension nd, with det(M ) = √ dn det(L). Minkowski’s second theorem asserts that i≤nd λi (L) ≤ dn det(L). Let c1 , . . . , cnd ∈ M be free over the integers such that ci = λi (L) holds for all i. For all i ≤ n, let φ(i) = min(j : rankK (c1 , . . . , cj ) = i). As OK has rank d as a Z-module, we have φ(i) ≤ (i − 1)d + 1. We conclude with the following sequence of inequalities: √ n 1 1 λi (M ) ≤ cφ(i) ≤ λ(i−1)d+1 (L) ≤ λi (L) d ≤ dn det(M ) d . i≤n

i≤n

i≤n

i≤dn

We now extend the concept of GSO. Let [(bi ) i , (bi )i ] be a pseudo-basis of an OK -module M . We deﬁne b∗i = argminbi + j _ := InvariantForm(N); // compute the form > SetVerbose("RLLL", 1); > O := Nice(N); > #Sprint(ActionGenerators(M)); 1359862 > #Sprint(ActionGenerators(N)); 327378 > #Sprint(ActionGenerators(O)); 4577 The function Nice implements the procedure outlined above. Note that the actual result can vary substantially as several parts use randomized algorithms. The Sprint statements are only used as a very crude indication of the output size, they simply give the number of characters neccessary to write the generating matrices for G.

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4. Bosma, W., Pohst, M.: Computations with ﬁnitely generated modules over Dedekind domains. In: Proc. ISSAC 1991, pp. 151–156. ACM, New York (1991) 5. Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1995) 6. Cohen, H.: Hermite and Smith normal form algorithms over Dedekind domains. Math. Comp. 65, 1681–1699 (1996) 7. Cohen, H.: Advanced topics in Computational Number Theory. Springer, Heidelberg (2000) 8. Evertse, J.-H.: Reduced bases of lattices over number ﬁelds. Indag. Mathem. N.S. 2(3), 153–168 (1992) 9. Fieker, C.: Minimizing representations over number ﬁelds II: Computations in the Brauer group. J. Algebra 3(322), 752–765 (2009) 10. Fieker, C., Pohst, M.E.: Lattices over number ﬁelds. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 147–157. Springer, Heidelberg (1996) 11. Gan, Y.H., Ling, C., Mow, W.H.: Complex lattice reduction algorithm for lowcomplexity full-diversity MIMO detection. IEEE Trans. Signal Processing 57, 2701– 2710 (2009) 12. Hoppe, A.: Normal forms over Dedekind domains, eﬃcient implementation in the computer algebra system KANT. PhD thesis, Technical University of Berlin (1998) 13. Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499–507 (1979) 14. Lagarias, J.C., Lenstra Jr., H.W., Schnorr, C.P.: Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10, 333–348 (1990) 15. Lenstra, A.K., Lenstra Jr., H.W., Lov´ asz, L.: Factoring polynomials with rational coeﬃcients. Math. Ann. 261, 515–534 (1982) 16. Lov´ asz, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. CBMSNSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1986) 17. Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006) 18. Magma. The Magma computational algebra system for algebra, number theory and geometry, http://magma.maths.usyd.edu.au/magma/ 19. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and eﬃcient oneway functions. Comput. Complexity 16(4), 365–411 (2007) 20. Micciancio, D., Goldwasser, S.: Complexity of lattice problems: a cryptographic perspective. Kluwer Academic Press, Dordrecht (2002) 21. Mollin, R.A.: Algebraic Number Theory. Chapman and Hall/CRC Press (1999) 22. Napias, H.: A generalization of the LLL-algorithm over Euclidean rings or orders. J. th´eorie des nombres de Bordeaux 2, 387–396 (1996) 23. O’Meara, O.T.: Introduction to Quadratic Forms. In: Grundlehren der Mathematischen Wissenschaften, vol. 117. Springer, Heidelberg (1963) 24. Peikert, C., Rosen, A.: Eﬃcient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006) 25. Peikert, C., Rosen, A.: Lattices that admit logarithmic worst-case to average-case connection factors. In: Proc. STOC 2007, pp. 478–487. ACM, New York (2007) 26. Stehl´e, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Eﬃcient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009)

On the Complexity of the Montes Ideal Factorization Algorithm David Ford and Olga Veres Concordia University, 1455 de Maisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1J1 [email protected], [email protected] Abstract. Let p be a rational prime and let Φ(X) be a monic irreducible polynomial in Z[X], with nΦ = deg Φ and δΦ = vp (disc Φ). In [13] Montes describes an algorithm for the decomposition of the ideal p OK in the algebraic number field K generated by a root of Φ. A simplified version of the Montes algorithm, merely testing Φ(X) for irreducibility over Qp , is given in [19], together with a full Maple implementation and a demonstration that in the worst case, when Φ(X) is irreducible over Qp , the expected 2+ number of bit operations for termination is O(n3+ Φ δΦ ). We now give a 2+ 2+ refined analysis that yields an improved estimate of O(n3+ Φ δΦ +nΦ δΦ ) bit operations. Since the worst case of the simplified algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the complete Montes algorithm.

1

Introduction

In an algebraic number ﬁeld K with ring of integers OK , factorization of the ideal pOK , for p prime, can be determined via polynomial factorization over the ﬁeld of p-adic numbers Qp [12]. If K = Q(α) for a given α ∈ OK such that the index OK : Z[α] is not divisible by p then the factorization of the ideal pOK can be determined by polynomial factorization modulo p [5,6,7]. In practice, eﬃcient techniques for polynomial factorization modulo p [1,2,4] combined with Hensel lifting [12,20] solve the problem of factoring pOK in a straightforward and eﬀective manner when p does not divide the index. The complications arising when p divides the index OK : Z[α] have been the subject of considerable study. Current ideas are derived from the “Round Four” algorithm of Zassenhaus [20], which has evolved into two main variations, the “one-element” method [8] and the “two-element” method [16]. Versions of the one-element method are used by Maple and PARI. The two-element method is used, e.g., by Magma. The algorithm of Montes [13] is in a separate category. Given a monic irreducible polynomial Φ(X) in Z[X], the Montes algorithm determines the number of irreducible factors of Φ(X) in Zp [X] and their respective degrees. The algorithm exploits classical results of Ore [15,14] on Newton G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 174–185, 2010. c Springer-Verlag Berlin Heidelberg 2010

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polygons and provides an alternative to the methods based on ideas of Zassenhaus. A familiar application of Newton polygons gives the p-adic valuations of roots of a polynomial in Zp [X]. If Φ(X) ∈ Zp [X] has two roots with diﬀerent p-adic values then Hensel-lifting techniques can be applied to construct a non-trivial p-adic factorization of Φ to any desired degree of precision. This process constitutes “level 0” of the Montes algorithm. For each factor of Φ revealed at level 0, the algorithm proceeds to higher levels, either to discover a reﬁned factorization or to establish irreducibility. At level r, with ϕr (X) an irreducible monic polynomial in Zp [X] and Vr a valuation of Qp [X], the algorithm constructs the ϕr -adic expansion of a given polynomial and then computes • a ﬁnite ﬁeld Fqr , • the Newton polygon Nr (Φ) of Φ with respect to the valuation Vr , • a slope −dr /er , with dr and er coprime positive integers, of an edge of Nr (Φ), (r)

• the “associated polynomial” ΨS,Φ (Y ) ∈ Fqr [Y ] for each segment S of Nr (Φ), (r)

• a monic irreducible factor ψr of ΨS,Φ with ξr a root of ψr and fr = deg ψr , • a valuation Vr+1 of Qp [X], • an irreducible monic polynomial ϕr+1 (X) ∈ Zp [X]. The number of edges of Nr (Φ) and the number of distinct irreducible factors of (r) ΨS,Φ give information for the factorization of Φ; if either is greater than one then Φ is reducible. Our goal being to give an estimate of the complexity of the worst case of the Montes algorithm, we have restricted the algorithm merely to decide the question of irreducibility of a given polynomial. When Φ is irreducible over Qp the Newton polygon at each level is a single segment. It is apparent that this is the most costly case, i.e., the case that reaches the highest level, for the full algorithm. So our restricted algorithm operates under the assumption that Nr (Φ) has just one edge at each level r; the failure of this condition terminates the restricted algorithm. In [19, Chapter 3] a complete Maple implementation of the restricted Montes algorithm is given, together with a demonstration that in the worst case, when Φ is irreducible over Qp , the expected number of bit operations for termination 2+ is O(n3+ Φ δΦ ), with nΦ = deg Φ and δΦ = vp (disc Φ). In the present paper we 2+ 2+ give a reﬁned analysis that yields an improved estimate of O(n3+ Φ δΦ + n Φ δΦ ) bit operations. Since the worst case of the simpliﬁed algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the full Montes algorithm.

2

Definitions and Notation

Definition 1. Let ϕ0 (X) = X and let V0 denote the standard p-adic valuation of Qp . For K(X) ∈ Qp [X] and r ≥ 1, the level-r Newton polygon of K, denoted

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Nr (K), is the Newton polygon of K with respect to the valuation Vr of Qp [X], which can be defined recursively as Vr (K) = min er−1 Vr−1 Ar−1,k + kVr ϕr−1 0 ≤ k ≤ n n with K(X) = k=0 Ar−1,k (X) ϕr−1 (X)k the ϕr−1 -adic expansion of K(X). Remark 1. Nr (K) is the lower convex hull of the set { (k, Vr (Ar,k ϕkr )) | 0 ≤ k ≤ n, Ar,k (X) = 0 } , and if deg K < deg ϕr then Nr (K) = {(0, Vr (K))} and Vr+1 (K) = er Vr (K). Definition 2. For r ≥ 1 and K(X) a nonzero polynomial in Zp [X] we define Sr,K to be the segment of Nr (K) having slope −dr /er . Definition 3. For positive integers r and ν we define αr,ν = ν d−1 r mod er , βr,ν = (ν − αr,ν dr )/er , Tr,ν = { (αr,ν + λer , βr,ν − λdr ) | 0 ≤ λ ≤ βr,ν /dr } . Remark 2. If L is the line through the point (0, ν/er ) with slope −dr /er then Tr,ν is the longest segment of L with endpoints having nonnegative integer coordinates. Definition 4. For r ≥ 0 we define μr = 0 ,

νr = 0 ,

if r = 0 ,

μr = dr−1 + er−1 ν r−1 ,

ν r = er−1 fr−1 μr ,

if r ≥ 1 .

Remark 3. For r ≥ 1 it is easily seen that μr = Vr (ϕr−1 ) and ν r = Vr (ϕr ). Definition 5 (Associated Polynomial ). Let r ≥ 0, let α and β be nonnegative integers, and let S be an arbitrary segment of slope −dr /er with left endpoint (α, β). Let m0 = 0 and for r ≥ 1 and k ≥ 0 define mr = (1/dr ) mod er ,

1 Ωr = er−1 fr−1 mr−1 fr−1 μr ξr−1 Ωr−1

if r = 1 , if r > 1 ,

(β − kdr ) − (α + ker ) ν r , Θ(S, r, k) = mr−1 er−1 Θ(S,r,k)

ΓS,r,k = Ωrα+ker ξr−1

∈ Fqr .

Let K(X) ∈ Zp [X] have ϕr -adic expansion K(X) = A0 (X) + A1 (X) ϕr (X) + · · · + An (X) ϕr (X)n

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with dr j + er Vr (Aj ϕjr ) ≥ dr α + er β for j = 0, . . . , n and let r ) ∈S . J = k 0 ≤ k ≤ (n − α)/er , α + ker , Vr (Aα+ker ϕα+ke r We define the level-r associated polynomial of K with respect to S to be (r) ΨS,K (Y ) = k∈J ηk Y k with ηk ∈ Fqr defined as ⎧ A ⎪ ⎪ ⎨ α+ke0 ηk = B k (ξ0 ) , ⎪ ⎪ ⎩ Γ −1 Ψ (r−1)

with Bk (X) = Aα+ke1 (X) p

S,r,k Tr−1,νk,Aα+ker (ξr−1 ) ,

if r = 0 , β−kd1

,

if r = 1 ,

with νk = Vr (Aα+ker ) ,

if r ≥ 2 .

We further define the natural level-r associated polynomial of K to be (r) (r) ΨK (Y ) = ΨSr,K ,K (Y ) . (r) Remark 4. The polynomial ΨK (Y ) has nonzero constant term.

3

Outline of the Restricted Montes Algorithm

A complete Maple implementation of the restricted Montes algorithm, with proofs and explanatory comments interspersed, is given in [19]. Here we give an outline showing the three major phases of the algorithm. The algorithm begins in phase M0 (level 0), then alternates between phase M1 and phase M2 (level r, for r = 1, 2, . . . ) until reaching a terminating condition. • input: Φ(X) ∈ Z[X] monic and irreducible, p ∈ Z prime

TRUE if Φ(X) is irreducible over Qp [X], • output: FALSE if Φ(X) is reducible over Qp [X]. M0 :

1. Factorize Φ modulo p: a

a0,κ

0,1 · · · ψ0,κ00 Φ ≡ ψ0,1

(mod p) .

2. If κ0 > 1 then return FALSE. If κ0 = 1 and a0,1 = 1 then return TRUE. 3. Deﬁne ϕ0 (X) = X, n0 = 1, d0 = 0, e0 = 1, ψ0 = ψ0,1 , f0 = deg ψ0 , ξ0 a root of ψ0 . 4. Set r ← 1. M1 :

5. If r = 1 let ϕ1 (X) be a monic polynomial in Z[X] such that ϕ1 = ψ0 . If r > 1 construct Hr−1 according to Algorithm 1 in Sect. 6 below and let er−1 fr−1 ϕr = ϕr−1 + Hr−1 .

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6. Deﬁne nr = er−1 fr−1 nr−1 = deg ϕr . 7. If r > 1 and er−1 fr−1 = 1 then replace ϕr−1 ← ϕr and r ← r − 1. M2 :

8. If ϕr = Φ then return TRUE. If ϕr | Φ and ϕr = Φ then return FALSE. 9. Let Sr,1 , . . . , Sr,λr be the segments of Nr (Φ) and let ζr,k + 1 be the number of points on Sr,k with integer coordinates, for k = 1, . . . , λr . 10. If λr > 1 then return FALSE. If λr = 1 and ζr,1 = 1 then return TRUE. 11. Let −dr /er be the slope of Sr,1 , with dr and er relatively prime and (r) er > 0, and construct ΨΦ (Y ) ∈ Fqr [ Y ]. 12. Factorize ar,1 ar,κr (r) ΨΦ = cr ψr,1 · · · ψr,κ r over Fqr , with cr ∈ Fqr a nonzero constant. 13. If κr > 1 then return FALSE. If κr = 1 and ar,1 = 1 then return TRUE. 14. Deﬁne ψr = ψr,1 , fr = deg ψr , ξr a root of ψr . 15. Replace r ← r + 1. Go to M1 .

4

Complexity of Fundamental Operations

Notation. We use alpha F and alpha Q to denote the number of operations p in Fp and Q respectively required for the execution of the procedure alpha. We use the notation f (n) ∈ O(nk+ ) as an alternative to the “soft-O” notation f (n) ∈ O∼(nk ) ≡ f (n) ∈ O(nk (ln n)c ) for some positive constant c (see [9]). For n ≥ 3 and q a prime power we deﬁne the following. L(n) = ln n ln ln n M(n) = n L(n)

F(n, q) = n M(n) ln(qn) K(q) = M(ln q) ln ln q

We are concerned with the reducibility of the monic polynomial Φ(X) ∈ Zp [X] ∗ for some prime p. We let δΦ denote vp (disc Φ) and we let pδΦ denote the p-adic ∗ reduced discriminant of Φ [8, Appendix A]. It is clear that δΦ ≤ δΦ . Magnitude of p. To simplify the subsequent discussion we impose the condition that p ∈ O(1), by which we mean that p is a small prime, not exceeding the magnitude of a single machine word.

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∗

Arithmetic in Zp . If F (X) ∈ Z[X] with F (X) ≡ Φ(X) (mod p2δΦ +1 Zp [X]) then Φ(X) is reducible in Zp [X] if and only if F (X) is reducible in Zp [X]. Thus in our computations p-adic integers are represented as rational approximations ∗ with 2δΦ + 1 p-adic digits of precision, i.e., as rational integers reduced modulo ∗ 2δΦ +1 p . Sch¨ onhage and Strassen have shown that the time required to perform an arithmetic operation on two rational integers of length m is O(M(m)); see [9, Ch.8, §8.3]. It follows that if we represent p-adic integers in this fashion then the cost of an arithmetic operation is O(ΔΦ ), with ∗ ln p) . ΔΦ = M(δΦ

Arithmetic in Fq . By [9, Ch.14, §14.7], a single operation in Fq can be per∗ formed in O(K(q)) word operations. If q = pf the assumption that ln p ∈ O(1) gives ln q = f ∗ ln p ∈ O(f ∗ ) and thus the cost of an operation in Fq is O(K(q)) = O M(ln q) ln ln q ⊆ O f ∗ (ln f ∗ )2 ln ln f ∗ ⊆ O f ∗ (1+) . For α ∈ Fq and any integer n the cost of computing αn is O(ln q K(q)) ⊆ O(f ∗ f ∗ (1+) ) = O(f ∗ (2+) ) since we may assume 0 ≤ n ≤ q − 1. By [18, Theorem 10], the asymptotic cost for constructing an irreducible polynomial of degree n over the ﬁnite ﬁeld Fq is O (n2 ln n + n ln q) L(n) . Polynomial Arithmetic. The number of operations required to evaluate a polynomial of degree n at a given point using Horner’s rule is O(n). By [17] and [3], the number of operations needed to multiply two polynomials of degree at most n is O(M(n)). It follows that the number of operations needed to compute the mth power of a polynomial of degree n is O nm ln2 (nm) ⊆ O (nm)1+ . By [9, Ch 14, §14.4 and §14.5], the expected number of operations in Fq needed to factorize a polynomial of degree n over Fq is O(F(n, q)) ⊆ O(n2+ ln q) . Let ϕ(X) be a monic polynomial in Zp [X] of degree nϕ , let f (X) be a polynomial in Zp [X] of degree n, and let kϕ = n/nϕ . Let E(f, kϕ ) denote the number of operations in Zp needed to compute the ϕ-adic expansion kϕ f (X) = i=1 ai (X) ϕi (X) . From [9, Ch 5, §5.11], we have E(f, kϕ ) ∈ O(kϕ (kϕ + 1)n2ϕ ) = O(n2ϕ kϕ2 ) = O(n2 ) .

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Complexity of the Algorithm

Finite Fields. For r ≥ 0 the ﬁnite ﬁeld Fqr+1 is implemented as Fp [ρr ], with • ρr of a root of ψr∗ , • ψr∗ (Y ) an arbitrary irreducible monic polynomial in Fp [Y ] of degree fr∗ , • fr∗ = f0 · · · fr .

∗

Thus Fqr+1 = Fqr [ξr ] = Fp [ξ0 , . . . , ξr ] = Fp [ρr ] and qr+1 = qrfr = pfr . Computing the Newton Polygon. It follows from [19, Theorem 15] that the recursive computation of Vr (Φ) requires O(n2+ Φ ΔΦ ) operations in Q and that this dominates the cost of constructing Nr (Φ). e

r−1 Computing ϕr . The construction of ϕr = ϕr−1 er−1 fr−1 Sect. 6 below. The cost of computing ϕr−1 is

fr−1

+ Hr−1 is explained in

er−1 fr−1 ϕr−1 = 0, Fp er−1 fr−1 ϕr−1 ∈ O (nr−1 er−1 fr−1 )1+ ΔΦ = O n1+ r ΔΦ . Q

A slight modiﬁcation of the proof of [19, Theorem 17] shows that the cost of constructing Hr−1 = Hr−1,ν r ,γr−1 is

∗ (3+) Hr−1 Fp ∈ O rfr−1 fr−2 ⊆ O(rn3+ ), r 1+ Hr−1 Q ∈ O rnr ΔΦ .

Thus the cost of computing ϕr is dominated by the cost of computing Hr−1 . Computing the Associated Polynomial. It follows from [19, Theorem 16] that if r ≥ 2 then (r) 2+ ΨΦ Fp ∈ O(nΦ n1+ r ) ⊆ O(nΦ ) ,

(r) ΨΦ

Q

2+ ∈ O nΦ n1+ r ΔΦ ⊆ O nΦ ΔΦ .

Total Complexity. The cost of phase M0 is dominated by the cost of factorizing Φ over Fp . Hence M0 Fp ∈ O(F(nΦ , p)) ⊆ O(n2+ Φ ), M0 Q ∈ O(1) .

On the Complexity of the Montes Ideal Factorization Algorithm

181

The cost of phase M1 is dominated by the cost of constructing ϕr . Hence ), M1 (r) Fp ∈ O(rn3+ r M1 (r) Q ∈ O rn1+ ΔΦ . r

The cost in Q-operations of phase M2 is dominated by the construction of the (r) Newton polygon Nr (Φ) and of the associated polynomial ΨΦ , each of which 2+ require O(nΦ ΔΦ ) operations in Q. Since Fqr+1 = Fp [ρr ], the necessity of expressing ξr and ρr−1 in terms of ρr arises. This is achieved in each case by ∗ factoring ψr−1 over Fp [ρr ], which requires O(fr∗ 3+ ) ⊆ O(n3+ Φ ) operations in Fp . These are the dominant ﬁnite-ﬁeld operations in M2 , hence M2 (r) Fp ∈ O(n3+ Φ ), M2 (r) Q ∈ O(n2+ Φ ΔΦ ) .

We now estimate the number of operations required for the chain of computations M0 (Φ) → M1 (1) → M2 (1) → M1 (2) → M2 (2) → · · · → M1 (m) → M2 (m) with the algorithm terminating at level m. We note that at level r we have n0 < n1 < · · · < nr with n0 | n1 | · · · | nr . Hence 2r ≤ nr and thus r ∈ O(ln nr ). It follows that m ∈ O(ln nΦ ) and we have m M0 (F ) Fp + r=1 M1 (r) Fp + M2 (r) Fp m m = M0 (F ) Fp + r=1 M1 (r) Fp + r=1 M2 (r) Fp ∈ O n2+ + m2 n3+ + mn3+ Φ Φ Φ ⊆ O n3+ , Φ m M0 (F ) Q + r=1 M1 (r) Q + M2 (r) Q m m = M0 (F ) Q + r=1 M1 (r) Q + r=1 M2 (r) Q 2+ ∈ O nΦ + m2 n1+ Φ ΔΦ + m nΦ ΔΦ ⊆ O n2+ Φ ΔΦ .

From [16, Proposition 4.1] it follows that the case er−1 fr−1 = 1 can occur at most e∗ 2 r−2 vp (disc Φ) ≤ 2 vp (disc Φ) nΦ times. Hence the sequence M1 (r) → M2 (r − 1) → M1 (r)

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can occur at most 2vp (disc Φ) times in the course of the computation. From the results above we have 3+ M1 (r) Fp + M2 (r − 1) Fp ∈ O(rn3+ + n3+ r Φ ) ⊆ O(nΦ ) , 2+ M1 (r) Q + M2 (r − 1) Q ∈ O(rn1+ + n2+ r Φ ΔΦ ) ⊆ O(nΦ ΔΦ ) . ∗ Since δΦ ≤ δΦ and ln p ∈ O(1) we have 1+ ∗ ln p) ∈ O(δΦ ). ΔΦ = M(δΦ

It now follows that the expected number of operations required for the restricted Montes algorithm to terminate is 3+ 2+ 2+ . + n2+ O 2δΦ (n3+ Φ Φ ΔΦ ) ⊆ O nΦ δΦ + nΦ δΦ 2+ Remark 5. This is a slight improvement on the estimate O(n3+ Φ δΦ ) from [19]. By way of comparison, Pauli [16] gives an estimate of 1+ 2+ O n3+ + n2+ Φ δΦ Φ δΦ

bit operations for factorization of a univariate polynomial over Qp via the “twoelement” method.

6

The Construction of ϕr

Algorithm 1 (Montes). Given ds , es , fs , etc., for 1 ≤ s ≤ r and given • an integer t in the range 1 ≤ t ≤ r, • an integer ν ≥ ν t+1 , • a nonzero polynomial δ(Y ) ∈ Fqt [Y ] of degree less than ft , to construct a polynomial Ht,ν,δ (X) ∈ Zp [X] such that • deg Ht,ν,δ < nt+1 , • Vt+1 (Ht,ν,δ ) = ν, (t)

• ΨTt,ν , Ht,ν,δ(Y ) = δ(Y ). Construction. Let ζ0 , . . . , ζft −1 in Fqt be such that ft −1 δ(Y ) = i=0 ζi Y i . Since δ(Y ) = 0 the set Jδ = { i | 0 ≤ i ≤ ft − 1, ζi = 0 } is not empty. For i ∈ Jδ we construct Ki (X) as follows. • We take δi (Y ) to be the unique polynomial in Fqt−1 [Y ] of degree less than ft−1 such that δi (ξt−1 ) = ΓTt,ν ,t,i ζi .

On the Complexity of the Montes Ideal Factorization Algorithm

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• If t = 1 we take Pi (X) to be a polynomial in Zp [X] of degree less than f0 such that P i (Y ) = δi (Y ) and we set Ki (X) = pβ1,ν −id1 Pi (X) . • If t ≥ 2 we let νi = (βt,ν − idt ) − (αt,ν + iet )ν t and we set Ki (X) = Ht−1, νi , δi(X) . Having constructed Ki (X) for i ∈ Jδ , we set Ht,ν,δ (X) = i∈Jδ Ki (X) ϕt (X)αt,ν +iet .

Remark 6. It follows from [13, Proposition 3.2] that Algorithm 1 correctly constructs the polynomial Ht,ν,δ with the indicated properties. The construction of δi (Y ) in Algorithm 1 being rather complicated, we provide some implementation details. f ∗ ×fr ×f ∗

r−1 Computing Υr . If r > 0 we construct Υr ∈ Fp r such that ∗ fr −1 ρkr−1 ξrj = h=0 (Υr )h,j,k ρhr

f ∗ ×fr∗

∗ for j = 0, . . . , fr − 1, k = 0, . . . , fr−1 − 1. In practice we construct Υr ∈ Fp r ∗ ∈ Fpfr such that and M

(Υr )1+h,1+j+kfr = (Υr )h,j,k ,

1+j+kfr = Mj,k , M

∗ for h = 0, . . . , fr∗ − 1, j = 0, . . . , fr − 1, k = 0, . . . , fr−1 − 1.

Deriving δi from Υt−1 . Given i ∈ Jδ and t ≥ 2, let f∗

t−1 ∗ ΓTt,ν ,t,i ζi = κi,0 + κi,1 ρt−1 + · · · + κi,ft−1 −1 ρt−1

∗ ft−2

For j = 0, . . . , ft−1 − 1, k = 0, . . . , ∗ ft−1 −1 ft−2 −1 j=0

k=0

−1

∈ Fp [ρt−1 ] = Fqt .

− 1, let Mj,k ∈ Fp satisfy

(Υt−1 )h,j,k Mj,k = κi,h

∗ for h = 0, . . . , ft−1 − 1, and let ∗ ft−1 −1 ft−2 −1 Mj,k ρkt−2 Y j . δi (Y ) = j=0 k=0

Then δi (Y ) ∈ Fp [ρt−2 ][Y ] = Fqt−1 [Y ] and ∗ ft−1 −1 ft−2 −1 j Mj,k ρkt−2 ξt−1 δi (ξt−1 ) = j=0 k=0 = = =

∗ ft−1 −1 ft−2 −1

j=0

k=0

Mj,k

∗ ft−1 −1

h=0

∗ ∗ ft−1 −1 ft−1 −1 ft−2 −1

j=0

h=0

∗ ft−1 −1

h=0

k=0

(Υt−1 )h,j,k ρht−1

(Υt−1 )h,j,k Mj,k ρht−1

κi,h ρht−1

= ΓTt,ν ,t,i ζi . The essential properties of ϕr are as follows (see [19, Proposition 9]).

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Proposition 1 (Montes). Let ds , es , fs , ϕs , ψs , etc., be given for 1 ≤ s ≤ r−1 and let −e

γr−1 (Y ) = Ωr−1r−1

fr−1

(ψr−1 (Y ) − Y fr−1 ),

ϕr (X) = ϕr−1 (X)er−1 fr−1 + Hr−1,ν r ,γr−1 (X) . Then ϕr (X) is a monic polynomial in Zp [X] with the following properties. • deg ϕr = nr . • Nr−1 (ϕr ) consists of the single segment Sr−1,ϕr . • Vr (ϕr ) = ν r . (r−1) −e f • Ψϕr (Y ) = Ω r−1 r−1 ψr−1 (Y ). r−1

• ϕr is irreducible over Zp .

7

Supplementary Remarks

The Maple code from [19], including an example, can be found at this URL. http://www.mathstat.concordia.ca/faculty/ford/Student/Veres/mmtest.mpl

Two recent monographs by Gu`ardia, Montes, and Nart give a thorough revision of the theory underlying the Montes algorithm [10] and a detailed description of the algorithm [11]. Algorithm 1 and Proposition 1 in Sect. 6 above appear in [10]. A simpler choice for Ωr (see Deﬁnition 5) is also given, but with no eﬀect on the complexity of the algorithm.

References 1. Berlekamp, E.R.: Factoring Polynomials over Finite Fields. Bell Systems Technical Journal 46, 1853–1859 (1967) 2. Berlekamp, E.R.: Factoring Polynomials over Large Finite Fields. Math. Comp. 24, 713–735 (1970) 3. Cantor, D.G., Kaltofen, E.: On Fast Multiplication of Polynomials over Arbitrary Algebras. Acta Informatica 28(7), 693–701 (1991) 4. Cantor, D.G., Zassenhaus, H.: A New Algorithm for Factoring Polynomials Over Finite Fields. Math. Comp. 36, 587–592 (1981) 5. Dedekind, R.: Supplement X to Vorlesungen u ¨ber Zahlentheorie von P.G. Lejeune Dirichlet (2nd ed.). Vieweg, Braunschweig (1871); Also Werke 3, 223–261 (1932) (in part) 6. Dedekind, R.: Sur la th´eorie des nombres entiers alg´ebriques. Gauthier-Villars (1877); Also Bull. des Sci. Math. Astron. 11(1), 278–288 (1876); 1(2), 17–41, 69–92, 144–164, 207–248 (1877) and Werke 3, 263–296 (1932) (in part) ¨ 7. Dedekind, R.: Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der h¨ oheren Kongruenzen. Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen 23, 1–23 (1878)

On the Complexity of the Montes Ideal Factorization Algorithm

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8. Ford, D., Pauli, S., Roblot, X.-F.: A Fast Algorithm for Polynomial Factorization over Qp . Journal de Th´eorie des Nombres de Bordeaux 14, 151–169 (2002) 9. von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999) 10. Gu` ardia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory (2008), arXiv:0807.2620v2[math.NT] 11. Gu` ardia, J., Montes, J., Nart, E.: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields (2008), arXiv:0807.4065v3[math.NT] 12. Hensel, K.: Theorie der algebraischen Zahlen. Teubner, Leipzig (1908) 13. Montes, J.: Pol´ıgonos de Newton de orden superior y aplicaciones aritm´eticas. PhD thesis, Universitat de Barcelona (1999) 14. Montes, J., Nart, E.: On a theorem of Ore. Journal of Algebra 146, 318–334 (1992) 15. Ore, Ø.: Newtonsche Polygone in der Theorie der algebraischen K¨ orper. Math. Ann. 99 (1928) 16. Pauli, S.: Factoring Polynomials over Local Fields. Journal of Symbolic Computation 32(5), 533–547 (2001) 17. Sch¨ onhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971) 18. Shoup, V.: Fast Construction of Irreducible Polynomials over Finite Fields. Journal of Symbolic Computation 17, 371–394 (1994) 19. Veres, O.: On the Complexity of Polynomial Factorization over p-adic Fields. PhD Dissertation, Concordia University (2009), http://www.mathstat.concordia.ca/faculty/ford/Student/Veres/vthp.pdf 20. Zassenhaus, H.: On Hensel factorization II. In: Symposia Mathematica XV, Instituto Di Alta Matematica, pp. 499–513. Academic Press, New York (1975)

Congruent Number Theta Coeﬃcients to 1012 William B. Hart1, , Gonzalo Tornar´ıa2, and Mark Watkins3, 1 2 3

Mathematics Institute, Warwick University, Coventry, United Kingdom Centro de Matem´ atica, Universidad de la Rep´ ublica, Montevideo, Uruguay Department of Mathematics and Statistics, University of Sydney, Australia

Abstract. We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 1012 . The computation involves multiplying long theta series as per Tunnell (1983). The method, which we describe in some detail, uses a multimodular disk based technique for multiplying polynomials out-ofcore which minimises expensive disk access by keeping data truncated.

1

History

The congruent number problem ﬁrst makes its appearance in the literature of the classical Islamic period, e.g. in al-Karaji’s text the al-Fakhri. Dickson [11] states that an anonymous Arab manuscript written before 972 A.D. contains reference to the problem. The problem was initially studied in terms of squares of rational numbers: a natural number n is congruent iﬀ there exist rational numbers x, y, z, w such that x2 + ny 2 = z 2 and x2 − ny 2 = w2 . In other words n is congruent iﬀ there exist three rational squares in arithmetic progression with common diﬀerence n. It suﬃces to consider squarefree n. Bachet, in translating Diophantus’ Arithmetica, wrote an appendix of problems on right triangles. Problem 20 was “to ﬁnd a right-angled triangle such that its area is equal to a given number”. This equivalent problem refers to right triangles with rational sides whose area n is a natural number. The problem was studied by Fermat and Fibonacci the latter of which referred to a common diﬀerence of squares in arithmetic progression as a congruum. Euler referred to such numbers as congruere meaning to “come together”. Many authors have contributed to the study of the properties of and computation of congruent numbers, including Alter, Curtz and Kubota [1] who conjectured that if n is congruent to 5, 6 or 7 modulo 8 then n is a congruent number. This was shown to be true, subject to the weak Birch and Swinnerton-Dyer conjecture by Stephens [35] in 1975.

Supported by EPSRC grant number EP/G004870/1. All authors were supported at workshops administered by AIMath under NSF Grant number DMS-0757627.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 186–200, 2010. c Springer-Verlag Berlin Heidelberg 2010

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The earliest computations of congruent numbers are due to the classical Islamic mathematicians, the congruent numbers 5, 6, 14, 15, 21, 30, 34, 65, 70, 110, 154, 190, 210, 221, 231, 246, 290, 390, 429, 546 and ten other substantially larger congruent numbers being known to them. Fibonacci, Genocchi and G´erardin added 7, 22, 41, 69, 77 and forty-three other values below 1000. Fermat showed that 1 is not congruent in 1659, something which had been stated but not proved by Fibonacci in 1225. By scaling this is equivalent to the fact that no square number can be congruent. Bastien [5] observed that numbers which are prime and 3 modulo 8, products of two such primes, twice a prime which is 5 modulo 8, twice a product of two such primes or twice a prime which is 9 modulo 16 are not congruent. Numerous congruent numbers were demonstrated by Alter, Curtz and Kubota [1] and by Jean Lagrange in his thesis [23]. See Guy [17] for further details on the history of the computation of congruent numbers. More recently Monsky [28] showed that, for example, two times the product of primes p ≡ 1 (mod 8) and q ≡ 7 (mod 8) with (p/q) = −1 is a congruent number. For a history of results along these lines see Feng [13]. Also see [27]. By 1980 there were numerous values below 1000 not yet decided either way. By 1986 Kramarz [26] had handled all cases up to 2000, and Noe’s list up to 10000 is included in Sloane’s database. Matsuno had reached 300000 in 2005. Subject to a conjecture of Birch and Swinnerton-Dyer (see Tunnell’s Criterion below), Rogers [32] had computed all congruent numbers up to 107 by the year 2000 and Mike Rubinstein (personal communication) had computed all congruent numbers up to 109 a few years prior to the current work. We had raised that limit to 2 × 1010 by 2008 and with this paper the current plateau is now 1012 . By counting representations of n or n/2 by ternary quadratic forms, previous 3 computations had the asymptotic running time O(N 2 ) for computing coeﬃcients up to a limit N . In this paper we describe a multimodular Fast Fourier Transform technique with quasilinear runtime. We demonstrate that the method is practical as it permits computations whose data is considerably larger than main memory.

2

Relating Congruent Numbers to Elliptic Curves

If three rational squares in arithmetic progression have common diﬀerence n, their product is a square: v 2 = (u2 − n)u2 (u2 + n) = (u2 )3 − n2 (u2 ). This shows immediately that if n is congruent then it corresponds to a point (u2 , v) on the elliptic curve En : y 2 = x3 − n2 x. Along similar lines, in 1877 Lucas showed that n is congruent iﬀ y 2 = x4 − n2 has a positive rational solution. The group of points on the curve En is isomorphic to (Z/2Z × Z/2Z) × Zr where r is the rank. The three non-trivial 2-torsion points do not yield congruent numbers and so n is congruent iﬀ En has positive rank.

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There has been considerable interest in verifying that the curves En for which n is thought to be congruent do in fact have positive rank. See for example the tables of Elkies [12]. As the sign of the functional equation of L(En /Q, s) is +1 for n ≡ 1, 2, 3 (mod 8) and −1 for n ≡ 5, 6, 7 (mod 8) [7] then by the Parity Conjecture (a special case of the Birch and Swinnerton-Dyer Conjecture) we expect that the rank of En is even in the +1 case and odd in the −1 case. This is an interesting test of the Birch and Swinnerton-Dyer Conjecture. 2.1

Tunnell’s Criterion

In 1983 Jerrold Tunnell gave the following criterion: Theorem 1 (Tunnell). Let n be an odd squarefree positive integer. Set a(n) = #{(x, y, z) ∈ Z3 | x2 + 2y 2 + 8z 2 = n} − 2 #{(x, y, z) ∈ Z3 | x2 + 2y 2 + 32z 2 = n}, b(n) = #{(x, y, z) ∈ Z3 | x2 + 4y 2 + 8z 2 = n} − 2 #{(x, y, z) ∈ Z3 | x2 + 4y 2 + 32z 2 = n}. If n is congruent then a(n) = 0. If 2n is congruent then b(n) = 0. Moreover, if the weak BSD conjecture is true for the curve y 2 = x3 − n2 x then the converses also hold: a(n) = 0 implies n is congruent and b(n) = 0 implies 2n is congruent. We explain brieﬂy the connection between the curves En and Tunnell’s criterion. The curve En is a quadratic twist of the curve E : y 2 = x3 − x. Associated to E is a weight 2 newform F (z) = η(4z)2 η(8z)2 ∈ S2new (Γ0 (32)) such that L(E, s) = L(F, s), where L(E, s) is the Hasse-Weil L-series of the elliptic curve E and L(F, s) is the Mellin transform of the modular form F . If we write L(E, s) = bm m−s then L(En , s) = LF (χD , s) = χD (m)bm m−s , where D = n if n ≡ 1 (mod 4) and D = 4n if n ≡ 2, 3 (mod 4). The importance of this fact is that the conjecture of Birch and SwinnertonDyer (applied to En ) then gives a condition on when n can be congruent: Conjecture 1 (Birch and Swinnerton-Dyer). If E is an elliptic curve deﬁned over Q then L(E, 1) = 0 iﬀ E has positive rank. The following theorem of Shimura gives a link between modular forms of half integer weight k/2 and forms of integer weight k − 1. The correspondence is called a Shimura lift. We are interested in this theorem in the case k = 3. m Theorem 2 (Shimura). Let f (z) = ∞ ∈ Sk/2 (4N, χ) be a modum=1 a(m)q lar form of weight k/2 for Γ0 (4N ) (actually Δ0 (4N )) with χ a Dirichlet character modulo 4N and suppose that Tp2 (f ) = ωp f for all primes p, where Tp2 are the

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m Hecke operators. Deﬁne F (z) = ∞ where the values A(m) are given m=1 A(m)q by ∞ A(m)m−s = (1 − ω p ps + χ(p)2 pk−2−2s )−1 . p

m=1

Then for some integer N0 divisible by the conductor of χ2 we have that F (z) ∈ Mk−1 (N0 , χ2 ), i.e. F (z) is an integer weight modular form of weight k − 1. As mentioned above, we are interested in whether or not the L-series L(En , s) vanishes at s = 1. Tunnell made use of a result of Waldspurger to access information about the value of these L-series at s = 1. The basic idea behind Waldspurger’s Theorem and related results is that if F (z) is the Shimura lift of f (z) as per the previous theorem, then the value of L(Fn , s) at s = (k − 1)/2 for squarefree n, is proportional to the n-th Fourier coeﬃcient of f (z). In particular if suitable forms f (z) can be identiﬁed then it is possible to determine when L(Fn , s) vanishes at the centre of the critical strip, s = (k − 1)/2. The following result (which is a reformulation of the theorem of Waldspurger, see [30]) formulates this more precisely. Theorem 3 (Waldspurger). If F (z) = ∞ a(m)q m ∈ S new (Γ (M )) and k−1

m=1

0

δ = ±1 is the sign of the functional equation of L(F, s) then there is a Dirichlet character χ modulo 4N , a positive integer M |N , a nonzero complex number ΩF and a nonzero Hecke eigenform ∞

f (z) =

bF (m)q m ∈ Sk/2 (Γ0 (4N ), χ)

m=1

such that there are fundamental disciminants n, coprime to 4N and with the same sign as δ that lie in arithmetic progressions and for which bF (n0 )2 = εn ·

k/2

L(Fn , (k − 1)/2)n0 ΩF

,

where εn is algebraic and n0 = |n| if n is odd, otherwise n0 = |n|/4. For all other n with the same sign as δ the Fourier coeﬃcients bF (n0 ) vanish. By careful examination of the conditions of Waldspurger’s Theorem, Tunnell was able to construct modular forms which allowed for identiﬁcation of the values of n for which L(En , s) vanishes at s = 1. Even better yet, he was able to write these weight 3/2 modular forms as the product of explicit theta series. ∞ 2 Following Tunnel we let g = (θ1 − θ4 )(θ8 − 2θ32 ), where θt = m=−∞ q tm . Then ∞ g θ2 = a(m)q m ∈ S 32 (Γ0 (128)), g θ4 =

m=1 ∞ m=1

b(m)q m ∈ S 32 (Γ0 (128), χ),

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where χ(r) = 8r . Note this agrees with the formulas for a(n) and b(n) given above for odd n. Tunnell proved that these were Hecke eigenforms whose Shimura lift was F (z). He then showed that if n is an odd positive squarefree integer then Ω L(En , 1) = a(n)2 · √ , 4 n

Ω and L(E2n , 1) = b(n)2 · √ , 2 2n

for a certain real period Ω. For further information on Tunnell’s approach, see Tunnell’s original paper [39] and the books by Ono [30] and Koblitz [25]. The above result of Tunnell allows us to determine congruent numbers, subject to the BSD conjecture, simply by checking whether the Fourier coeﬃcients a(n) and b(n) are zero. Thus the entire problem of determining congruent numbers is reduced to computing the theta series g and θt and performing power series multiplications. We actually use slight modiﬁcations of these θ-functions, which allow us to exploit additional information on arithmetic progressions. 2.2

Our Θ-Functions

Rather than use the modular forms of Tunnell given above, we note (as suggested to us by N. D. Elkies) that we can split the problem(s) up by a factor of two. The series g θ2 and g θ4 can each be split into a sum of two similar products, each of which is supported on (approximately) half as many coeﬃcients. Indeed, we have the following product expressions: θ8 (θ1 − θ4 ) × (θ8 − 2θ32 ) = a(n) q n , n≡1

(θ2 − θ8 )(θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡3

θ16 (θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡1

(θ4 − θ16 )(θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡5

(mod 8)

a(n) q n ,

(mod 8)

b(n) q n ,

(mod 8)

b(n) q n .

(mod 8)

As each factor above is a (shifted) power series in q 8 , our complexity reduces by a factor of 8. Indeed, the second factor above is θ8 − 2θ32 = C(q 8 ) where C = θ1 − 2θ4 is a sparse power series which can be quickly computed. For the ﬁrst factor, we can easily compute theta series A1 , A3 , B1 and B5 such that θ8 (θ1 − θ4 ) = q A1 (q 8 ),

(θ2 − θ8 )(θ1 − θ4 ) = q 3 A3 (q 8 ),

θ16 (θ1 − θ4 ) = q B1 (q 8 ),

(θ4 − θ16 )(θ1 − θ4 ) = q 5 B5 (q 8 ).

These series can be computed directly by counting lattice points in 2 dimensions, taking approximately linear time. So we only need one convolution for each of

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the four cases: two convolutions of 1.25 × 1011 coeﬃcients (for the a(n),) and two convolutions of 6.25 × 1010 coeﬃcients (for the b(n).) The computation of the Θ-series can be done eﬃciently in intervals, √ √ taking essentially N time to compute the coeﬃcients between N and N + N . For N up to 1.25×1011 this ensures each interval includes less than 500, 000 coeﬃcients, ﬁtting comfortably in a typical L2 cache. This cache locality is essential for the computation.

3

“Out-of-Core” Fast Fourier Transform Methods

The complex FFT algorithm was essentially known to Gauss in 1805 (see [19]) but developed in its current form by Cooley and Tukey in 1965 [9]. In 1971 Sch¨ onhage and Strassen presented two algorithms for multiplication of large integers based on the FFT [33]. One of these methods, where the ﬁeld of complex numbers is replaced by a ﬁnite ring Z/pZ containing a principal root of unity of order 2K , has become known as the Sch¨onhage-Strassen method. It can multiply two n bit numbers in asymptotic time O(n logn log logn). Power series multiplication can be eﬀected by truncating a full polynomial multiplication of two n term polynomials to length n and by encoding the polynomial multiplication as an integer multiplication using Kronecker Segmentation. The latter technique is that of evaluating the polynomials at a power of 2 chosen suﬃciently large that the product coeﬃcients can be identiﬁed from their binary representation in the output of the large integer multiplication. In the literature, FFT computations whose data exceeds the size of available memory are referred to as out-of-core FFT methods. The literature is replete with many references to methods for defunct vector architectures, or for distributed memory systems, including those with tree, mesh or hypercube architectures (see [2], [8], [24], [36] and [38] for examples), where the emphasis is often on minimising interprocess communication. In our case, we used a shared memory system where available memory was a limiting factor for the computation, forcing an “out-of-core” computation. The principal issue with standard FFT algorithms in a hierarchical memory system (e.g. where disk is one level of the hierarchy) is that at least K complete passes over the data are required for a convolution of length 2K . However disk access is typically a couple of orders of magnitude slower than memory access, making such algorithms prohibitively slow. The ﬁrst FFT technique to deal with a memory hierarchy is that of Gentleman and Sande [20]. The method has become known as Bailey’s Four Step method (in the context of complex FFT’s), see [3]. The idea is to break the data into a two dimensional array and perform small FFT’s in the horizontal and then in the vertical directions, with certain “twiddle factors” applied between the two stages. A ﬁnal transpose stage then follows. This basic strategy is also sometimes referred to as the Matrix Fourier Algorithm. Bailey’s method can be extended to a six (or ﬁve) step three dimensional method and beyond. See the above cited paper of Bailey’s for older references,

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or [31] for a more recent reference. For applications to integer multiplication, see for example [21]. Some other algorithms for out-of-core FFT’s include the algorithm of Cormen [10] based on the in-core method of Swarztrauber, the method of Takahashi [37] for the Parallel Disk Model (PDM) of Vitter and Shriver and the parallel FFT method of Vitter and Shriver [40] for a two level memory system. Another technique commonly used for out-of-core FFT computations is the method of performing Number Theoretic Transforms (NTTs) with Chinese Remainder Theorem reconstitution. A Number Theoretic Transform is an FFT in the ring R = Z/pZ for a specially chosen small prime p sometimes called an “FFT prime”. Usually p is chosen to ﬁt into a single machine word, i.e. 32 or 64 bits. For this to work, R must have suﬃciently many roots of unity to support the convolution. FFT primes p can be chosen to be of the form p = m2K + 1 for some small value m. Let x be a primitive root modulo p, i.e. a value x such that xp−1 ≡ 1 (mod p), but such that xa is not 1 (mod p) for any value of a dividing p − 1. Then xm is a 2K -th root of unity, supporting convolutions of length 2K . In order to perform an out-of-core polynomial multiplication h(x) = fA (x) × gB (x) using NTTs the coeﬃcients of the two polynomials are ﬁrst reduced modulo a number of FFT primes. Then the Chinese Remainder Algorithm can be used to reconstitute the full product from the results of the NTTs. The NTT transform method is a standard one for computing large numbers of digits of π. See for example the paper of Bailey, [4] where two FFT primes were used, in that case to avoid the necessity of quad-precision arithmetic in a complex FFT. The same paper also mentions a proposal to use three FFT primes, even avoiding double precision arithmetic in the NTT’s, but imposing severe restriction on the length of convolution possible for machines of that era. More recently Carey Bloodworth’s record-holding programs used eight NTTs and CRT, and were topped in 2004 by the program of Xavier Gourdon [16] for greatest number of digits of π computed on a home computer. Gourdon’s program uses an unspeciﬁed number of NTTs. More recent than our theta computation is the record π computation of Fabrice Bellard [6], using NTTs and a home computer. For out-of-core operations, his computation made use of eight 64 bit moduli, however for in-core components he made use of ﬂoating point arithmetic and unproven, heuristically chosen error bounds on the precision required.

4

The Power Series Multiplication

For any method using FFTs, optimised for out-of-core operation, the main bottleneck becomes disk I/O. To minimise this, it is not only important to minimise the number of passes over the data, but also to minimise the amount of data that must be traversed.

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Two issues arise. Firstly, techniques such as the Sch¨ onhage-Strassen technique are diﬃcult to optimise for convolution lengths which are not a power of two, in the worst case increasing the disk I/O by a factor of two. Secondly, when performing a large FFT or a small number of very large NTTs that do not ﬁt into memory, even when combined with Bailey’s technique, truncation of the polynomial multiplications occurs after each large FFT computation. In other words, the disk I/O occurs for the entire untruncated FFT computation. For multiplication of integers of n bits, these methods require a total disk I/O of 12n bits with a peak usage of 8n bits. Our technique reduces this to a total disk I/O of just over 6n bits with a peak usage of just over 4n bits. This is achieved by eﬃcient multimodular reduction and CRT recombination using a large number of small primes p with truncation occurring in-core. One advantage of using NTTs is that the primes p can be chosen in such a way that reduction modulo p can be performed very eﬃciently. E.g. for primes p of the form 2K + 1 reduction modulo p can be performed with subtractions rather than expensive divisions. More generally, many primes of the form p = m2K + 1 for small values of m can be used. Reduction modulo p can still be computed relatively eﬃciently. For our computation we chose to use many general word sized primes p and an alternative method of performing polynomial multiplications over Z/pZ. For the largest polynomial multiplications, in the 1 (mod 8) and 3 (mod 8) cases, we used just over 500 primes. The main reason for this choice was the existence of well-tested, high performance packages for doing such computations, such as FLINT [18] and zn poly [22]. There was also an advantage in having two separate implementations of arithmetic in Z/pZ[x] in that comparisons could be made between the two implementations whilst testing. The implementation of multiplication in Z/pZ[x] in zn poly is highly optimised. It oﬀers a thread-safe, cache-eﬃcient, truncated, Sch¨onhageNussbaumer convolution [21], which performs signiﬁcantly better than other implementations for general primes p. In contrast, Victor Shoup’s NTL package [34] was the only library we were aware of with asymptotically fast NTTs. However NTL is not threadsafe. Also, numerous recent improvements in polynomial arithmetic are not reﬂected in NTL, which is no longer under active development. Our implementation made use of 16 CPU cores. The data for all 16 threads must be in memory simultaneously, and thus to beneﬁt from the disk-to-memory ratio of the multimodular approach it was necessary to use a number of primes signiﬁcantly larger than this. One disadvantage of using so many primes is that multimodular reduction and CRT reconstruction constitute a signiﬁcant part of the runtime. The naive approach is to reduce the large coeﬃcients of the polynomials in Z[x] modulo each of the primes p in turn and to similarly reconstruct each coeﬃcient one prime at a time. However for n1 coeﬃcients in Z of n2 bits, reconstruction using this approach will take time O(n1 n22 ). This is asymptotically much worse than the time required to do the actual polynomial multiplications over Z/pZ.

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In order to avoid this, a divide-and-conquer approach was used for the multimodular reduction and recombination phases. This completes the CRT recombination in time O(n2 log2 n1 n2 ) ignoring smaller log log factors. Note that this is asymptotically a log factor greater than the time for the multiplications, however the running time is still quasilinear in the input size. The extra theoretical complexity of our approach is oﬀset by the “embarrasingly parallel” nature of the multiplications, multimodular reduction and recombinations and the large saving in disk I/O (by far the bottleneck for our computation). For a straightforward description of the divide-and-conquer approach to the CRT algorithm see [41], pages 57–58. Similar preconditioning and a divide-andconquer approach was of course applied to the multimodular reduction phase. A slight adjustment was also made to both the reduction and CRT phases to cope with a number of primes which is not a power of 2. 4.1

The Algorithm in Pseudocode

We now describe our algorithm in full. We make use of two sets of disk ﬁles, F = {Fi : i = 0, 1, .., F ILES − 1} and G = {Gj : j = 0, 1, .., F ILES − 1} . In our implementation we used FILES = 500 for the 1 (mod 8) and 3 (mod 8) computations and half that in the 2 (mod 16) and 10 (mod 16) computations. We also set: LIMIT (the length of the theta functions), BLOCK (number of theta coeﬃcients computed at a time), BUNDLE (number of theta coeﬃcients bundled, using Kronecker Segmentation, into each large polynomial coeﬃcient) and THREADS (number of threads used), PRIMES (number of primes used in multimodular reduction and CRT). We experimented with various values for BUNDLE from 500 to 1000. To simplify the computation, PRIMES was rounded up to a multiple THREADS. The value LIMIT, (1012 /8 in the 1, 3 (mod 8) cases and 1012 /16 in the 2, 10 (mod 16) cases), was chosen to be a multiple of FILES×BUNDLE, and a multiple of FILES×BLOCK. Coeﬃcients of the product of our θ-series comfortably ﬁt into 16 signed bits. Thus the Kronecker Segmentation phase used zero-padded ﬁelds of 16 bits. Throughout the following we write FOR i = 0 to A and similar expressions, by which we mean i in 0 0, each vertex in Vi has exactly one edge leading to a vertex in Vi−1 , and every edge not on the crater is of this form. (b) For i < h, each vertex in Vi has degree + 1. We call the level Vh the ﬂoor of the volcano. Vertices lying on the ﬂoor have degree 1. The following proposition [23] follows essentially from [14, Prop. 23]. Proposition 1. Let p be a prime number, q = pr , and dπ = t2 − 4q. Take = p another prime number. Let G be the undirected graph with vertex set Ellt (Fq ) and edges -isogenies deﬁned over Fq . We denote by h the largest power of dividing the conductor of dπ . Then the connected components of G that do not contain curves with j-invariant 0 or 1728 are -volcanoes of height h and for each component V , we have : (a) The elliptic curve whose j-invariants lie in V0 have endomorphism rings isomorphic to some Od0 ⊇ Odπ whose conductor is not divisible by . (b) The elliptic curve whose j-invariants lie in Vi have endomorphism rings isomorphic to Odi , where di = 2i d0 . Elliptic curves are determined by their j-invariant, up to a twist1 . Throughout the paper, we refer to a vertex in a volcano by giving the curve or its j-invariant. Exploring the volcano. Given a curve E on an -volcano, two methods are known to ﬁnd its neighbours. The ﬁrst method relies on the use of modular polynomials. The -th modular polynomial, denoted by Φ (X, Y ) is a polynomial with integer coeﬃcients. It satisﬁes the following property: given two elliptic curves E and E with j-invariants j(E) and j(E ) in Fq , there is an -isogeny deﬁned over Fq , if and only if, #E(Fq ) = #E (Fq ) and Φ (j(E), j(E )) = 0. As a consequence, the curves related to E via an -isogeny can be found by solving Φ (X, j(E)) = 0. As stated in [20], this polynomial2 may have 0, 1, 2 or + 1 roots in Fq . In order to ﬁnd an edge on the volcano, it suﬃces to ﬁnd a root j of this polynomial. Finally, if we need the equation of the curve E with j-invariant j , we may use the formula in [20]. The second method to build -isogenous curves constructs, given a point P of order on E, the -isogeny I : E → E whose kernel G is generated by P using 1 2

For a deﬁnition of twists of elliptic curves, refer to [21]. The case where the modular polynomial does not have any root corresponds to a degenerate case of isogeny volcanoes containing a single curve and no -isogenies.

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V´elu’s classical formulae [24] in an extension ﬁeld Fqr . To use this approach, we need the explicit coordinates of points of order on E. We denote by Gi , 1 ≤ i ≤ + 1, the + 1 subgroups of order of E. In [17], Miret and al. give the degree ri of the smallest extension ﬁeld of Fq such that Gi ⊂ Fqri , 1 ≤ i ≤ + 1. This degree is related to the order of q in the group F∗ , that we denote by ord (q). Proposition 2. Let E deﬁned over Fq be an elliptic curve with k rational isogenies, > 2, and let Gi , 1 ≤ i ≤ k, be their kernels, and let ri be the minimum value for which Gi ⊂ E(Fqri ). (a) If k = 1 then r1 = ord (q) or r1 = 2ord (q). (b) If k = + 1 then either ri = ord (q) for all i, or ri = 2ord (q) for all i. (c) If k = 2 then ri | − 1 for i = 1, 2. We also need the following corollary [17]. ˜ its twist. If E/Fq Corollary 1. Let E/Fq be an elliptic curve over Fq and E ˜ has 1 or + 1 rational -isogenies, then #E(Fqord q ) or #E(Fqord q ) is a multiple of . Moreover, if there are + 1 rational isogenies, then it is a multiple of 2 .

Z n 1 Z

×

Z n 2 Z

Z n1 +1 Z

×

Z n2 −1 Z

Z n1 +n2 −1 Z

×

Z Z

Z n1 +n2 Z

Fig. 1. A regular volcano

The group structure of the elliptic curve on the volcano. Lenstra [13] relates the group structure of an elliptic curve to its endomorphism ring by proving that E(Fq ) OE /(π − 1) as OE -modules. It is thus natural to see how this structure relates to the isogeny volcano. From Lenstra’s equation, we can deduce that E(Fq ) Z/M Z × Z/N Z. We write π = a + gω, with: 1+√d K (t − g)/2 if dK ≡ 1 (mod 4) a= and ω = √ 2 t/2 dK if dK ≡ 2, 3 (mod 4) where dK is the discriminant of the quadratic imaginary ﬁeld containing OE . Note that N is maximal such that E[N ] ⊂ E(Fq ) and by [19, Lemma 1] we get that N = gcd(a − 1, g/f ). Note moreover that N |M , N |(q − 1) and M N = #E(Fq ). This implies that on a -volcano the structure of all the curves in a given level is the same.

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Let E be a curve on the isogeny volcano such that v (N ) < v (M ). As explained in [16] (in the case = 2, but the result is general), a is such that v (a − 1) ≥ min {v (g), v (#E(Fq ))/2} . Since N = gcd(a−1, g/f ) and v (N ) ≤ v (#E(Fq ))/2, it follows that v (N ) = v (g/f ). As we descend, the valuation at of the conductor f increases by 1 at each level (by proposition 1b). This implies that the -valuation of N for curves at each level decreases by 1 and is equal to 0 for curves lying on the ﬂoor. Note that if v (#E(Fq )) is even and the height h of the volcano is greater than v (#E(Fq )), the structure of the -torsion group is unaltered from the crater down to the level h − v (#E(Fq ))/2. From this level down, the structure of the -torsion groups starts changing as explained above. In the sequel, we call this level the ﬁrst stability level.3 A volcano with ﬁrst stability level equal to 0, i.e. on the crater, is called regular. Notations. Let n ≥ 0. We denote by E[n ] the n -torsion subgroup, i.e. the ¯ q ), by E[n ](Fqk ) the subgroup subgroup of points of order n on the curve E(F n of points of order deﬁned over an extension ﬁeld of Fq and by E[∞ ](Fq ) the -Sylow subgroup of E(Fq ). Given a point P ∈ E[n ](Fq ), we also need to know the degree of the smallest extension ﬁeld containing an n+1 -torsion point such that P˜ = P . The following result is taken from [7]. Proposition 3. Let E/Fq be an elliptic curve which lies on a -volcano whose height h(V ) is diﬀerent from 0. Then the height of V , the -volcano of the curve E/Fqs is h(V ) = h(V ) + v (s). From this proposition, it follows easily that if the structure of -torsion on the curve E/Fq is Z/n1 Z × Z/n2 Z, then the smallest extension in which the structure of the -torsion changes is Fq . We sketch here the proof in the case n1 = n2 = n, which is the only case in which we consider volcanoes over extension ﬁelds in this paper4 . First of all, note that E lies on a -volcano V /Fq of height at least n. We consider a curve E lying on the ﬂoor of V /Fq such that there is a descending path of isogenies between E and E . Obviously, we have E [∞ ](Fq ) Z/2n Z. By proposition 3, V /Fq has one extra down level, which means that the curve E is no longer on the ﬂoor, but on the level just above the ﬂoor. Consequently, we have that E [] ⊂ E (Fq ) and, moreover, E [∞ ](Fq ) Z/2n+Δ Z × Z/Z. By ascending on the volcano from E to E, we deduce that the structure of the -torsion of E over Fq is necessarily E[∞ ](Fq ) Z/n+Δ Z × Z/n+1 Z. Moreover, Δ ≥ 1, because if it were 0, the height of V /Fq would be n.

3 4

Miret et al. call it simply the stability level. For the proof in the general case, see [11].

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Background on Pairings

Let E be an elliptic curve deﬁned over some ﬁnite ﬁeld Fq , m a number such that m| gcd(#E(Fq ), q − 1). Let P ∈ E[m](Fq ) and Q ∈ E(Fq )/mE(Fq ). Let fm,P be the function whose divisor5 is m(P ) − m(O), where O is the point at inﬁnity of the curve E. Take R a random point in E(Fq ) such as the support of the divisor D = (Q + R) − (R) is disjoint from the support of fm,P . Then we can deﬁne the Tate pairing as follows: tm : E[m] × E(Fq )/mE(Fq ) → F∗q /(F∗q )m (P, Q) → fm,P (Q + R)/fm,P (R). The Tate pairing is a bilinear non-degenerate application, i.e. for all P ∈ E[m](Fq ) diﬀerent from O there is a Q ∈ E(Fq )/mE(Fq ) such that Tm (P, Q) = 1. The output of the pairing is only deﬁned up to a coset of (F∗q )m . However, for implementation purposes, it is useful to have a uniquely deﬁned value and to use the reduced Tate pairing, i.e. Tm (P, Q) = tm (P, Q)(q−1)/m ∈ μm , where μm denotes the group of m-th roots of unity. Pairing computation can be done in time O(log m) using Miller’s algorithm [15]. For more details and properties of pairings, the reader can refer to [9]. Note that in the recent years, in view of cryptographic applications, many implementation techniques have been developed and pairings on elliptic curves can be computed very eﬃciently6 . Suppose now that m = n , with n ≥ 1 and prime. Now let P and Q be two n -torsion points on E. We deﬁne the following symmetric pairing [12] 1

S(P, Q) = (Tn (P, Q) Tn (Q, P )) 2 .

(1)

Note that for any point P , Tn (P, P ) = S(P, P ). In the remainder of this paper, we call S(P, P ) the self-pairing of P . We focus on the case where the pairing S is non-constant. Suppose now that P and Q are two linearly independent n torsion points. Then all n -torsion points R can be expressed as R = aP + bQ. Using bilinearity and symmetry of the S-pairing, we get log(S(R, R)) = a2 log(S(P, P )) + 2ab log(S(P, Q)) + b2 log(S(Q, Q)) (mod n ), where log is a discrete logarithm function in μn . We denote by k the largest integer such that the polynomial P(a, b) = a2 log(S(P, P )) + 2ab log(S(P, Q)) + b2 log(S(Q, Q))

(2)

is identically zero modulo k and nonzero modulo k+1 . Obviously, since S is non-constant we have 0 ≤ k < n. Dividing by k , we may thus view P as a polynomial in F [a, b]. When we want to emphasize the choice of E and n , we write PE,n instead of P. 5 6

For background on divisors, see [21]. See [10] for a fast recent implementation.

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Since P is a non-zero quadratic polynomial, it has at most two homogeneous roots, which means that that from all the + 1 subgroups of E[n ]/E[n−1 ]

(Z/Z)2 , at most 2 have self-pairings in μk (see also [12]). In the remainder of this paper, we denote by NE,n the number of zeros of PE,n . Note that this number does not depend on the choice of the two generators P and Q of the n -torsion subgroup E[n ]. Moreover, we say that a n -torsion point R has degenerate self-pairing if Tn (R, R) is a k -th root of unity and that R has nondegenerate self-pairing if Tn (R, R) is a primitive k+1 -th root of unity. Also, if Tn (R, R) is a primitive n -th root of unity, we say that R has primitive selfpairing.

4

Determining Directions on the Volcano

In this section, we explain how we can distinguish between diﬀerent directions on the volcano by making use of pairings. We give some lemmas explaining the relations between pairings on two isogenous curves. Lemma 1. Suppose E/Fq is an elliptic curve and P, Q are points in E(Fq ) of ˜ ∈ E[F ¯ q ] the points such that P˜ = P and order n , n ≥ 1. Denote by P˜ , Q ˜ Q = Q. We have the following relations for the Tate pairing ˜ 2 = Tn (P, Q). ˜ ∈ E[Fq ], then Tn+1 (P˜ , Q) (a) If P˜ , Q ˜ ∈ E[Fq ]\E[Fq ], then Tn+1 (P˜ , Q) ˜ = Tn (P, Q). (b) Suppose ≥ 3. If Q Proof. a. By writing down the divisors of the functions fn+1 ,P˜ , fn ,P˜ , fn ,P , one can easily check that n fn+1 ,P˜ = (f,P˜ ) · fn ,P . We evaluate these functions at some points Q + R and R (where R is carefully chosen) and raise the equality to the power (q − 1)/n . b. Due to the equality on divisors div(fn+1 ,P ) = div(fn ,P ), we have ) q ˜ = T (F ˜ Tn+1 (P˜ , Q) (P, Q), n (F

where Tn q show that

)

is the n -Tate pairing for E deﬁned over Fq . It suﬃces then to

(F ) ˜ Tn q (P, Q)

= Tn (P, Q). We have (1+q+···+q−1 )(q−1)

(F ) ˜ = fn ,P ([Q ˜ + R] − [R]) n Tn q (P, Q) ˜ + R) + (π(Q) ˜ + R) + (π 2 (Q) ˜ + R) + . . . = fn ,P ((Q

˜ + R) − (R)) + (π −1 (Q)

(q−1) n

(3)

where R is a random point deﬁned over Fq . It is now easy to see that for ≥ 3, ˜ + . . . + π −1 (Q) ˜ = Q ˜ = Q, ˜ + π(Q) ˜ + π 2 (Q) Q

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˜ = Q ˜ + T , where T is a point of order . By applying Weil’s because π(Q) reciprocity law [21, Ex. II.2.11], it follows that the equation (3) becomes: (F ) ˜ Tn q (P, Q)

=

fn ,P (Q + R) fn ,P (R)

q−1 n

f ((P ) − (O))q−1 ,

˜ + R) + (π(Q) ˜ + R) + (π 2 (Q) ˜ + R) + ... + where f is such that div(f ) = (Q −1 ˜ (π (Q) + R) − (Q + R) − ( − 1)(R). Note that this divisor is Fq -rational, so f ((P ) − (O))q−1 = 1. This concludes the proof. Lemma 2. (a) Let φ : E → E be a separable isogeny of degree d deﬁned over Fq , P a -torsion on the curve E such that φ(P ) is a -torsion point on E , and Q a point on E. Then we have T (φ(P ), φ(Q)) = T (P, Q)d . (b) Let φ : E → E be a separable isogeny of degree deﬁned over Fq , P a -torsion point such that Ker φ = P and Q a point on the curve E. Then we have T (φ(P ), φ(Q)) = T (P, Q) . Proof. Proof omitted for lack of space. See [3, Th. IX.9.4] for (a), [11] for (b). Proposition 4. Let E be an elliptic curve deﬁned a ﬁnite ﬁeld Fq and assume that E[∞ ](Fp ) is isomorphic to Z/n1 Z × Z/n2 Z (with n1 ≥ n2 ). Suppose that there is a n2 -torsion point P such that Tn2 (P, P ) is a primitive n2 -th root of unity. Then the -isogeny whose kernel is generated by n2 −1 P is descending. Moreover, the curve E does not lie above the ﬁrst stability level of the corresponding -volcano. Proof. Let I1 : E → E1 be the isogeny whose kernel is generated by n2 −1 P and suppose this isogeny is ascending or horizontal. This means that E1 [n2 ] is deﬁned over Fq . Take Q another n2 -torsion point on E, such that E[n2 ] = P, Q and denote by Q1 = I1 (Q). One can easily check that the dual of I1 has kernel generated by n2 −1 Q1 . It follows that there is a point P1 ∈ E1 [n2 ] such that P = Iˆ1 (P1 ). By Lemma 2 this means that T (P, P ) ∈ μn2 −1 , which is false. This proves not only that the isogeny is descending, but also that the structure of the -torsion is diﬀerent at the level of E1 . Hence E cannot be above the stability level. Proposition 5. Let ≥ 3 a prime number and suppose that E/Fq is a curve which lies in a -volcano and on the ﬁrst stability level. Suppose E[∞ ](Fq )

Z/n1 Z×Z/n2 Z, n1 ≥ n2 . Then there is at least one n2 -torsion point R ∈ E(Fq ) with primitive self-pairing. Proof. Let P be a n1 -torsion point and Q be a n2 -torsion point such that {P, Q} generates E[∞ ](Fq ). I

1 E1 be a descending -isogeny and Case 1. Suppose n1 ≥ n2 ≥ 2. Let E −→ n1 +1 n2 −1 denote by P1 and Q1 the and -torsion points generating E1 [∞ ](Fp ). Moreover, without loss of generality, we may assume that I1 (P ) = P1 and I1 (Q) = Q1 . If Tn2 −1 (Q1 , Q1 ) is a primitive n2 −1 -th root of unity, Tn2 (Q, Q) is

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a primitive n2 -th root of unity by Lemma 2. If not, from the non-degeneration of the pairing, we deduce that Tn2 −1 (Q1 , P1 ) is a primitive n2 −1 -th root of unity, which means that Tn2 −1 (Q1 , P1 ) is a n2 −2 -th primitive root of unity. By applying Lemma 2, we get Tn2 (Q, P ) ∈ μn2 −1 at best. It follows that Tn2 (Q, Q) ∈ μn2 by the non-degeneracy of the pairing. Case 2. If n2 = 1, then consider the volcano deﬁned over the extension ﬁeld Fq . ˜ ∈ E(Fq ) with Q = Q. ˜ We obviously have 2 |q − 1 There is a 2 -torsion point Q and from Lemma 1, we get T2 (P˜ , P˜ ) = T (P, P ). By applying Case 1, we get that T2 (P˜ , P˜ ) is a primitive 2 -th root of unity, so T (P, P ) is a primitive -th root of unity. Two stability levels. Remember that in any irregular volcano, v (#E(Fq )) is even and the height h of the volcano is greater than v (#E(Fq )). Moreover, all curves at the top of the volcano have E[∞ ](Fq ) Z/n2 Z × Z/n2 Z with n2 = v (#E(Fq )). The existence of a primitive self-pairing of a n2 -torsion point on any curve lying on the ﬁrst stability level implies that the polynomial P is nonzero at every level from the ﬁrst stability level up to the level max(h + 1 − 2n2 , 0) (by Lemma 2). We call this level the second level of stability. On the second stability level there is at least one point of order n2 with pairing equal to a primitive -th root of unity. At every level above the second stability level all polynomials PE,n2 may be zero7 . Consider now E a curve on the second stability level and I : E → E1 an ascending isogeny. Let P be a n2 -torsion point on E and assume that Tn2 (P, P ) ∈ μ∗ . We denote by P˜ ∈ E(Fq )\E(Fq ) the point such that P˜ = P . By Lemma 1 we get Tn2 +1 (P˜ , P˜ ) is a primitive 2 -th root of unity. It follows by Lemma 2 that Tn2 (I(P ), I(P )) is a primitive -th root of unity. We deduce that PE1 ,n2 +1 corresponding to E1 /Fq is non-zero. Applying this reasoning repeatedly, we conclude that for every curve E above the second stability level there is an extension ﬁeld Fqs such that the polynomial PE,n2 +s associated to the curve deﬁned over Fqs is non-zero. When the second stability level of a volcano is 0, we say that the volcano is almost regular. We now make use of a result on the representation of ideal classes of orders in imaginary quadratic ﬁelds. This is Corollary 7.17 from [5]. Lemma 3. Let O be an order in an imaginary quadratic ﬁeld. Given a nonzero integer M , then every ideal class in Cl(O) contains a proper O-ideal whose norm is relatively prime to M . Proposition 6. We use the notations and assumptions from Proposition 1. Furthermore, we assume that for all curves Ei lying at a ﬁxed level i in V the curve structure is Z/n1 Z × Z/n2 Z, with n1 ≥ n2 . The value of NEi ,n2 , the number of zeros of the polynomial deﬁned at 2, is constant for all curves lying at level i in the volcano. Proof. Let E1 and E2 be two curves lying at level i in the volcano V . Then by Proposition 1 they both have endomorphism ring isomorphic to some order Odi . 7

In all the examples we considered for this case, P is always 0.

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Now by taking into account the fact that the action of Cl(Odi ) on Ell di (Fq ) is transitive, we consider an isogeny φ : E1 → E2 of degree 1 . By applying Lemma 3, we may assume that (1 , ) = 1. Take now P and Q two independent n2 -torsion points on E1 and denote by PE1 ,n2 the quadratic polynomial corresponding to the n2 -torsion on E1 as in (2). We use Lemma 2 to compute S(φ(P ), φ(P )), S(φ(P ), φ(Q)) and S(φ(Q), φ(Q)) and deduce that a polynomial PE2 ,n2 (a, b) on the curve E2 computed from φ(P ) and φ(Q) is such that PE1 ,n2 (a, b) = PE2 ,n2 (a, b). This means that NE1 ,n2 and NE2 ,n2 coincide, which concludes the proof. Moreover, we have showed that the value of k for two curves lying on the same level of a volcano is the same. Proposition 7. Let E be an elliptic curve deﬁned a ﬁnite ﬁeld Fq and let E[∞ ](Fq ) be isomorphic to Z/n1 Z × Z/n2 Z with ≥ 3 and n1 ≥ n2 ≥ 1. Suppose NE,n2 ∈ {1, 2} and let P be a n2 -torsion point with degenerate selfpairing. Then the -isogeny whose kernel is generated by n2 −1 P is either ascending or horizontal. Moreover, for any n2 -torsion point Q whose self-pairing is non-degenerate, the isogeny with kernel spanned by n2 −1 Q is descending. Proof. Case 1. Suppose Tn2 (P, P ) ∈ μk , k ≥ 1 and that Tn2 (Q, Q) ∈ μk+1 \μk . Denote by I1 : E → E1 the isogeny whose kernel is generated by n2 −1 P and I2 : E → E2 the isogeny whose kernel is generated by n2 −1 Q. By repeatedly applying Lemmas 1 and 2, we get the following relations for points generating the n2 −1 -torsion on E1 and E2 : Tn2 −1 (I1 (P ), I1 (P )) ∈ μk−1 , Tn2 −1 (I1 (Q), I1 (Q)) ∈ μk−2 \μk−3 Tn2 −1 (I2 (P ), I2 (P )) ∈ μk−3 , Tn2 −1 (I2 (Q), I2 (Q)) ∈ μk \μk−1 with the convention that μh = ∅ whenever h ≤ 0. From the relations above, we deduce that on the -volcano having E, E1 and E2 as vertices, E1 and E2 do not lie at the same level. Given the fact that there are at least − 1 descending rational -isogenies parting from E and that Q is any of the − 1 (or more) n2 torsion points with non-degenerate self-pairing, we conclude that I1 is horizontal or ascending and that I2 is descending. Case 2. Suppose now that k = 0. Note that the case n2 = 1 was already treated in proposition 4. Otherwise, consider the curve E deﬁned over Fq . By lemma 1 we have k = 1 for points on E/Fq , and we may apply Case 1. A special case. If E is a curve lying under the ﬁrst stability level and that E[∞ ](Fq ) Z/n1 Z × Z/n2 Z, with n1 > n2 , then it suﬃces to ﬁnd a point P1 of order n1 and the point n1 −1 P1 generates the kernel of an horizontal or ascending isogeny (P1 has degenerate self-pairing). Crater detection. Assume that P = 0. When is split in OE , there are two horizontal isogenies from E and this is equivalent, by propositions 6 and 7, to NE,n2 = 2. Similarly, when is inert in OE , there are neither ascending nor

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horizontal isogenies and NE,n2 = 0. In these two cases, we easily detect that the curve E is on the crater. Note. All statements in the proof of Case 1 are true for = 2 also. The statement in Proposition 4 is also true for = 2. The only case that is not clear is what happens when k = 0 and n2 ≥ 1. We did not ﬁnd a proof for the statement in proposition 5 for = 2, but in our computations with MAGMA we did not ﬁnd any counterexamples either. We conclude this section by presenting an algorithm which determines the group structure of the ∞ -torsion group of a curve E and also an algorithm which outputs the kernel of an horizontal (ascending) isogeny from E, when E[∞ ](Fq ) is given. Algorithm 1. Computing the structure of the ∞ -torsion of E over Fq (assuming volcano height ≥ 1) Require: A curve E deﬁned over Fq , a prime Compute: Structure Z/n1 Z × Z/n2 Z, generators P1 and P2 1: Check that q ≡ 1 (mod ) (if not need to move to extension ﬁeld: abort) 2: Let t be the trace of E(Fq ) 3: Check q + 1 − t ≡ 0 (mod ) (if not consider twist or abort) 4: Let dπ = t2 − 4q, let z be the largest integer such that z |dπ and h = z2 5: Let n be the largest integer such that n |q + 1 − t and N = q+1−t n 6: Take a random point R1 on E(Fq ), let P1 = N · R1 7: Let n1 be the smallest integer such that n1 P1 = 0 8: if n1 = n then 9: Output: Structure is nZZ , generator P1 . Exit (E is on the ﬂoor, ascending isogeny with kernel n−1 P1 ) 10: end if 11: Take a random point R2 on E(Fq ), let P2 = N · R2 and n2 = n − n1 12: Let α = logn2 P1 (n2 P2 ) (mod n1 −n2 ) 13: if α is undeﬁned then 14: Goto 6 (n2 P2 does not belong to n2 P1 ) 15: end if 16: Let P2 = P2 − αP1 17: If WeilPairing (n1 −1 P1 , n2 −1 P2 ) = 1 goto 6 (This checks linear independence) 18: Output: Structure is nZ1 Z × nZ2 Z , generators (P1 , P2 )

We assume that the height of the volcano is h ≤ 2n2 + 1, or, equivalently, that the curve E lies on or below the second stability level, which implies that the polynomial P is non-zero at every level in the volcano. This allows us to distinguish between diﬀerent directions of -isogenies parting from E. Of course, similar algorithms can be given for curves lying above the second stability level, but in this case we are compelled to consider the volcano over an extension ﬁeld Fqs . Since computing points deﬁned over extension ﬁelds of degree greater than is expensive, our complexity analysis in section 5 will show that it is more eﬃcient to use Kohel’s and Fouquet-Morain algorithms to explore the volcano until the second level of stability is reached and to use algorithms 1 and 2

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Algorithm 2. Finding the kernel of ascending or horizontal isogenies (Assuming curve not on ﬂoor and below the second stability level) Require: A curve E, its structure nZ1 Z × nZ2 Z and generators (P1 , P2 ) 1: if n1 > n2 then 2: The isogeny with kernel n1 −1 P1 is ascending or horizontal 3: To check whether there is another, continue the algorithm 4: end if 5: Let g be a primitive -th root of unity in Fq 6: Let Q1 = n1 −n2 P1 7: Let a = Tn2 (Q1 , Q1 ), b = Tn2 (Q1 , P2 ) · Tn2 (P2 , Q1 ) and c = Tn2 (P2 , P2 ) 8: If (a, b, c) = (1, 1, 1) abort (Above the second stability level) 9: repeat 10: Let a = a, b = b and c = c 11: Let a = a , b = b and c = c 12: until a = 1 and b = 1 and c = 1 13: Let La = log g (a ), Lb = logg (b ) and Lc = logg (c ) (mod ) 14: Let P(x, y) = La x2 + Lb xy + Lc y 2 (mod ) 15: If P has no roots modulo , Output: No isogeny (a single point on the crater) 16: If single root (x1 , x2 ) Output: One isogeny with kernel n2 −1 (x1 Q1 + x2 P2 ) 17: if P has two roots (x1 , x2 ) and (y1 , y2 ) then 18: Two isogenies with kernel n2 −1 (x1 Q1 + x2 P2 ) and n2 −1 (y1 Q1 + y2 P2 ) 19: end if

afterwards. We assume ≥ 3, even though in many cases these methods work also for = 2.

5

Walking the Volcano: Modified Algorithms

As mentioned in the introduction, several applications of isogeny volcanoes have recently been proposed. These applications require the ability to walk descending and ascending paths on the volcano and also to walk on the crater of the volcano. We recall that a path is a sequence of isogenies that never backtracks. We start this section with a brief description of existing algorithms for these tasks, based on methods given by Kohel [14] and by Fouquet and Morain in [8]. We present modiﬁed algorithms, which rely on the method presented in Algorithm 2 to ﬁnd ascending or horizontal isogenies. Then, we give complexity analysis for these algorithms and show that in many cases our method is competitive. Finally, we give two concrete examples in which the new algorithms can walk the crater of an isogeny volcano very eﬃciently compared to existing algorithms. A brief description of existing algorithms. Existing algorithms rely on three essential properties in isogeny volcanoes. Firstly, it is easy to detect that a curve lies on the ﬂoor of a volcano, since in that case, there is a single isogeny from this curve. Moreover, this isogeny can only be ascending (or horizontal if the height is 0). Secondly, if in an arbitrary path in a volcano there is a descending isogeny,

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then all the subsequent isogenies in the path are also descending. Thirdly, from a given curve, there is either exactly one ascending isogeny or at most two horizontal ones. As a consequence, ﬁnding a descending isogeny from any curve is easy: it suﬃces to walk three paths in parallel until one path reaches the ﬂoor. This shortest path is necessarily descending and its length gives the level of the starting curve in the volcano. To ﬁnd an ascending or horizontal isogeny, the classical algorithms try all possible isogenies until they ﬁnd one which leads to a curve either at the same level or above the starting curve. This property is tested by contructing descending paths from the all the neighbours of the initial curve and picking the curve which gave the longest path. Note that alternatively, one could walk in parallel all of the +1 paths starting from the initial curve and keep the (two) longest as horizontal or ascending. As far as we know, this has not been proposed in the literature, but this variant of existing algorithms oﬀers a slightly better asymptotic time complexity. For completeness, we give a pseudo-code description of this parallel variant of Kohel and Fouquet-Morain algorithms as Algorithm 3.

Algorithm 3. Parallel variant of ascending/horizontal step (using modular polynomials) Require: A j-invariant j0 in Fq , a prime , the modular polynomial Φ (X, Y ). 1: Let f (x) = Φ (X, j0 ) 2: Compute J0 the list of roots of f (x) in Fq 3: If #J0 = 0 Output: “Trivial volcano” Exit 4: If #J0 = 1 Output: “On the ﬂoor, step leads to:”, J0 [1] Exit 5: If #J0 = 2 Output: “On the ﬂoor, two horizontal steps to:”, J0 [1] and J0 [2] Exit 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

Let J = J0 . Let J and K be empty lists. Let Done = false. repeat Perform multipoint evaluation of Φ (X, j), for each j ∈ J. Store in list F for i from 1 to + 1 do Perform partial factorization of F [i], computing at most two roots r1 and r2 if F [i] has less than two roots then Let Done = true. Append ⊥ to K (Reaching ﬂoor) else If r1 ∈ J then append r1 to K else append r2 to K. (Don’t backtrack) end if end for Let J = J, J = K and K be the empty list until Done for each i from 1 to + 1 such that J[i] = ⊥ append J0 [i] to K Output: “Possible step(s) lead to:” K (One or two outputs)

Basic idea of the modiﬁed algorithms. In our algorithms, we ﬁrst need to choose a large enough extension ﬁeld to guarantee that the kernels of all required isogenies are spanned by -torsion points deﬁned on this extension ﬁeld. As explained in

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Corollary 1, the degree r of this extension ﬁeld is the order of q modulo and it can be computed very quickly after factoring q − 1. As usual, we choose an arbitrary irreducible polynomial of degree r to represent Fqr . The necessary points of ∞ -torsion are computed in Algorithm 1, multiplying random points over Fqr by the cardinality of the curve divided by the highest possible power of . Once this is done, assuming that we are starting from a curve below the second level of stability, we use Algorithms 1 and 2 to ﬁnd all ascending or horizontal isogenies from the initial curve. In order to walk a descending path, it suﬃces to choose any other isogeny. Note that, in the subsequent steps of a descending path, in the cases where the group structure satisﬁes n1 > n2 , it is not necessary to run Algorithm 2 as a whole. Indeed, since we know that we are not on the crater, there is a single ascending isogeny and it is spanned by n1 −1 P1 . Finally, above the second stability level, we have two options. In theory, we can consider curves over larger extension ﬁelds (in order to get polynomials P = 0. Note that this is too costly in practice. Therefore, we use preexisting algorithms, but it is not necessary to follow descending paths all the way to the ﬂoor. Instead, we can stop these paths at the second stabilty level, where our methods can be used. 5.1

Complexity Analysis

Computing a single isogeny. Before analyzing the complete algorithms, we ﬁrst compare the costs of taking a single step on a volcano by using the two methods existing in the literature: modular polynomials and classical V´elu’s formulae. Suppose that we wish to take a step from a curve E. With the modular polynomial approach, we have to evaluate the polynomial f (X) = Φ (X, j(E)) and ﬁnd its roots in Fq . Assuming that the modular polynomial (modulo the characteristic of Fq ) is given as input and using asymptotically fast algorithms to factor f (X), the cost of a step in terms of arithmetic operations in Fq is O(2 + M () log q), where M () denotes the operation count of multiplying polynomials of degree . In this formula, the ﬁrst term corresponds to evaluation of Φ (X, j(Ei−1 )) and the second term to root ﬁnding8 . With V´elu’s formulae, we need to take into account the fact that the required -torsion points are not necessarily deﬁned over Fq . Let r denotes the smallest integer such that the required points are all deﬁned over Fqr . We know that 1 ≤ r ≤ − 1. Using asymptotically eﬃcient algorithms to perform arithmetic operations in Fqr , multiplications in Fqr cost M (r) Fq -operations. Given an torsion point P in E(Fqr ), the cost of using V´elu’s formulae is O() operations in Fqr . As a consequence, in terms of Fq operations, each isogeny costs O(M (r)) operations. As a consequence, when q is not too large and r is close to , using V´elu formulae is more expensive by a logarithmic factor. 8

Completely splitting f (X) to ﬁnd all its roots would cost O(M () log log q), but this is reduced to O(M () log q) because we only need a constant number of roots for each polynomial f (X).

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Computing an ascending or horizontal path. With the classical algorithms, each step in an ascending or horizontal path requires to try O() steps and test each by walking descending paths of height bounded by h. The cost of each descending path is O(h(2 + M () log q)) and the total cost is O(h(3 + M () log q)) (see [14,23]). When >> log q, this cost is dominated by the evaluations of the polynomial Φ at each j-invariant. Thus, by walking in parallel + 1 paths from the original curve, we can amortize the evaluation of Φ (X, j) over many j-invariants using fast multipoint evaluation, see [18, Section 3.7] or [25], thus replacing 3 by M () log and reducing the complexity of a step to O(h M ()(log + log q)). However, this increases the memory requirements. With our modiﬁed algorithms, we need to ﬁnd the structure of each curve, compute some discrete logarithms in -groups, perform a small number of pairing computations and compute the roots of PE,n2 . Except for the computation of discrete logarithms, it is clear that all these additional operations are polynomial in n2 and log and they take negligible time in practice √ (see Section 5.2). Using generic algorithms, the discrete logarithms cost O( ) operations, and this can be reduced to log by storing a sorted table of precomputed logarithms. After this is done, we have to compute at most two isogenies, ignoring the one that backtracks. Thus, the computation of one ascending or horizontal step is dominated by the computation of isogenies and costs O(M (r)). For completeness, we also mention the complexity analysis of Algorithm 1. The dominating step here is the multiplication by N of randomly chosen points. When we consider the curve over an extension ﬁeld Fqr , this costs O(r log q) operations in Fqr , i.e. O(rM (r) log q) operations in Fq . Finally, comparing the two approaches on a regular volcano, we see that even in the less favorable case, we gain a factor h compared to the classical algorithms. More precisely, the two are comparable, when the height h is small and r is close to . In all the other cases, our modiﬁed algorithms are more eﬃcient. This analysis is summarized in Table 1. For compactness O(·)s are omitted from the table. Table 1. Walking the volcano: Order of the cost per step

[14,8] Parallel evaluation Regular volcanoes Best case Worst case r ≈ /2 Regular volcanoes Best case Worst case r ≈ /2 Irregular volcanoes (worst case)

Descending path Ascending/Horizontal One step Many steps h(2 + M () log q) (2 + M () log q) h(3 + M () log q) – – h M ()(log + log q) Structure determination log q log q r M (r) log q r M (r) log q Isogeny construction r M (r) r M (r) No improvement

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Irregular volcanoes. Consider a ﬁxed value of q and let s = v (q − 1). First of all, note that all curves lying on irregular volcanoes satisfy 2s |q + 1 − t and 2s+2 |t2 − 4q. For traces that satisfy only the ﬁrst condition, we obtain a regular volcano. We estimate the total number of diﬀerent traces of elliptic curves lying √ √ √ 4 q on -volcanoes by #{t s.t. 2s |q + 1 − t and t ∈ [−2 q, 2 q]} ∼ 2s . Next, we estimate traces of curves lying on irregular volcanoes by √ √ √ 4 q #{t s.t. 2s |q + 1 − t , 2s+2 |t2 − 4q and t ∈ [−2 q, 2 q]} ∼ 2s+2 . Indeed, by writing q = 1 + γs and t = 2 + γs + μ2s , and imposing the condition 2s+2 |t2 − 4q, we ﬁnd that t ∼ = t0 (γ, μ)(mod 2s+2 ). Thus, we estimate the probability of picking a curve whose volcano is not regular, among curves lying on volcanoes of height greater than 0, by 12 . (This is a crude estimate because the number of curves for each trace is proportional to the Hurwitz class number9 H(t2 − 4q)). This probability is not negligible for small values of . However, since our method also works everywhere on almost regular volcano, the probability of ﬁnding a volcano where we need to combine our modiﬁed algorithm with the classical algorithms is even lower. Furthermore, in some applications, it is possible to restrict ourselves to regular volcanoes. 5.2

Two Practical Examples

A favorable case. In order to demonstrate the potential of the modiﬁed algorithm, we consider the favorable case of a volcano of height 2, where all the necessary -torsion points are deﬁned over the base ﬁeld Fp , where p = 619074283342666852501391 is prime. We choose = 100003. Let E be the elliptic curve whose Weierstrass equation is y 2 = x3 + 198950713578094615678321 x + 32044133215969807107747. The group E[∞ ] over Fp has structure 4ZZ . It is spanned by the point P = (110646719734315214798587, 521505339992224627932173). Taking the -isogeny I1 with kernel 3 P , we obtain the curve E1 : y 2 = x3 + 476298723694969288644436 x + 260540808216901292162091, with structure of the ∞ -torsion Z3 × Z and generators P1 = (22630045752997075604069, 207694187789705800930332) and Q1 = (304782745358080727058129, 193904829837168032791973). The -isogeny I2 with kernel 2 P1 leads to the curve E2 : y 2 = x3 + 21207599576300038652790 x + 471086215466928725193841, on the volcano’s crater and with structure 2ZZ × 2ZZ and generators P2 = (545333002760803067576755, 367548280448276783133614) and Q2 = (401515368371004856400951, 225420044066280025495795). Using pairings on these points, we construct the polynomial: P(x, y) = 97540 x2 + 68114 x y + 38120 y 2, having homogeneous roots (x, y) = (26568, 1) and (72407, 1). As a consequence, we have two horizontal isogenies with kernels (26568 P2 + Q2 ) and (72407 P2 + Q2 ). We can continue and make a complete walk around the 9

See [5, Th. 14.18] for q prime.

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crater which contains 22 diﬀerent curves. Using a simple implementation under Magma 2.15-15, a typical execution takes about 134 seconds10 on a single core of an Intel Core 2 Duo at 2.66 GHz. Most of the time is taken by the computation of V´elu’s formulas (132 seconds) and the computation of discrete logarithms (1.5 seconds) which are not tabulated in the implementation. The computation of pairings only takes 20 milliseconds. A less favorable example. We have also implemented the computation for = 1009 using an elliptic curve with j-invariant j = 34098711889917 in the prime ﬁeld deﬁned by p = 953202937996763. The -torsion appears in a extension ﬁeld of degree 84. The -volcano has height two and the crater contains 19 curves. Our implementation walks the crater in 20 minutes. More precisely, 750 seconds are needed to generate the curves’ structures, 450 to compute V´elu’s formulas, 28 seconds for the pairings and 2 seconds for the discrete logarithms.

6

Conclusion and Perspectives

In this paper, we have proposed a method which allows, in the regular part of an isogeny volcano, to determine, given a curve E and a -torsion point P , the type of the -isogeny whose kernel is spanned by P . In addition, this method also permits, given a basis for the -torsion, to ﬁnd the ascending isogeny (or horizontal isogenies) from E. We expect that this method can be used to improve the performance of several volcano-based algorithms, such as the computation of the Hilbert class polynomial [23] or of modular polynomials [4]. Acknowledgments. The authors thank Jean-Marc Couveignes for the idea in the proof of Lemma 1 and two anonymous reviewers for their helpful comments. The ﬁrst author is grateful to Ariane M´ezard for many discussions on number theory and isogeny volcanoes, prior to this work.

References 1. Belding, J., Broker, R., Enge, A., Lauter, K.: Computing Hilbert Class Polynomials. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 282–295. Springer, Heidelberg (2008) 2. Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a ﬁnite ﬁeld. Journal of Number Theory (to appear 2010) 3. Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005) 4. Broker, R., Lauter, K., Sutherland, A.: Computing modular polynomials with the chinese remainder theorem (2009), http://arxiv.org/abs/1001.0402 10

This timing varies between executions. The reason that we ﬁrst try one root of P, if it backtracks on the crater, we need to try the other one. On average, 1.5 root is tried for each step, but this varies depending on the random choices.

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5. Cox, D.A.: Primes of the Form x2 + ny 2 : Fermat, class ﬁeld theory, and complex multiplication. John Wiley & Sons, Inc., Chichester (1989) 6. Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkorper. Abh. Math. Sem. Hansischen Univ., vol. 14 (1941) 7. Fouquet, M.: Anneau d’endomorphismes et cardinalit´e des courbes elliptiques: aspects algorithmiques. PhD thesis, Ecole Polytechnique (2001) 8. Fouquet, M., Morain, F.: Isogeny Volcanoes and the SEA Algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002) 9. Frey, G.: Applications of arithmetical geometry to cryptographic constructions. In: Proceedings of the Fifth International Conference on Finite Fields and Applications, pp. 128–161. Springer, Heidelberg (2001) 10. Grabher, P., Großsch¨ adl, J., Page, D.: On software parallel implementation of cryptographic pairings. In: Avanzi, R.M., Keliher, L., Sica, F. (eds.) SAC 2008. LNCS, vol. 5381, pp. 35–50. Springer, Heidelberg (2009) 11. Ionica, S.: Algorithmique des couplages et cryptographie. PhD thesis, Universit´e de Versailles St-Quentin-en-Yvelines (2010) 12. Joux, A., Nguyen, K.: Separating decision Diﬃe–Hellman from computational Diﬃe–Hellman in cryptographic groups. Journal of Cryptology 16(4), 239–247 (2003) 13. Lenstra Jr., H.W.: Complex multiplication structure of elliptic curves. Journal of Number Theory 56(2), 227–241 (1996) 14. Kohel, D.: Endomorphism rings of elliptic curves over ﬁnite ﬁelds. PhD thesis, University of California, Berkeley (1996) 15. Miller, V.S.: The Weil pairing, and its eﬃcient calculation. Journal of Cryptology 17(4), 235–261 (2004) 16. Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: An algorithm to compute volcanoes of 2-isogenies of elliptic curves over ﬁnite ﬁelds. Applied Mathematics and Computation 176(2), 739–750 (2006) 17. Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: Computing the height of volcanoes of l-isogenies of elliptic curves over ﬁnite ﬁelds. Applied Mathematics and Computation 196(1), 67–76 (2008) 18. Montgomery, P.L.: A FFT extension of the elliptic curve method of factorization. PhD thesis, University of California (1992) 19. Ruck, H.-G.: A note on elliptic curves over ﬁnite ﬁelds. Mathematics of Computation 179, 301–304 (1987) 20. Schoof, R.: Counting points on elliptic curves over ﬁnite ﬁelds. Journal de Theorie des Nombres de Bordeaux 7, 219–254 (1995) 21. Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986) 22. Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, Heidelberg (1994) 23. Sutherland, A.: Computing Hilbert Class Polynomials with the Chinese Remainder Theorem. Mathematics of Computation (2010) 24. V´elu, J.: Isogenies entre courbes elliptiques. Comptes Rendus De L’Academie Des Sciences Paris, Serie I-Mathematique, Serie A. 273, 238–241 (1971) 25. von zur Gathen, J., Shoup, V.: Computing Frobenius maps and factoring polynomials. Computational Complexity 2, 187–224 (1992)

A Subexponential Algorithm for Evaluating Large Degree Isogenies David Jao and Vladimir Soukharev Department of Combinatorics and Optimization University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada {djao,vsoukhar}@math.uwaterloo.ca

Abstract. An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves eﬃciently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.

1

Introduction

A well known theorem of Tate [29] states that two elliptic curves deﬁned over the same ﬁnite ﬁeld Fq are isogenous (i.e. admit an isogeny between them) if and only if they have the same number of points over Fq . Using fast point counting algorithms such as Schoof’s algorithm and others [9,25], it is very easy to check whether this condition holds, and thus whether or not the curves are isogenous. However, constructing the actual isogeny itself is believed to be a hard problem due to the nonconstructive nature of Tate’s theorem. Indeed, given an ordinary curve E/Fq and an ideal of norm n in the endomorphism ring, the fastest previously known algorithm for constructing the unique (up to isomorphism) isogeny having this ideal as kernel has a running time of O(n3+ε ), except in a certain very small number of special cases [4,16,17]. In this paper, we present a new probabilistic algorithm for evaluating such isogenies, which in the vast majority of cases runs (heuristically) in subexponential time. Speciﬁcally, we show that for ordinary curves, one can evaluate isogenies of degree n between curves of √ nearly equal endomorphism ring over Fq in time less than Lq ( 12 , 23 ) log(n), provided n has no large prime divisors in common with the endomorphism ring discriminant. Although this running time is not polynomial in the input length, our algorithm is still much faster than the (exponential) previous best known algorithm, and in practice allows for the evaluation of isogenies of cryptographically sized degrees, some examples of which we present here. We emphasize that, G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 219–233, 2010. c Springer-Verlag Berlin Heidelberg 2010

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in contrast with the previous results of Br¨ oker et al. [4], our algorithm is not limited to special curves such as pairing friendly curves with small discriminant. If an explicit equation for the isogeny as a rational function is desired, our approach in combination with the algorithm of Bostan et al. [3] can produce the equation in time O(n1+ε ) given E and an ideal of norm n, which is quasi-optimal in the sense that (up to log factors) it is equal to the size of the output. To our knowledge, this method is the only known algorithm for computing rational function expressions of large degree isogenies in quasi-optimal time in the general case, given only the starting curve and the kernel. Apart from playing a central role in the implementation of the point counting algorithms mentioned above, isogenies have been used in cryptography to transfer the discrete logarithm problem from one elliptic curve to another [9,16,17,20,23,30]. In many of these applications, our algorithm cannot be used directly, since in cryptography one is usually given two isogenous curves, rather than one curve together with the isogeny degree. However, earlier results [16,17,20] have shown that the problem of computing isogenies between a given pair of curves can be reduced to the problem of computing isogenies of prime degree starting from a given curve. It is therefore likely that the previous best isogeny construction algorithms in the cryptographic setting can be improved or extended in light of the work that we present here.

2

Background

Let E and E be elliptic curves deﬁned over a ﬁnite ﬁeld Fq of characteristic p. An isogeny φ : E → E deﬁned over Fq is a non-constant rational map deﬁned over Fq which is also a group homomorphism from E(Fq ) to E (Fq ). This deﬁnition diﬀers slightly from the standard deﬁnition in that it excludes constant maps [27, §III.4]. The degree of an isogeny is its degree as a rational map, and an isogeny of degree is called an -isogeny. Every isogeny of degree greater than 1 can be ¯ q [11]. factored into a composition of isogenies of prime degree deﬁned over F For any elliptic curve E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 deﬁned over Fq , the Frobenius endomorphism is the isogeny πq : E → E of degree q given by the equation πq (x, y) = (xq , y q ). The characteristic polynomial of πq is X 2 − tX + q where t = q + 1 − #E(Fq ) is the trace of E. An endomorphism of E is an isogeny E → E deﬁned over the algebraic closure ¯ q of Fq . The set of endomorphisms of E together with the zero map forms F a ring under the operations of pointwise addition and composition; this ring is called the endomorphism ring of E and denoted End(E). The ring End(E) is isomorphic either to an order in a quaternion algebra or to an order in an imaginary quadratic ﬁeld [27, V.3.1]; in the ﬁrst case we say E is supersingular and in the second case we say E is ordinary. Two elliptic curves E and E deﬁned over Fq are said to be isogenous over Fq if there exists an isogeny φ : E → E deﬁned over Fq . A theorem of Tate states that two curves E and E are isogenous over Fq if and only if #E(Fq ) = #E (Fq ) [29, §3]. Since every isogeny has a dual isogeny [27, III.6.1], the property of being

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¯ q -isomorphism isogenous over Fq is an equivalence relation on the ﬁnite set of F classes of elliptic curves deﬁned over Fq . Moreover, isomorphisms between elliptic curves can be classiﬁed completely and computed eﬃciently in all cases [16]. Accordingly, we deﬁne an isogeny class to be an equivalence class of elliptic ¯ q -isomorphism, under this equivalence relation. curves, taken up to F Curves in the same isogeny class are either all supersingular or all ordinary. The vast majority of curves are ordinary, and indeed the number of isomorphism classes of supersingular curves is ﬁnite for each characteristic. Also, ordinary curves form the majority of the curves of interest in applications such as cryptography. Hence, we assume for the remainder of this paper that we are in the ordinary case. Let K denote the imaginary quadratic ﬁeld containing End(E), with maximal order OK . For any order O ⊆ OK , the conductor of O is deﬁned to be the integer [OK : O]. The ﬁeld K is called the CM ﬁeld of E. We write cE for the conductor of End(E) and cπ for the conductor of Z[πq ]. It follows from [12, §7] that End(E) = Z + cE OK and Δ = c2E ΔK , where Δ (respectively, ΔK ) is the discriminant of the imaginary quadratic order End(E) (respectively, OK ). Furthermore, the characteristic polynomial has discriminant Δπ = t2 − 4q = disc(Z[πq ]) = c2π ΔK , with cπ = cE · [End(E) : Z[πq ]]. Following [14] and [16], we say that an isogeny φ : E → E of prime degree deﬁned over Fq is “down” if [End(E) : End(E )] = , “up” if [End(E ) : End(E)] = , and “horizontal” if End(E) = End(E). Two curves in an isogeny class are said to “have the same level” if their endomorphism rings are equal. Within each isogeny class, the property of having the same level is an equivalence relation. A horizontal isogeny always goes between two curves of the same level; likewise, an up isogeny enlarges the endomorphism ring and a down isogeny reduces it. Since there are fewer elliptic curves at higher levels than at lower levels, the collection of elliptic curves in an isogeny class visually resembles a “pyramid” or a “volcano” [14], with up isogenies ascending the structure and down isogenies descending. If we restrict to the graph of -isogenies for a single , then in general the -isogeny graph is disconnected, having one -volcano for each intermediate order Z[πq ] ⊂ O ⊂ OK such that O is maximal at (meaning [OK : O]). The “top level” of the class consists of curves E with End(E) = OK , and the “bottom level” consists of curves with End(E) = Z[πq ]. We say that is an Elkies prime [2, p. 119] if cE and Δ = −1, or equivalently if and only if E admits a horizontal isogeny of degree . The number of -isogenies of each type can easily be determined explicitly [14,16,21]. In particular, for all but the ﬁnitely many primes dividing [OK : Z[πq ]], we have that every rational -isogeny admitted by E is horizontal.

3

The Br¨ oker-Charles-Lauter Algorithm

Our algorithm is an extension of the algorithm developed by Br¨ oker, Charles, and Lauter [4] to evaluate large degree isogenies over ordinary elliptic curves with

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endomorphism rings of small class number, such as pairing-friendly curves [15]. In this section we provide a summary of their results. The following notation corresponds to that of [4]. Let E/Fq be an ordinary elliptic curve with endomorphism ring End(E) isomorphic to an imaginary quadratic order OΔ of discriminant Δ < 0. Identify End(E) with OΔ via the unique isomorphism ι such that ι∗ (x)ω = xω for all invariant diﬀerentials ω and all x ∈ OΔ . Then every horizontal separable isogeny on E of prime degree corresponds (up to isomorphism) to a unique prime ideal L ⊂ OΔ of norm for some Elkies prime . We denote the kernel of this isogeny by E[L]. Any two distinct isomorphic horizontal isogenies (i.e., pairs of isogenies where one is equal to the composition of the other with an isomorphism) induce diﬀerent maps on the space of diﬀerentials of E, and a separable isogeny is uniquely determined by the combination of its kernel and the induced map on the space of diﬀerentials. A normalized isogeny is an isogeny φ : E → E for which φ∗ (ωE ) = ωE where ωE denotes the invariant diﬀerential of E. Algorithm 1 (identical to Algorithm 4.1 in [4]) evaluates, up to automorphisms of E, the unique normalized horizontal isogeny of degree corresponding to a given kernel ideal L ⊂ OΔ . The following theorem, taken verbatim from [4], shows that the running time of Algorithm 1 is polynomial in the quantities log(), log(q), n, and |Δ|. Theorem 3.1. Let E/Fq be an ordinary elliptic curve with Frobenius πq , given by a Weierstrass equation, and let P ∈ E(Fqn ) be a point on E. Let Δ = disc(End(E)) be given. Assume that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and let L = (, c + dπq ) be an End(E)-ideal of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Algorithm 1 computes the unique elliptic curve E such that there exists a normalized isogeny φ : E → E with kernel E[L]. Furthermore, it computes the x-coordinate of φ(P ) if End(E) does not equal Z[i] or Z[ζ3 ] and the square, respectively cube, of the x-coordinate of φ(P ) otherwise. The running time of the algorithm is polynomial in log(), log(q), n and |Δ|.

4

A Subexponential Algorithm for Evaluating Horizontal Isogenies

As was shown in Sections 2 and 3, any horizontal isogeny can be expressed as a composition of prime degree isogenies, one for each prime factor of the kernel, and any prime degree isogeny is a composition of a normalized isogeny and an isomorphism. Therefore, to evaluate a horizontal isogeny given its kernel, it suﬃces to treat the case of horizontal normalized prime degree isogenies. Our objective is to evaluate the unique horizontal normalized isogeny on a given elliptic curve E/Fq whose kernel ideal is given as L = (, c + dπq ), at a given point P ∈ E(Fqn ), where is an Elkies prime. As in [4], we must also impose the additional restriction that [End(E) : Z[πq ]]; for Elkies primes, an equivalent restriction is that [OK : Z[πq ]], but we retain the original formulation for consistency with [4].

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Algorithm 1. The Br¨oker-Charles-Lauter algorithm Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ and a point P ∈ E(Fqn ) such that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and an End(E)-ideal L = (, c+dπq ) of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Output: The unique elliptic curve E admitting a normalized isogeny φ : E → E with kernel E[L], and the x-coordinate of φ(P ) for Δ = −3, −4 and the square (resp. cube) of the x-coordinate otherwise. 1: Compute the direct sum decomposition Pic(OΔ ) = [Ii ] of Pic(OΔ ) into cyclic groups generated by the degree 1 prime ideals Ii of smallest norm that are coprime to the product p · #E(Fqn ) · [End(E) : Z[πq ]]. 2: Using brute force1 , ﬁnd e1 , e2 , . . . , ek such that [L] = [I1e1 ] · [I2e2 ] · · · [Ikek ]. e 3: Find α (using Cornacchia’s algorithm) and express L = I1e1 · I2e2 · · · Ikk · (α). 4: Compute a sequence of isogenies (φ1 , . . . , φs ) such that the composition φc : E → Ec has kernel E[I1e1 · I2e2 · · · Ikek ] using the method of [4, § 3]. 5: Evaluate φc (P ) ∈ Ec (Fqn ). ∼ 6: Write α = (u + vπq )/(zm). Compute the isomorphism η : Ec → E with η ∗ (ωE ) = (u/zm)ωEc . Compute Q = η(φc (P )). 7: Compute (zm)−1 mod #E(Fqn ), and compute R = ((zm)−1 (u + vπq ))(Q). ∗ 8: Put r = x(R)|OΔ | /2 and return (E , r).

In practice, one is typically given instead of L, but since it is easy to calculate the list of (at most two) possible primes L lying over (cf. [6]), these two interpretations are for all practical purposes equivalent, and we switch freely between them when convenient. When is small, one can use modular polynomial based techniques [4, §3.1], which have running time O(3 log()4+ε ) [13]. However, for isogeny degrees of cryptographic size (e.g. 2160 ), this approach is impractical. The Br¨oker-Charles-Lauter algorithm sidesteps this problem, by using an alternative factorization of L. However, the running time of Br¨ oker-Charles-Lauter is polynomial in |Δ|, and therefore even this method only works for small values of |Δ|. In this section we present a modiﬁed version of the Br¨ oker-Charles-Lauter algorithm which is suitable for large values of |Δ|. We begin by giving an overview of our approach. In order to handle large values of |Δ|, there are two main problems to overcome. One problem is that we need a fast way to produce a factorization L = I1e1 I2e2 · · · Ikek · (α)

(1)

as in lines 2 and 3 of Algorithm 1. The other problem is that the exponents ei in Equation (1) need to be kept small, since the running times of lines 3 and 4 of Algorithm 1 are proportional to i |ei | Norm(Ii )2 . The ﬁrst problem, that of ﬁnding a factorization of L, can be solved in subexponential time using the index calculus algorithm of Hafner and McCurley [18] (see also [6, Chap. 11]). 1

Br¨ oker, Charles, and Lauter mention that this computation can be done in “various ways” [4, p. 107], but the only explicit method given in [4] is brute force. The use of brute force limits the algorithm to elliptic curves for which |Δ| is small, such as pairing-friendly curves.

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Algorithm 2. Computing a factor base Input: A discriminant Δ, a bound N . Output: The set I consisting of split prime ideals of norm less than N , together with the corresponding set F of quadratic forms. 1: Set F ← ∅. 2: Set I ← ∅. 3: Find all primes p < N such that ( Δ ) = 1. Call this set P . Let k = |P |. p 4: For each prime pi ∈ P , ﬁnd an ideal pi of norm pi (using Cornacchia’s algorithm). 5: For each i, ﬁnd a quadratic form fi = [(pi , bi , ci )] corresponding to pi in Cl(OΔ ), using the technique of [26, §3]. 6: Output I = {p1 , p2 , . . . , pk } and F = {f1 , f2 , . . . , fk }.

To resolve the second problem, we turn to an idea which was ﬁrst introduced by Galbraith et. al [17], and recently further reﬁned by Bisson and Sutherland [1]. The idea is that, in the process of sieving for smooth norms, one can arbitrarily restrict the input exponent vectors to sparse vectors (e1 , e2 , ..., ek ) such that 2 i |ei |N (Ii ) is kept small. This restriction is implemented in line 6 of Algorithm 3. As in [1], one then assumes heuristically that the imposition of this restriction does not aﬀect the eventual probability of obtaining a smooth norm in the Hafner and McCurley algorithm. Note that, unlike the input exponents, the exponents appearing in the factorizations of the ensuing smooth norms (that is, the values of yi in Algorithm 3) are always small, since the norm in question is derived from a reduced quadratic form. We now describe the individual components of our algorithm in detail. 4.1

Finding a Factor Base

Let Cl(OΔ ) denote the ideal class group of OΔ . Algorithm 2 produces a factor base consisting of split primes in OΔ of norm less than some bound N . The optimal value of N will be determined in Section 4.4. 4.2

“Factoring” Large Prime Degree Ideals

Algorithm 3, based on the algorithm of Hafner and McCurley, takes as input a discriminant Δ, a curve E, a prime ideal L of prime norm in OΔ , a smoothness bound N , and an extension degree n. It outputs a factorization L = I1e1 I2e2 · · · Ikek · (α) as in Equation 1, where the Ii ’s are as in Algorithm 1, the exponents ei are positive, sparse, and small (i.e., polynomial in N ), and the ideal (α) is a principal fractional ideal generated by α. 4.3

Algorithm for Evaluating Prime Degree Isogenies

The overall algorithm for evaluating prime degree isogenies is given in Algorithm 4. This algorithm is identical to Algorithm 1, except that the factorization of L is performed using Algorithm 3. To maintain consistency with [4], we

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Algorithm 3. “Factoring” a prime ideal Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ , a smoothness bound N , a prime ideal L of norm inOΔ , an extension degree n. Output: Relation of the form L = (α) · ki=1 Iiei , where (α) is a fractional ideal, Ii are as in Algorithm 1, and ei > 0 are small and sparse. 1: Run Algorithm 2 on input Δ and N to obtain I = {p1 , p2 , . . . , pk } and F = {f1 , f2 , . . . , fk }. Discard any primes dividing p · #E(Fqn ) · [End(E) : Z[πq ]]. 2: Set pi ← Norm(pi ). (These values are also calculated in Algorithm 2.) 3: Obtain the reduced quadratic form [L] corresponding to the ideal class of L. 4: repeat 5: for i = 1, . . . , k do 6: Pick exponents xi in the range [0, (N/pi )2 ] such that at most k0 are nonzero, where k0 is a global absolute constant (in practice, k0 = 3 suﬃces). 7: end for 8: Compute the reduced quadratic form a = (a, b, c) for which the ideal class [a] is equivalent to [L] · ki=1 fixi . 9: until The integer primes pi , and the relation derived a factors completely into the from [a] = [L] · ki=1 fixi contains fewer than log(|Δ|/3)/z nonzero exponents. 10: Write a = ki=1 pui i . 11: for i=1, . . . , k do 12: Using the technique of Seysen ([26, Theorem 3.1]), determine the signs of the exponents yi = ±ui for which a = ki=1 fiyi . 13: Let ei = yi − xi . (These exponents satisfy [L] = ki=1 fiei .) 14: if ei ≥ 0 then 15: Set Ii ← ¯ pi 16: else 17: Set Ii ← pi 18: end if 19: end for |e | 20: Compute the principal ideal I = L · ki=1 Ii i . 21: Using Cornacchia’s algorithm, ﬁnd a generator β ∈ OΔ of I. |e | β . 22: Set m ← ki=1 pi i and α ← m |e | |e | |e | 23: Output L = (α) · I¯1 1 · I¯2 2 · · · I¯k k .

have included the quantities Δ and End(E) as part of the input to the algorithm. However, we remark that these quantities can be computed from E/Fq √ in Lq ( 12 , 23 ) operations using the algorithm of Bisson and Sutherland [1], even if they are not provided as input. 4.4

Running Time Analysis

In this section, we determine the theoretical running time of Algorithm 4, as well as the optimal value of the smoothness bound N to use in line 1 of the algorithm. As is typical for subexponential time factorization algorithms involving a factor base, these two quantities depend on each other, and hence both are calculated simultaneously.

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Algorithm 4. Evaluating prime degree isogenies Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ and a point P ∈ E(Fqn ) such that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and an End(E)-ideal L = (, c+dπq ) of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Output: The unique elliptic curve E admitting a normalized isogeny φ : E → E with kernel E[L], and the x-coordinate of φ(P ) for Δ = −3, −4 and the square (resp. cube) of the x-coordinate otherwise. 1: Choose a smoothness bound N (see Section 4.4). 2: Using Algorithm 3 on input (Δ, E, N, L, n), obtain a factorization of the form L = I1e1 · I2e2 · · · Ikek · (α). 3: Compute a sequence of isogenies (φ1 , . . . , φs ) such that the composition φc : E → e Ec has kernel E[I1e1 · I2e2 · · · Ikk ] using the method of [4, § 3]. 4: Evaluate φc (P ) ∈ Ec (Fqn ). ∼ 5: Write α = (u + vπq )/(zm). Compute the isomorphism η : Ec → E with η ∗ (ωE ) = (u/zm)ωEc . Compute Q = η(φc (P )). 6: Compute (zm)−1 mod #E(Fqn ), and compute R = ((zm)−1 (u + vπq ))(Q). ∗ 7: Put r = x(R)|OΔ | /2 and return (E , r).

As in [9], we deﬁne2 Ln (α, c) by Ln (α, c) = O(exp((c + o(1))(log(n))α (log(log(n)))1−α )). The quantity Ln (α, c) interpolates between polynomial and exponential size as α ranges from 0 to 1. We set N = L|Δ| ( 12 , z) for an unspeciﬁed value of z, and in the following paragraphs we determine the optimal value of z which minimizes the running time of Algorithm 4. (The fact that α = 12 is optimal is clear from the below analysis, as well as from prior experience with integer factorization algorithms.) For convenience, we will abbreviate L|Δ| (α, c) to L(α, c) throughout. Line 2 of Algorithm 4 involves running Algorithm 3, which in turn calls Algorithm 2. As it turns out, Algorithm 2 is almost the same as Algorithm 11.1 from [6], which requires L( 21 , z) time, as shown in [6]. The only diﬀerence is that we add an additional step where we obtain the quadratic form corresponding to each prime ideal in the factor base. This extra step requires O(log(Norm(I))1+ε ) time for a prime ideal I, using Cornacchia’s Algorithm [19]. Thus, the overall running time for Algorithm 2 is bounded above by L( 12 , z) · log(L( 12 , z))1+ε = L( 12 , z). Line 2 of Algorithm 3 takes log() time using standard algorithms [12]. The loop in lines 4–9 of Algorithm 3 is very similar to the FindRelation algorithm in [1], except that we only use one discriminant, and we omit the requirement that #R/D1 > #R/D2 (which in any case is meaningless when there is only one discriminant). Needless to say, this change can only speed up the algorithm. 2

The deﬁnition of Ln (α, c) in [6] diﬀers from that of [9] in the o(1) term. We account for this discrepancy in our text.

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√ Taking μ = 2z in [1, Prop. 6], we ﬁnd that the (heuristic) expected running 1 time of the loop in lines 4–9 of Algorithm 3 is L( 21 , 4z ). The next step in Algorithm 3 having nontrivial running time is the computation of the ideal product in line 20. To exponentiate an element of an arbitrary semigroup to a power e requires O(log e) semigroup multiplication operations [10, §1.2]. To multiply two ideals I and J in an imaginary quadratic order (via composition of quadratic forms) requires O(max(log(Norm(I)), log(Norm(J)))1+ε ) bit operations using fast multiplication [24, §6]. Each of the expressions |Ii ||ei | therefore requires O(log |ei |) ideal multiplication operations to compute, with each individual multiplication requiring ⎛

1+ε ⎞ 2 N ⎠ = O(N 2+ε ) O((|ei | log(Norm(Ii )))1+ε ) = O ⎝ log(pi ) pi bit operations, for a total running time of (log ei )O(N 2+ε ) = L( 12 , 2z) for each i. This calculation must be performed once for each nonzero exponent ei . By line 9, the number of nonzero exponents appearing in the relation is at|emost log(|Δ|/3)/z, so the amount of time required to compute all of the |Ii | i | for all i is ( log(|Δ|/3)/z)L( 21 , 2z) = L( 12 , 2z). Afterward, the values |Ii ||ei | must all be multiplied together, a calculation which entails at most log(|Δ|/3)/z ideal multiplications where the log-norms of the input multiplicands are bounded above by |ei |

log Norm(Ii

) = |ei | log Norm(Ii ) ≤

N pi

2 log pi ≤ N 2 = L( 12 , 2z),

and thus each of the (at most) log(|Δ|/3)/z multiplications in the ensuing product can be completed in time at most ( log(|Δ|/3)/z)L( 21 , 2z) = L( 12 , 2z). Finally, we must multiply this end result by L, an operation which requires O(max(log , L( 21 , 2z))1+ε ) time. All together, the running time of step 20 is L( 12 , 2z) + O(max(log , L( 21 , 2z))1+ε ) = max((log )1+ε , L( 12 , 2z)), and the norm of the resulting ideal I is bounded above by · exp(L( 12 , 2z)). Obtaining the generator β of I in line 21 of Algorithm 3 using Cornacchia’s algorithm requires O(log(Norm(I))1+ε ) = (log + L( 12 , 2z))1+ε time. We remark that ﬁnding β given I is substantially easier than the usual Cornacchia’s algorithm, which entails ﬁnding β given only Norm(I). The usual algorithm requires ﬁnding all the square roots of Δ modulo Norm(I), which is very slow when Norm(I) has a large number of prime divisors. This time-consuming step is unnecessary when the ideal I itself is given, since the embedding of the ideal I in End(E) already provides (up to sign) the correct square root of Δ mod I. A detailed description of this portion of Cornacchia’s algorithm in the context of the full algorithm, together with running time ﬁgures speciﬁc to each

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sub-step, is given by Hardy et al. [19]; for our purposes, the running time of a single iteration of Step 6 in [19, §4] is the relevant ﬁgure. This concludes our analysis of Algorithm 3. Returning to Algorithm 4, we ﬁnd that (as in [4]) the computation of the individual isogenies φi in line 3 of Algorithm 4 is limited by the time required to compute the modular polynomials Φn (x, y). Using the Chinese remainder theorembased method of Br¨oker et al. [5], these polynomials can be computed mod q in time O(n3 log3+ε (n)), and the resulting polynomials require O(n2 (log2 n+log q)) space. For each ideal Ii , the corresponding modular polynomial of level pi only needs to be computed once, but the polynomial once computed must be evaluated, diﬀerentiated, and otherwise manipulated ei times, at a cost of O(p2+ε ) i ﬁeld operations in Fq per manipulation, or O(p2+ε )(log q)1+ε bit operations using i fast multiplication. The total running time of line 3 is therefore

N 2 3+ε 2+ε 1+ε 3+ε p2+ε |ei |pi (log q) ≤ O(N )+ (log q)1+ε O(pi ) + i p i i i log(|Δ|/3) 2+ε ≤ O(N 3+ε ) + (log q)1+ε = L( 21 , 3z) + L( 12 , 2z)(log q)1+ε . N z Similarly, the evaluation of φc in line 4 requires |ei |p2+ε = L( 21 , 2z) i i

ﬁeld operations in Fqn , which corresponds to L( 12 , 2z)(log q n )1+ε bit operations using fast multiplication. Combining all the above quantities, we obtain a total running time of L( 12 , z) +

(algorithm 2)

1 ) L( 12 , 4z

(lines 4–9, algorithm 3)

+ max((log )

1+ε

+ + +

, L( 21 , 2z)) 1+ε

(log + L( 12 , 2z)) L( 12 , 3z) + L( 12 , 2z)(log q)1+ε L( 12 , 2z)(log q n )1+ε

(line 20, algorithm 3) (line 21, algorithm 3) (line 3, algorithm 4) (line 4, algorithm 4)

1 ) + (log + L( 12 , 2z))1+ε + L( 12 , 3z) + L( 21 , 2z)(log q n )1+ε . = L( 12 , 4z

When |Δ| is large, we may impose the reasonable assumption that log() L( 12 , z) and log(q n ) L( 12 , z). In this case, the running time of Algorithm 4 is 1 dominated by the expression L( 12 , 4z ) + L( 12 , 3z), which attains a minimum at 1 z = 2√3 . Taking this value of z, we ﬁnd that the running time of Algorithm 4 is equal to L|Δ| ( 12 ,

√ 3 2 ).

Since the maximum value of |Δ| ≤ |Δπ | = 4q − t2 is 4q,

we can alternatively express this running time as simply Lq ( 12 ,

√

3 2 ).

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In the general case, log() and log(q n ) might be non-negligible compared to L( 12 , z). This can happen in one of two ways: either |Δ| is small, or (less likely) is very large and/or n is large. When this happens, we can still bound the 1 running time of Algorithm 4 by taking z = 2√ in the foregoing calculation, 3 although such a choice may fail to be optimal. We then ﬁnd that the running time of Algorithm 4 is bounded above by (log() + L( 12 , √13 ))1+ε + L( 12 ,

√

3 2 )

+ L( 12 , √13 )(log q n )1+ε .

We summarize our results in the following theorem. Theorem 4.1. Let E/Fq be an ordinary elliptic curve with Frobenius πq , given by a Weierstrass equation, and let P ∈ E(Fqn ) be a point on E. Let Δ = disc(End(E)) be given. Assume that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and let L = (, c + dπq ) be an End(E)-ideal of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Under the heuristics of [1, §4], Algorithm 4 computes the unique elliptic curve E such that there exists a normalized isogeny φ : E → E with kernel E[L]. Furthermore, it computes the x-coordinate of φ(P ) if End(E) does not equal Z[i] or Z[ζ3 ] and the square, respectively cube, of the x-coordinate of φ(P ) otherwise. The running time of the algorithm is bounded above by (log() + L( 12 , √13 ))1+ε + L( 12 ,

√

3 2 )

+ L( 12 , √13 )(log q n )1+ε .

The running time of the algorithm is subexponential in log |Δ|, and polynomial in log(), log(q), and n.

5 5.1

Examples Small Example

Let p = 1010 +19 and let E/Fp be the curve y 2 = x3 +15x+129. Then E(Fp ) has cardinality 10000036491 = 3 · 3333345497 and trace t = −36471. To avoid any bias in the selection of the prime , we set to be the smallest Elkies prime of E larger than p/2, namely = 5000000029. We will evaluate the x-coordinate of φ(P ), where φ is an isogeny of degree , and P is chosen arbitrarily to be the point (5940782169, 2162385016) ∈ E(Fp ). We remark that, although this example is designed to be artiﬁcially small for illustration purposes, the evaluation of this isogeny would already be infeasible if we were using prior techniques based on modular functions of level . √ The discriminant Δ of E is Δ = t2 − 4p = −38669866235. Set w = 1+2 Δ and O = OΔ . The quadratic form (5000000029, −2326859861, 270713841) represents a prime ideal L of norm , and we show how to calculate the isogeny φ having kernel corresponding to E[L]. Using an implementation of Algorithm 3 β in MAGMA [22], we ﬁnd immediately the relation L = ( m ) · p19 · p24 31

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where β = 588048307603210005w − 235788727470005542279904, m = 19 · 3124 , p19 = (19, 2w + 7), and p31 = (31, 2w + 5). Using this factorization, we can then evaluate φ : E → E using the latter portion of Algorithm 4. We ﬁnd that E is the curve with Weierstrass equation y 2 = x3 + 3565469415x + 7170659769, and φ(P ) = (7889337683, ±3662693258). We omit the details of these steps, since this portion of the algorithm is identical to the algorithm of Br¨ oker, Charles and Lauter, and the necessary steps are already extensively detailed in their article [4]. We can check our computations for consistency by performing a second computation, starting from the curve E : y 2 = x3 + 3565469415x + 7170659769, the ¯ which point P = (7889337683, 3662693258) ∈ E (Fp ), and the conjugate ideal L, is represented by the quadratic form (5000000029, 2326859861, 270713841). Let ¯ Up to φ¯ : E → E denote the unique normalized isogeny with kernel E [L]. ¯ a normalization isomorphism ι : E → E , the isogeny φ should equal the dual ¯ isogeny φˆ of φ, and the composition φ(φ(P )) should yield ι(P ). Indeed, upon performing the computation, we ﬁnd that E has equation y 2 = x3 + (15/4 )x + (129/6), which is isomorphic to E via the isomorphism ι : E → E deﬁned by ι(x, y) = (x/2 , y/3 ), and ¯ φ(φ(P )) = (3163843645, 8210361642) = (5551543736/2, 6305164567/3), in agreement with the value of P , which is (5551543736, 6305164567). 5.2

Medium Example

Let E be the ECCp-109 curve [8] from the Certicom ECC Challenge [7], with equation y 2 = x3 + ax + b over Fp where p = 564538252084441556247016902735257 a = 321094768129147601892514872825668 b = 430782315140218274262276694323197 As before, to avoid any bias in the choice √ of , we set to be the least Elkies prime greater than p/2, and we deﬁne w = 1+2 Δ where Δ = disc(End(E)). Let L be the prime ideal of norm in End(E) corresponding to the reduced quadratic form (, b, c) of discriminant Δ, where b = −105137660734123120905310489472471. For each Elkies prime p, let pp denote the unique prime ideal corresponding to the reduced quadratic form (p, b, c) where b ≥ 0. Our smoothness bound in this 1 case is N = L( 12 , 2√ ) ≈ 200. Using Sutherland’s smoothrelation package [28], 3 which implements the FindRelation algorithm of [1], one ﬁnds in a few seconds β I, where (using an initial seed of 0) the relation L = m p100 p14 p247 p¯273 ¯ p103 p179 p191 I=¯ p72 7 ¯ 13 ¯ 23 ¯ m = 772 13100 2314 472 732 1031 1791 1911

A Subexponential Algorithm for Evaluating Large Degree Isogenies

231

and β = 3383947601020121267815309931891893555677440374614137047492987151\ 2226041731462264847144426019711849448354422205800884837 − 1713152334033312180094376774440754045496152167352278262491589014\ 097167238827239427644476075704890979685 · w We ﬁnd that the codomain E of the normalized isogeny φ : E → E of kernel E[L] has equation y 2 = x3 + a x + b where a = 84081262962164770032033494307976 b = 506928585427238387307510041944828 and that the base point P = (97339010987059066523156133908935, 149670372846169285760682371978898)

of E given in the Certicom ECC challenge has image (450689656718652268803536868496211, ±345608697871189839292674734567941).

under φ. As with the ﬁrst example, we checked the computation for consistency by using the conjugate ideal. 5.3

Large Example

Let E be the ECCp-239 curve [8] from the Certicom ECC Challenge [7]. Then E has equation y 2 = x3 + ax + b over Fp where p = 862591559561497151050143615844796924047865589835498401307522524859467869 a = 820125117492400602839381236756362453725976037283079104527317913759073622 b = 545482459632327583111433582031095022426858572446976004219654298705912499

Let L be the prime ideal whose norm is the least Elkies prime greater than p/2 and whose ideal class is represented by the quadratic form (, b, c) with 1 b ≥ 0. We have N = L( 12 , 2√ ) ≈ 5000, and one ﬁnds in a few hours using 3 smoothrelation [28] that L is equivalent to I=¯ p27 p11 p19 p237 ¯ p271 ¯ p131 p211 ¯ p389 ¯ p433 ¯ p467 ¯ p18 p1019 ¯p1151 ¯p1597 ¯p62143 ¯p52207 ¯p3359 859 p863 ¯ where each ideal pp is represented by the reduced quadratic form (p, b, c) having b ≥ 0 (this computation can be reconstructed with [28] using the seed 7). The quotient L/I is generated by β/m where m = Norm(I) and β is −923525986803059652225406070265439117913488592374741428959120914067053307\ 4585317 − 917552768623818156695534742084359293432646189962935478129227909w.

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D. Jao and V. Soukharev

Given this relation, evaluating isogenies of degree is a tedious but routine computation using Elkies-Atkin techniques [4, §3.1]. Although we do not complete it here, the computation is well within the reach of present technology; indeed, Br¨ oker et al. [5] have computed classical modular polynomials mod p of level up to 20000, well beyond the largest prime of 3389 appearing in our relation.

6

Related Work

Bisson and Sutherland [1] have developed an algorithm to compute the endomorphism ring of an elliptic curve in subexponential time, using relation-ﬁnding techniques which largely overlap with ours. Although our main results were obtained independently, we have incorporated their ideas into our algorithm in several places, resulting in a simpler presentation as well as a large speedup compared to the original version of our work. Given two elliptic curves E and E over Fq admitting a normalized isogeny φ : E → E of degree , the equation of φ as a rational function contains O() coeﬃcients. Bostan et al. [3] have published an algorithm which produces this equation, given E, E , and . Their algorithm has running time O(1+ε ), which is quasi-optimal given the size of the output. Using our algorithm, it is possible √ to compute E from E and in time log()L|Δ| ( 12 , 23 ) for large . Hence the combination of the two algorithms can produce the equation of φ within a quasioptimal running time of O(1+ε ), given only E and (or E and L), without the need to provide E in the input.

Acknowledgments We thank the anonymous referees for numerous suggestions which led to substantial improvements in our main result.

References 1. Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a ﬁnite ﬁeld. Journal of Number Theory (to appear 2009) 2. Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (2000); Reprint of the 1999 original (1999) ´ Fast algorithms for computing iso3. Bostan, A., Morain, F., Salvy, B., Schost, E.: genies between elliptic curves. Math. Comp. 77(263), 1755–1778 (2008) 4. Br¨ oker, R., Charles, D., Lauter, K.: Evaluating large degree isogenies and applications to pairing based cryptography. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 100–112. Springer, Heidelberg (2008) 5. Br¨ oker, R., Lauter, K., Sutherland, A.: Modular polynomials via isogeny volcanoes (2010) 6. Buchmann, J., Vollmer, U.: Binary quadratic forms. Algorithms and Computation in Mathematics, vol. 20. Springer, Berlin (2007); An algorithmic approach 7. Certicom ECC Challenge, http://www.certicom.com/images/pdfs/cert_ecc_challenge.pdf.

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8. Certicom ECC Curves List, http://www.certicom.com/index.php/curves-list 9. Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications. Chapman & Hall/CRC (2006) 10. Cohen, H.: A course in computational algebraic number theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993) 11. Couveignes, J.-M., Morain, F.: Schoof’s algorithm and isogeny cycles. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 43–58. Springer, Heidelberg (1994) 12. Cox, D.A.: Primes of the form x2 + ny 2 . A Wiley-Interscience Publication, John Wiley & Sons Inc., New York (1989); Fermat, class ﬁeld theory and complex multiplication 13. Enge, A.: Computing modular polynomials in quasi-linear time. Math. Comp. 78(267), 1809–1824 (2009) 14. Fouquet, M., Morain, F.: Isogeny volcanoes and the SEA algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002) 15. Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptology (to appear 2010) 16. Galbraith, S.D.: Constructing isogenies between elliptic curves over ﬁnite ﬁelds. LMS J. Comput. Math. 2, 118–138 (1999) (electronic) 17. Galbraith, S.D., Hess, F., Smart, N.P.: Extending the GHS Weil descent attack. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 29–44. Springer, Heidelberg (2002) 18. Hafner, J., McCurley, K.: A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2(4), 837–850 (1989) 19. Hardy, K., Muskat, J.B., Williams, K.S.: A deterministic algorithm for solving n = f u2 + gv 2 in coprime integers u and v. Math. Comp. 55(191), 327–343 (1990) 20. Jao, D., Miller, S.D., Venkatesan, R.: Do all elliptic curves of the same order have the same diﬃculty of discrete log? In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 21–40. Springer, Heidelberg (2005) 21. Kohel, D.: Endomorphism rings of elliptic curves over ﬁnite ﬁelds. PhD thesis, University of California, Berkeley (1996) 22. MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/ 23. Menezes, A., Teske, E., Weng, A.: Weak ﬁelds for ECC. In: Okamoto, T. (ed.) CT-RSA 2004. LNCS, vol. 2964, pp. 366–386. Springer, Heidelberg (2004) 24. Sch¨ onhage, A.: Fast reduction and composition of binary quadratic forms. In: ISSAC 1991: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, pp. 128–133. ACM, New York (1991) 25. Schoof, R.: Counting points on elliptic curves over ﬁnite ﬁelds. J. Th´eor. Nombres Bordeaux 7(1), 219–254 (1995); Les Dix-huiti`emes Journ´ees Arithm´etiques (Bordeaux, 1993) 26. Seysen, M.: A probabilistic factorization algorithm with quadratic forms of negative discriminant. Math. Comp. 48(178), 757–780 (1987) 27. Silverman, J.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1992); Corrected reprint of the 1986 original (1986) 28. Sutherland, A.:Smoothrelation, http://math.mit.edu/~ drew/smoothrelation_v1.tar 29. Tate, J.: Endomorphisms of abelian varieties over ﬁnite ﬁelds. Invent. Math. 2, 134–144 (1966) 30. Teske, E.: An elliptic curve trapdoor system. J. Cryptology 19(1), 115–133 (2006)

Huﬀ’s Model for Elliptic Curves Marc Joye1 , Mehdi Tibouchi2, , and Damien Vergnaud2 1 Technicolor, Security & Content Protection Labs 1 avenue de Belle Fontaine, 35576 Cesson-S´evign´e Cedex, France [email protected] 2 ´ Ecole Normale Sup´erieure – C.N.R.S. – I.N.R.I.A. 45, Rue d’Ulm – 75230 Paris CEDEX 05 – France {mehdi.tibouchi,damien.vergnaud}@ens.fr

Abstract. This paper revisits a model for elliptic curves over Q introduced by Huﬀ in 1948 to study a diophantine problem. Huﬀ’s model readily extends over ﬁelds of odd characteristic. Every elliptic curve over such a ﬁeld and containing a copy of Z/4Z × Z/2Z is birationally equivalent to a Huﬀ curve over the original ﬁeld. This paper extends and generalizes Huﬀ’s model. It presents fast explicit formulæ for point addition and doubling on Huﬀ curves. It also addresses the problem of the eﬃcient evaluation of pairings over Huﬀ curves. Remarkably, the so-obtained formulæ feature some useful properties, including completeness and independence of the curve parameters. Keywords: Elliptic curves, Huﬀ’s model, uniﬁed addition law, complete addition law, explicit formulæ, scalar multiplication, Tate pairing, Miller’s algorithm.

1

Introduction

Elliptic curves have been extensively studied in algebraic geometry and number theory since the middle of the nineteenth century. More recently, they have been used to devise eﬃcient algorithms for factoring large integers [19,22] or for primality proving [2,13,23]. They also revealed useful in the construction of cryptosystems [18,20]. In this paper, we develop an elliptic curve model introduced by Huﬀ in 1948 to study a diophantine problem. We present fast explicit formulæ for adding or doubling points on Huﬀ curves. We also devise a couple of extensions and generalizations upon this model. We analyze the impact of these curves in cryptographic applications. Some of our addition formulæ are uniﬁed; i.e., they remain valid for doubling a point. Even better, they achieve completeness (i.e., are valid for all inputs) when restricted to a cyclic subgroup, as is customary in cryptographic settings. We also consider the problem of pairing computation over Huﬀ curves.

This research was completed while the second author was visiting the Okamoto Research Laboratory at the NTT Information Sharing Platform (Tokyo, Japan).

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 234–250, 2010. c Springer-Verlag Berlin Heidelberg 2010

Huﬀ’s Model for Elliptic Curves

1.1

235

Background

Elliptic curves and cryptography. In 1985, Koblitz [18] and Miller [20] independently proposed the use of elliptic curves in public-key cryptography. The main advantage of elliptic curve systems stems from the absence of a subexponentialtime algorithm to compute discrete logarithms on general elliptic curves over ﬁnite ﬁelds. Consequently, one can use an elliptic curve group that is smaller in size compared with systems based on either integer factorization or the discrete log problem in the multiplicative group of a ﬁnite ﬁeld, while maintaining the same (heuristic) level of security (see [17] for a recent survey on elliptic curve cryptography). The use of elliptic curves in cryptography makes the key sizes smaller but the arithmetic of the underlying group is more tedious (for example, with the widely-used Jacobian coordinates, the general addition of two points on an elliptic curve typically requires 16 ﬁeld multiplications). Therefore a huge amount of research has been devoted to the analysis of the performance of various forms of elliptic curves proposed in the mathematical literature: Weierstraß cubics, Jacobi intersections, Hessian curves, Jacobi quartics, or the more recent forms of elliptic curves due to Montgomery, Doche-Icart-Kohel or Edwards (see [6] for an encyclopedic overview of these models). For instance, since 2007, there has been a rapid development of the curves introduced by Edwards in [12] and their use in cryptology. Bernstein and Lange proposed a more general version of these curves in [7] and the inverted Edwards coordinates in [8]. Bernstein, Birkner, Joye, Lange, and Peters studied twisted Edwards curves in [5]. Hisil, Wong, Carter and Dawson proposed extended twisted Edwards coordinates in [14]. Bernstein, Lange, and Farashahi covered the binary case in [9]. The ﬁrst formulæ for computing pairings over Edwards curves were published by Das and Sarkar [11]. They were subsequently improved by Ionica and Joux [16]. The best implementation to date is due to Ar`ene, Lange, Naehrig, and Ritzenhaler [1]. The present paper is aimed at providing a similar study for a forgotten model of elliptic curves hinted by Huﬀ in 1948. A diophantine problem. Huﬀ [15] considered rational distance sets S (i.e., subsets S of the plane R2 such that for all s, t ∈ S, the distance between s and t is a rational number) of the following form: given distinct a, b ∈ Q, S contains the four points (0, ±a) and (0, ±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q. Such a point (x, 0) must then satisfy the equations x2 +a2 = u2 and x2 + b2 = v 2 with u, v ∈ Q. The system of associated homogeneous equations x2 + a2 z 2 = u2 and x2 + b2 z 2 = v 2 deﬁnes a curve of genus 1 in P3 . Huﬀ, and later his student Peeples [24], provided examples where this curve has positive rank over Q, thus exhibiting examples of arbitrarily large rational distance sets of cardinality k > 4 such that exactly k − 4 points are on one line. The above mentioned genus 1 curve is birationally equivalent to the curve ax(y 2 − 1) = by(x2 − 1)

(1)

for some parameters a and b in Q. It is easily seen that, over any ﬁeld K of odd characteristic, Equation (1) deﬁnes an elliptic curve if a2 = b2 and a, b = 0.

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Indeed, if ab = 0, the gradient of the curve F (X, Y, Z) = aX(Y 2 −Z 2 )−bY (X 2 − Z 2 ) in the projective plane P2 (K) is ∂F ∂F ∂F , , = a(Y 2 − Z 2 ) − 2bXY, 2aXY − b(X 2 − Z 2 ), 2(−aX + bY )Z , ∂X ∂Y ∂Z which does not vanish at the three points at inﬁnity (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0) and vanishes at a ﬁnite point (x : y : 1) if and only if ax = by, which together with Eq. (1) implies that x2 = y 2 and therefore a2 = b2 . It is worth noting that in characteristic 2, the point (1 : 1 : 1) is always singular and therefore the family of curves deﬁned by (1) does not contain any smooth curve. As will be shown in Section 3, we can extend our study to even characteristic by considering a generalized model. 1.2

Contributions of the Paper

Our ﬁrst contribution is a detailed study of Huﬀ’s form for elliptic curves over ﬁnite ﬁelds of odd characteristic and a statement of the addition law in these groups. We show in particular that all elliptic curves over non-binary ﬁnite ﬁelds with a subgroup isomorphic to Z/4Z × Z/2Z can be transformed to Huﬀ’s form. We then analyze their arithmetic and investigate several generalizations and extensions. In particular, we present explicit formulæ (i.e., as a series of ﬁeld operations) that – – – –

compute compute compute compute

a a a a

complete addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : Z2 ) using 12m; uniﬁed addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : Z2 ) using 11m; mixed addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : 1) using 10m; doubling [2](X1 : Y1 : Z1 ) using 6m + 5s

where m and s denote multiplications and squarings in the base ﬁeld K. As a further contribution, since bilinear pairings have found numerous applications in cryptography, we also present formulæ for computing Tate pairings using Huﬀ’s form. Speciﬁcally, we present explicit formulæ that – compute a full Miller addition using 1M + (k + 15)m; – compute a mixed Miller addition using 1M + (k + 13)m; – compute a Miller doubling using 1M + 1S + (k + 11)m + 6s on a Huﬀ curve over K = Fq of embedding degree k. M and S denote multiplications and squarings in the larger ﬁeld Fqk while m and s are operations in Fq as before. Outline. The rest of this paper is organized as follows. The next section introduces Huﬀ’s model. We develop eﬃcient uniﬁed addition formulæ and discuss the applicability of the model. We explicit the class of elliptic curves covered by Huﬀ’s model. In Section 3, we present several generalizations and extensions. We oﬀer dedicated addition formulæ. We generalize Huﬀ’s model to cover a larger class of elliptic curves. We also extend the model to the case of binary ﬁelds. Section 4 deals with pairings over Huﬀ curves. We exploit the relative simplicity of the underlying group law to devise eﬃcient formulæ for the evaluation of the Tate pairing. Finally, we conclude in Section 5.

Huﬀ’s Model for Elliptic Curves

2

237

Huﬀ’s Model

Let K denote a ﬁeld of characteristic = 2. Consider the set of projective points (X : Y : Z) ∈ P2 (K) satisfying the equation E/K : aX(Y 2 − Z 2 ) = bY (X 2 − Z 2 )

(2)

where a, b ∈ K× and a2 = b2 . This form is referred to as Huﬀ ’s model of an elliptic curve.

Fig. 1. Example of a Huﬀ curve (over R)

The tangent line at (0 : 0 : 1) is aX = bY , which intersects the curve with multiplicity 3, so that O = (0 : 0 : 1) is an inﬂection point of E. (E, O) is therefore an elliptic curve with O as neutral element and whose group law, denoted ⊕, has the following property: for any line intersecting the cubic curve E at the three points P1 , P2 and P3 (counting multiplicities), we have P1 ⊕ P2 ⊕ P3 = O. In particular, the inverse of point P1 = (X1 : Y1 : Z1 ) is P1 = (X1 : Y1 : −Z1 ) and the sum of P1 and P2 is P1 ⊕ P2 = P3 . We note that a point at inﬁnity is its own inverse. Hence, the three points at inﬁnity (i.e., on the line Z = 0 in P2 ) — namely, (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0), are exactly the three primitive 2-torsion points of E. The sum of any two of them is equal to the third one. More generally, (X1 : Y1 : Z1 ) ⊕ (1 : 0 : 0) is the inverse of the point of intersection of the “horizontal” line passing through (X1 : Y1 : Z1 ) with E. When Z1 = 0, we have (X1 : Y1 : Z1 ) ⊕ (1 : 0 : 0) = (Z1 2 : −X1 Y1 : X1 Z1 ) , and analogously, (X1 : Y1 : Z1 ) ⊕ (0 : 1 : 0) = (−X1 Y1 : Z1 2 : Y1 Z1 ) .

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From (a : b : 0) = (1 : 0 : 0) ⊕ (0 : 1 : 0), when Z1 = 0, we get (X1 : Y1 : Z1 ) + (a : b : 0) = (Z1 2 : −X1 Y1 : X1 Z1 ) ⊕ (0 : 1 : 0) and therefore (a : b : 0) if (X1 : Y1 : Z1 ) = (0 : 0 : 1) . (X1 : Y1 : Z1 ) ⊕ (a : b : 0) = (Y1 Z1 : X1 Z1 : −X1 Y1 ) otherwise We remark that adding (a : b : 0) to any of the points (±1 : ±1 : 1) transforms it into its inverse. It follows that these four points are the four solutions to the equation [2]P = (a : b : 0) and so are primitive 4-torsion points. The eight remarkable points we identiﬁed form a subgroup isomorphic to Z/4Z × Z/2Z. When K = Q, this must be the full torsion since, according to a theorem by Mazur, the torsion subgroup is of order at most 12 (and thus exactly 8 here). Remark 1. In [15, p. 445], it is noted that the inverse projective transformations Υ : P2 (K) → P2 (K) :

(X : Y : Z) → (U : V : W ) = ab(bX − aY ) : ab(b2 − a2 )Z : −aX + bY

and Υ −1 : P2 (K) → P2 (K) :

(U : V : W ) → (X : Y : Z) = b(U + a2 W ) : a(U + b2 W ) : V

induce a correspondence between Eq. (2) and the Weierstraß equation V 2 W = U (U + a2 W )(U + b2 W ) . Observe that point at inﬁnity (0 : 1 : 0) on the Weierstraß curve is mapped to (0 : 0 : 1) on the Huﬀ curve through Υ −1 . Observe also that map Υ −1 is a line-preserving transformation. This is another way to see that the group law on a Huﬀ curve E follows the chord-and-tangent rule [25, § 2] with O = (0 : 0 : 1) as neutral element. 2.1

Aﬃne Formulæ

We give explicit formulæ for the group law. Excluding the 2-torsion, we use the non-homogeneous form ax(y 2 − 1) = by(x2 − 1). Let y = λ x + μ denote the secant line passing through two diﬀerent points P1 = (x1 , y1 ) and P2 = (x2 , y2 ). This line intersects the curve at a third point P3 = (−x3 , −y3 ). Plugging the line equation into the curve equation, we get ax (λx+ μ)2 − 1 = b(λx+ μ)(x2 − 1) =⇒ λ(aλ− b)x3 + μ(2aλ− b)x2 + · · · = 0 . Whenever deﬁned, we so obtain ⎧ ⎨x = x + x + μ(2aλ − b) 3 1 2 λ(aλ − b) ⎩ y3 = λx3 − μ

Huﬀ’s Model for Elliptic Curves

with λ =

239

y1 − y2 and μ = y1 − λx1 . After simpliﬁcation, we have x1 − x2 (x1 y2 − x2 y1 ) 2a(y1 − y2 ) − b(x1 − x2 ) x3 = x1 + x2 + (y1 − y2 ) a(y1 − y2 ) − b(x1 − x2 ) (x1 − x2 ) a(y1 2 − y2 2 ) − b(x1 y1 − x2 y2 ) = (y1 − y2 ) a(y1 − y2 ) − b(x1 − x2 )

and (y1 − y2 ) b(x1 2 − x2 2 ) − a(x1 y1 − x2 y2 ) y3 = − . (x1 − x2 ) a(y1 − y2 ) − b(x1 − x2 ) The above formulæ can be further simpliﬁed by reusing the curve equation. A simple calculation shows that a(y1 − y2 ) − b(x1 − x2 ) (x1 + x2 )y1 y2 = a(x2 y1 − x1 y2 )(y1 y2 − 1) . Hence, we can write

2a(y1 − y2 ) − b(x1 − x2 ) (x1 + x2 )y1 y2 (y1 − y2 )a(y1 y2 − 1) x2 y1 − x1 y2 (x1 + x2 )y1 y2 − = x1 + x2 − y1 − y2 y1 y2 − 1 (x1 + x2 )y1 y2 x1 y1 − x2 y2 . − = y1 − y2 y1 y2 − 1

x3 = x1 + x2 −

Furthermore, as easily shown b(x1 y1 − x2 y2 )(x1 x2 + 1) = (y1 − y2 ) ax1 x2 (y1 + y2 ) + b(x1 + x2 ) , it thus follows that ax1 x2 (y1 + y2 ) + b(x1 + x2 ) (x1 + x2 )y1 y2 − b(x1 x2 + 1) y1 y2 − 1 (x1 + x2 )(1 + y1 y2 ) , = (1 + x1 x2 )(1 − y1 y2 )

x3 =

(3)

since ax1 x2 (y1 + y2 )(1 − y1 y2 ) = by1 y2 (x1 + x2 )(1 − x1 x2 ). Likewise, by symmetry, we have y3 =

(y1 + y2 )(1 + x1 x2 ) . (1 − x1 x2 )(1 + y1 y2 )

(4)

Equations (3) and (4) are deﬁned whenever x1 x2 = ±1 and y1 y2 = ±1. Advantageously, curve parameters are not involved. Moreover, this addition law is uniﬁed : it can be used to double a point (i.e., when P2 = P1 ).

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Projective Formulæ

Previous aﬃne formulæ involve inversions in K. To avoid these operations and get faster arithmetic, projective coordinates may be preferred. We let m and s represent the cost of a multiplication and of a squaring in K, respectively. The projective form of Eqs (3) and (4) is ⎧ 2 ⎪ ⎨X3 = (X1 Z2 + X2 Z1 )(Y1 Y2 + Z1 Z2 ) (Z1 Z2 − X1 X2 ) . (5) Y3 = (Y1 Z2 + Y2 Z1 )(X1 X2 + Z1 Z2 )2 (Z1 Z2 − Y1 Y2 ) ⎪ ⎩ 2 2 2 2 2 2 2 2 Z3 = (Z1 Z2 − X1 X2 )(Z1 Z2 − Y1 Y2 ) In more detail, this can be evaluated as m1 = X1 X2 , m2 = Y1 Y2 , m3 = Z1 Z2 , m4 = (X1 + Z1 )(X2 + Z2 ) − m1 − m3 , m5 = (Y1 + Z1 )(Y2 + Z2 ) − m2 − m3 , m6 = (m2 + m3 )(m3 − m1 ), m7 = (m1 + m3 )(m3 − m2 ), m8 = m4 (m2 + m3 ), m9 = m5 (m1 + m3 ), X3 = m8 m6 , Y3 = m9 m7 , Z3 = m6 m7 , that is, with 12m. 2.3

Applicability

If (x1 , y1 ) = (0, 0) then (x1 , y1 ) ⊕ (a : b : 0) = −( x11 , y11 ). Observe that Equation (5) remains valid for doubling point (a : b : 0) or for adding point (a : b : 0) to another ﬁnite point (i.e., which is not at inﬁnity) diﬀerent from O; we get (X1 : Y1 : Z1 ) ⊕ (a : b : 0) = (−Y1 Z1 : −X1 Z1 : X1 Y1 ) as expected. The addition formula is however not valid for adding (0 : 1 : 0) or (1 : 0 : 0). More generally, we have: Theorem 1. Let K be a ﬁeld of characteristic = 2. Let P1 = (X1 : Y1 : Z1 ) and P2 = (X2 : Y2 : Z2 ) be two points on a Huﬀ curve over K. Then the addition formula given by Eq. (5) is valid provided that X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 . Proof. If P1 and P2 are ﬁnite, we can write P1 = (x1 , y1 ) and P2 = (x2 , y2 ). The above aﬃne formula for (x3 , y3 ) as given by Eqs (3) and (4) is deﬁned whenever x1 x2 = ±1 and y1 y2 = ±1. This translates into X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 for their projective coordinates. It remains to analyze points at inﬁnity. The points with their Z-coordinate equal to 0 are (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0). If P1 or P2 ∈ {(1 : 0 : 0), (0 : 1 : 0)}, the condition X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 is not satisﬁed. Suppose now P2 = (a : b : 0). The condition becomes X1 = 0 and Y1 = 0, which corresponds to P1 ∈ / {O, (1 : 0 : 0), (0 : 1 : 0)}. As aforementioned, the addition law is then valid for adding P1 to (a : b : 0).

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The previous theorem says that the addition on a Huﬀ curve is almost complete. However, the exceptional inputs are easily prevented in practice. Cryptographic applications typically involve (large) prime-order subgroups. More speciﬁcally, we state: Corollary 1. Let E be a Huﬀ curve over a ﬁeld K of odd characteristic. Let also P ∈ E(K) be a point of odd order. Then the addition law in the subgroup generated by P is complete. Proof. All points in P are of odd order and thus are ﬁnite (remember that points at inﬁnity are of order 2). It remains to show that for any points P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ P , we have x1 x2 = ±1 and y1 y2 = ±1. Note that x1 , y1 , x2 , y2 = ±1 since this corresponds to points of order 4 (and thus not in P ). Suppose that x1 x2 = ±1. Then ax1 (y1 2 − 1) = by1 (x1 2 − 1) =⇒ a x11 (y1 2 − 1) = by1 (1 − x11 2 ) =⇒ ±ax2 (y1 2 − 1) = −by1 (x2 2 − 1). Hence, since ax2 (y2 2 − 1) = by2 (x2 2 − 1), it follows that ∓y2 (y1 2 − 1) = y1 (y2 2 − 1) =⇒ (y1 ± y2 )(1 ∓ y1 y2 ) = 0 =⇒ y2 = ∓y1 or y1 y2 = ±1. As a result, when x1 x2 = ±1, we have (x2 , y2 ) ∈ ( x11 , −y1 ), ( x11 , y11 ), (− x11 , y1 ), (− x11 , − y11 ) . In all cases, one of (x1 , y1 ) ⊕ (x2 , y2 ) or (x1 , y1 ) (x2 , y2 ) is a 2-torsion point, a contradiction. Likewise, it can be veriﬁed that the case y1 y2 = ±1 leads to a contradiction, which concludes the proof.

The completeness of the addition law is very useful as it yields a natural protection against certain side-channel attacks (e.g., see [10]). Another useful feature is that the addition law is independent of the curve parameters. 2.4

Universality of the Model

The next theorem states that every elliptic curve over a ﬁeld of characteristic = 2 containing a copy of Z/4Z×Z/2Z can be put in Huﬀ’s form. Generalizations and extensions are discussed in Section 3. Theorem 2. Any elliptic curve (E, O) over a perfect ﬁeld K of characteristic = 2 such that E(K) contains a subgroup G isomorphic to Z/4Z × Z/2Z is birationally equivalent over K to a Huﬀ curve. Proof. The Riemann-Roch theorem implies that if D = a1 P1 + · · · + ar Pr is a divisor of degree 0 on E then the dimension of the vector space L (D) = {f ∈ K(E)× | div(f ) −D} ∪ {0} is equal to 1 when a1 P1 ⊕ · · · ⊕ ar Pr = O, and to 0 otherwise. Let H++ , H+− , H−+ and H−− denote the four points of G of order exactly 4 (with the convention H++ ⊕ H−− = O). Doubling these points produces a unique primitive 2-torsion point that we denote R. We further let P and Q denote the other two 2-torsion points; say, P = H++ ⊕ H+− and Q = H++ ⊕ H+− . We have P ⊕ R Q O = O; so there exists a nonzero

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rational function x with divisor exactly Q + O − P − R. In particular, x is well-deﬁned and nonzero at H++ and thus without loss of generality we may assume that x(H++ ) = 1. Similarly, there exists a rational function y with divisor P + O − Q − R such that y(H++ ) = 1. The rational function x − 1 has the same poles as x and vanishes at H++ . Its divisor div(x − 1) is thus given by H++ + X − P − R for some point X. Since this divisor is principal, we have H++ ⊕ X P R = O. Hence, it follows that X = P ⊕ R H++ = H++ ⊕ H+− ⊕ R H++ = H+− . Consequently, we have x(H+− ) = 1. Likewise, it is veriﬁed that y(H−+ ) = 1. Now, consider the map ι taking a rational function f to ιf : M → f (M). This is an endomorphism of the vector space L (P + R − Q − O). Indeed, the poles of ιf are P = P and R = R and its zeros are Q = Q and O = O. Moreover, since ι2 = id and since L (P + R − Q − O) is a onedimensional vector space, ι is the multiplication map by 1 or −1. The equality ιx = x would imply x(H−− ) = x(H++ ) = 1, which contradicts the previous calculation of div(x − 1). As a result, we must have ιx = −x. In particular, noting that H−+ = H+− , we obtain x(H−+ ) = ιx(H+− ) = −x(H+− ) = −1 , and similarly for H−− . Since x + 1 has the same poles as x, its divisor is then given by div(x+1) = H−+ +H−− −P −R. Analogously, we obtain div(y +1) = H+− + H−− − Q − R. Finally, consider the rational functions u = x(y 2 − 1) and v = y(x2 − 1). We have: div(u) = div(x) + div(y − 1) + div(y + 1) = (Q + O − P − R) + (H++ + H−+ − Q − R) + (H+− + H−− − Q − R) = H++ + H+− + H−+ + H−− + O − P − Q − 3R and div(v) = div(y) + div(x − 1) + div(x + 1) = (P + O − Q − R) + (H++ + H+− − P − R) + (H−+ + H−− − P − R) = H++ + H+− + H−+ + H−− + O − P − Q − 3R . But the vector space L (P + Q + 3R − O − H++ − H+− − H−+ − H−−) is of dimension 1, so there exists a linear relation between u and v. In other words, there exist a, b ∈ K× such that au = bv; i.e., such that ax(y 2 − 1) = by(x2 − 1). The rational map E → P2 (K) given by M → (x(M ) : y(M ) : 1) extends to a morphism deﬁned on all of E, and its image is contained in Ea,b in view of the previous relation (and Ea,b itself is a smooth irreducible curve as seen in §1.1). We therefore have a non-constant — and hence surjective — morphism of curves E → Ea,b . Moreover, its degree is at most 1: indeed, if a point (x0 : y0 : 1) ∈ Ea,b (K) has two distinct pre-images M = M ∈ E(K), the functions x − x0 and

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y − y0 vanish at M and M . Since they have the same poles as x and y, their divisors are respectively M + M − P − R and M + M − Q − R, which yields P ⊕ R = M ⊕ M = Q ⊕ R, a contradiction. As a surjective morphism of degree 1, the map E → Ea,b is thus an isomorphism.

3

Generalizations and Extensions

This section presents dedicated addition formulæ. It also presents a generalization of the model as originally introduced by Huﬀ so that it covers more curves and extends to binary ﬁelds. 3.1

Faster Computations

Dedicated doubling. The doubling formula can be sped up by evaluating squarings in K with a specialized implementation. The cost of a point doubling then becomes 7m + 5s. When s > 34 m, an even faster way for doubling a point is given by m1 = X1 Y1 , m2 = X1 Z1 , m3 = Y1 Z1 , s1 = Z1 2 , m4 = (m2 − m3 )(m2 + m3 ), m5 = (m1 − s1 )(m1 + s1 ), m6 = (m1 − s1 )(m2 − m3 ), m7 = (m1 + s1 )(m2 + m3 ), X([2]P1 ) = (m6 − m7 )(m4 + m5 ), Y ([2]P1 ) = (m6 + m7 )(m4 − m5 ), Z([2]P1 ) = (m4 + m5 )(m4 − m5 ), that is, with 10m + 1s. Moving the origin. Choosing O = (0 : 1 : 0) as the neutral element results in translating the group law. If we let ⊕ denote the corresponding point addition, we have P1 ⊕ P2 = (P1 O ) ⊕ (P2 O ) ⊕ O = P1 ⊕ P2 ⊕ O . Hence, we get ⎧ ⎪ ⎨X3 = (X1 Z2 + X2 Z1 )(Y1 Y2 + Z1 Z2 )(Y1 Z2 + Y2 Z1 ) Y3 = (X1 X2 − Z1 Z2 )(Z1 2 Z2 2 − Y1 2 Y2 2 ) ⎪ ⎩ Z3 = (Y1 Z2 + Y2 Z1 )(X1 X2 + Z1 Z2 )(Y1 Y2 − Z1 Z2 )

.

This can be evaluated with 11m as m1 = X1 X2 , m2 = Y1 Y2 , m3 = Z1 Z2 , m4 = (X1 + Z1 )(X2 + Z2 ) − m1 − m3 , m5 = (Y1 + Z1 )(Y2 + Z2 ) − m2 − m3 , X3 = m4 (m2 + m3 )m5 , Y3 = (m1 − m3 )(m3 − m2 )(m3 + m2 ), Z3 = m5 (m1 + m3 )(m2 − m3 ) . (6) This addition formula is uniﬁed: it can be used for doubling as well.

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For a mixed point addition (i.e., when Z2 = 1), we have m3 = Z1 and the number of required multiplications drops to 10m. When used for dedicated doubling, the above addition formula requires 6m + 5s, which can equivalently be obtained as s1 = X1 2 , s2 = Y1 2 , s3 = Z1 2 , s4 = (X1 + Y1 )2 − s1 − s2 , s5 = (Y1 + Z1 )2 − s2 − s3 , X([2]P1 ) = 2s3 s4 (s2 + s3 ), Y ([2]P1 ) = (s1 − s3 )(s3 − s2 )(s3 + s2 ),

(7)

Z([2]P1 ) = s5 (s1 + s3 )(s2 − s3 ) . Note that the expression for the inverse of point P1 is unchanged: P1 = (P1 O ) ⊕ O = P1 = (X1 : Y1 : −Z1 ). 3.2

More Formulæ

Alternative addition formulæ can be derived using the curve equation. For example, whenever deﬁned, we can write (x3 , y3 ) = (x1 , y1 ) ⊕ (x2 , y2 ) with x3 =

(x1 − x2 )(y1 + y2 ) (y1 − y2 )(1 − x1 x2 )

and y3 =

(y1 − y2 )(x1 + x2 ) . (x1 − x2 )(1 − y1 y2 )

In projective coordinates, this gives ⎧ 2 ⎪ ⎨X3 = (X1 Z2 − X2 Z1 ) (Y1 Z2 + Y2 Z1 )(Z1 Z2 − Y1 Y2 ) Y3 = (Y1 Z2 − Y2 Z1 )2 (X1 Z2 + X2 Z1 )(Z1 Z2 − X1 X2 ) ⎪ ⎩ Z3 = (X1 Z2 − X2 Z1 )(Y1 Z2 − Y2 Z1 )(Z1 Z2 − X1 X2 )(Z1 Z2 − Y1 Y2 )

,

which can be evaluated with 13m as m1 = X1 Z2 , m2 = X2 Z1 , m3 = Y1 Z2 , m4 = Y2 Z1 , m5 = (Z1 − X1 )(Z2 + X2 ) + m1 − m2 , m6 = (Z1 − Y1 )(Z2 + Y2 ) + m3 − m4 , m7 = (m1 − m2 )m6 , m8 = (m3 − m4 )m5 , X3 = (m1 − m2 )(m3 + m4 )m7 , Y3 = (m1 + m2 )(m3 − m4 )m8 , Z3 = m7 m8 . Although not as eﬃcient as the usual addition, this alternative formula is useful in some pairing computations (see Section 4.2). 3.3

Twisted Curves

As shown in Theorem 1, the group of points of a Huﬀ elliptic curve contains a copy of Z/4Z×Z/2Z. This implies that the curve order is a multiple of 8. Several cryptographic standards, however, require elliptic curves with group order of the form h n where h ∈ {1, 2, 3, 4} and n is a prime. We can generalize Huﬀ’s model to accommodate the case h = 4. Let P ∈ K[t] denote a monic polynomial of degree 2, with non-zero discriminant, and such that P(0) = 0. We can then introduce the cubic curve axP(y) = byP(x)

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where a, b ∈ K× . The set of points {(0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0), (a : b : 0)} ∼ = Z/2Z × Z/2Z belongs to the curve. Moreover, when P factors in K — i.e., when P(t) = (t − ω1 )(t − ω2 ) with ω1 , ω2 ∈ K× , the four points (±ω1 : ±ω2 : 1) are also on the curve. When Char K = 2, we consider P(t) = t2 − d for some d ∈ K× . So we deal with the set of projective points (X : Y : Z) ∈ P2 (K) satisfying the non-singular cubic equation (8) Eˆd : aX(Y 2 − dZ 2 ) = bY (X 2 − dZ 2 ) where a, b, d ∈ K× and a2 = b2 . This equation corresponds to Weierstraß equa2 2 (X : Y : tion V 2 W = U (U + ad W )(U + bd W ) under the inverse transformations Z) = b(dU + a2 W ) : a(dU + b2 W ) : dV and (U : V : W ) = ab(bX − aY ) : √ ab(b2 − a2 )Z : d(−aX + bY ) . The transformation (X : Y : Z) ← (X : Y : Z d) √ ˆd over K( d). Curves Eˆd are therefore induces an isomorphism from E = Eˆ1 to E quadratic twists of Huﬀ curves. In aﬃne coordinates, we consider the curve equation ax(y 2 − d) = by(x2 − d). The sum of two ﬁnite points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) such that x1 x2 = ±d and y1 y2 = ±d is given by (x3 , y3 ) where x3 =

d(x1 + x2 )(d + y1 y2 ) (d + x1 x2 )(d − y1 y2 )

and y3 =

d(y1 + y2 )(d + x1 x2 ) . (d − x1 x2 )(d + y1 y2 )

(9)

Extending the computations of § 2.2, it is readily veriﬁed that the sum of two points can be evaluated with 12m (plus a couple of multiplications by constant d) using projective coordinates. The faster computations of the previous section also generalize to twisted curves. 3.4

Binary Fields

Huﬀ’s form can be extended to a binary ﬁeld as ax(y 2 + y + 1) = by(x2 + x + 1) . This curve is birationally equivalent to Weierstraß curve v(v + (a + b)u) = u(u + a2 )(u + b2 ) under the inverse maps b(u + a2 ) a(u + b2 ) (x, y) = , v v + (a + b)u

and (u, v) =

ab ab(axy + b) , xy x2 y

.

The neutral element is O = (0, 0).

4 4.1

Pairings Preliminaries

Let (E, O) be an elliptic curve over K = Fq , with q odd. Suppose that #E(Fq ) = hn where n is a prime such that gcd(n, q) = 1. Let further k denote the

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embedding degree with respect to n, namely the smallest extension Fqk of Fq containing all n-th roots of unity. In other words, k is the smallest positive integer k such that n | q k − 1. For better eﬃciency, we further assume that k > 1 is even. For any point P ∈ E(Fq )[n], we let fP denote a rational function on E deﬁned over Fq such that div(fP ) = nP − nO; it exists and is unique up to a multiplicative constant, according to the Riemann-Roch theorem. The group of n-th roots of unity in Fqk is denoted by μn . The (reduced) Tate pairing is then deﬁned as Tn : E(Fq )[n] × E(Fqk )/[n]E(Fqk ) → μn : (P , Q) → fP (Q)(q

k

−1)/n

.

This deﬁnition does not depend on the choice of fP with the appropriate divisor, nor on the class of Q mod [n]E(Fqk ). In practice, Tn can be computed using a technique due to Miller [21], in terms of rational functions gR,P depending on P and on a variable point R. Function gR,P is the so-called line function with divisor R + P − O − (R ⊕ P ), which arises in addition formulæ when E is represented as a plane cubic. The core idea is to derive function fP iteratively. Letting fi,P be the function with divisor div(fi,P ) = iP − ([i]P ) − (i − 1)O, it is easily veriﬁed that fi+j,P = fi,P · fj,P · g[i]P ,[j]P . Observe that f1,P = 1 and fn,P = fP . Hence, if n = n−1 n−1 · · · n0 2 is the binary representation of n, the Tate pairing can be computed as follows.

Algorithm 1. Miller’s algorithm 1: 2: 3: 4: 5: 6: 7: 8:

f ← 1; R ← P for i = − 2 down to 0 do f ← f 2 · gR,R (Q); R ← [2]R if (ni = 1) then f ← f · gR,P (Q); R ← R ⊕ P end if end for k return f (q −1)/n

Contrary to Edwards curves or Jacobi quartics, Huﬀ curves are represented as plane cubics. This makes Miller’s algorithm, along with a number of improvements proposed for Weierstraß curves (e.g., as presented in [3]), directly applicable to the computation of pairings over Huﬀ curves. 4.2

Pairing Formulæ for Huﬀ Curves

Throughout the for-loop of Algorithm 1, the line function is always evaluated at the same point Q ∈ E(Fqk ) \ E(Fq ). It is therefore customary to represent

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this point in aﬃne coordinates. In our case, it is most convenient to choose the coordinates of Q as Q = (y, z) = (1 : y : z). Indeed, since the embedding degree k is even, the ﬁeld Fqk can be represented as Fqk/2 (α), where α is any quadratic non-residue in Fqk/2 . As a result, Q can be chosen of the form Q = (yQ , zQ α) with yQ , zQ ∈ Fqk/2 [4]. To do so, it suﬃces to pick a point on a quadratic twist of E over Fqk/2 and take its image under the isomorphism over Fqk . Now, for any two points R, P in E(Fq ), let R,P denote the rational function vanishing on the line through R and P . In general, we have R,P (Q) =

(zXP − ZP ) − λ(yXP − YP ) YP

where λ is the “(y, z)-slope” of the line through R and P . Then, the divisor of R,P is div(R,P ) = R + P + T − (1 : 0 : 0) − (0 : 1 : 0) − (a : b : 0) where T is the third point of intersection (counting multiplicities) of the line through R and P with the elliptic curve. In particular, if the neutral element of the group law ⊕ is denoted by U , the line function gR,P can be written as gR,P =

R,P R⊕P ,U

.

We concentrate on the case when U = O = (0 : 0 : 1). Then for any Q = (yQ , zQ α), we have R⊕P ,O (Q) = yQ −

YR⊕P ∈ Fqk/2 . XR⊕P

Since this quantity lies in a proper subﬁeld of Fqk , it goes to 1 after the ﬁnal exponentiation in Miller’s algorithm, which means that it can be discarded altogether. Similarly, divisions by XP can be omitted, and denominators in the expression of λ can be canceled. In other words, if λ = A/B, we can compute the line function as gR,P (Q) = (zXP − ZP ) · B − (yXP − YP ) · A and get the required result. We can now detail precise formulæ for the addition and doubling steps in the so-called Miller loop (i.e., the main for-loop in Algorithm 1). We let M and S represent the cost of a multiplication and of a squaring in Fqk while m and s are operations in Fq as before. Addition step. In the case of addition, the (y, z)-slope of the line through R = (XR : YR : ZR ) and P = (XP : YP : ZP ) is λ=

ZR XP − ZP XR . YR XP − YP XR

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Therefore, the line function to be evaluated is of the form gR,P (Q) = (zQ α·XP −ZP )(YR XP −YP XR )−(yQ ·XP −YP )(ZR XP −ZP XR ) . Since P and Q are constant throughout the loop, the values depending only on P and Q — in this case yQ = yQ · XP − YP and z Q = zQ α · XP , can be precomputed. Then, each Miller addition step requires computing R ⊕ P (one addition on the curve over Fq ), evaluating gR,P (Q), and computing f · gR,P (Q) (one multiplication in the ﬁeld Fqk ). We consider two types of Miller addition steps: full addition, for which no assumption is made on the representation of P , and mixed addition, for which we further assume that P is given in aﬃne coordinates (i.e., XP = 1). Both steps start with computing R ⊕ P , including all intermediate results. Full addition. Computing R ⊕ P requires 13m using the dedicated addition formula from §3.1, including all intermediate results m1 , . . . , m8 . Compute further m9 = (XR + YR )(XP − YP ). We then have gR,P (Q) = (z Q − ZP )(m9 + m5 − m6 ) − yQ (m1 − m2 ) where the ﬁrst term requires ( k2 + 1)m and the second term k2 m. With the ﬁnal multiplication over Fqk , the total cost of full addition is thus of 1M + (k + 15)m. Mixed addition. Now that XP = 1, computing R ⊕ P using the formula from §2.2, including all the intermediate results m1 , . . . , m9 , only requires 11m, since the computation of m1 is free. We then have gR,P (Q) = (z Q − ZP )(YR − YP XR ) − y Q (2ZR − m4 ) where both terms require the same number of multiplications as before, plus one for YP XR . The total cost of mixed addition is thus of 1M + (k + 13)m. Doubling step. In the case of doubling, the (y, z)-slope of the tangent line at R = (XR : YR : ZR ) is λ=

a(ZR )2 − 2bYR ZR − a(XR )2 A . = b(YR )2 − 2aYR ZR − b(XR )2 B

Thus, the line function is of the form gR,R (Q) = zQ α · XR B − ZR B − yQ · XR A + YR A . Miller’s doubling involves computing the point [2]R, which we do using the formulæ from §2.2 in 7m + 5s. Then the quantities A and B are obtained by computing the additional product m10 = 2YR ZR = (YR + ZR )2 − m2 − m3 using a single squaring. Computing gR,R (Q) requires multiplying those two values by XR and YR (resp. XR and ZR ), hence an additional 4m. And ﬁnally, multiplications by yQ and zQ α both require k2 m. Taking into account the multiplication and the squaring in Fqk needed to complete the doubling step, the total cost of Miller doubling is thus of 1M + 1S + (k + 11)m + 6s.

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249

Conclusion

This paper introduced and studied Huﬀ’s model, a new representation of elliptic curves to be considered alongside previous models such as Montgomery, Doche-Icart-Kohel and Edwards. This new model provides eﬃcient arithmetic, competitive with some of the fastest known implementations (although not quite as fast as “inverted Edwards” for now). Moreover, it has a number of additional desirable properties, including uniﬁed/complete addition laws and formulæ that do not depend on curve parameters (both properties are useful in cryptographic applications to thwart certain implementation attacks). It is also suitable to other computations on elliptic curves, such as the evaluation of pairings. We believe that this model is worthy of consideration by the community, and hope our contribution might spark further research into eﬃcient implementations of elliptic curve arithmetic. Acknowledgments. We are grateful to an anonymous referee for useful comments. This work was partly supported by the French ANR-07-TCOM-013-04 PACE Project and by the European Commission through the IST Program under Contract ICT-2007-216646 ECRYPT II.

References 1. Ar`ene, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster computation of the Tate pairing. In: Cryptology ePrint Archive, Report 2009/155 (2009), http://eprint.iacr.org/ 2. Atkin, A.O.L., Morain, F.: Elliptic curves and primality proving. Math. Comp. 61(203), 29–68 (1993) 3. Barreto, P.S.L.M., Lynn, B., Scott, M.: Eﬃcient implementation of pairing-based cryptosystems. J. Cryptology 17(4), 321–334 (2004) 4. Barreto, P.S.L.M., Lynn, B., Scott, M.: On the selection of pairing-friendly groups. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 17–25. Springer, Heidelberg (2004) 5. Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389– 405. Springer, Heidelberg (2008) 6. Bernstein, D.J., Lange, T.: Explicit-formulas database, http://www.hyperelliptic.org/EFD/ 7. Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007) 8. Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: Bozta¸s, S., Lu, H.F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007) 9. Bernstein, D.J., Lange, T., Farashahi, R.R.: Binary Edwards curves. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 244–265. Springer, Heidelberg (2008) 10. Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography, ch. V. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005)

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11. Das, M.P.L., Sarkar, P.: Pairing computation on twisted Edwards form elliptic curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 192–210. Springer, Heidelberg (2008) 12. Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc., New Ser. 44(3), 393–422 (2007) 13. Goldwasser, S., Kilian, J.: Primality testing using elliptic curves. J. ACM 46(4), 450–472 (1999) 14. Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards curves revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326–343. Springer, Heidelberg (2008) 15. Huﬀ, G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15, 443–453 (1948) 16. Ionica, S., Joux, A.: Another approach to pairing computation in Edwards coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008) 17. Koblitz, A.H., Koblitz, N., Menezes, A.: Elliptic curve cryptography: The serpentine course of a paradigm shift. J. Number Theory (to appear) 18. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987) 19. Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. 126(2), 649– 673 (1987) 20. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986) 21. Miller, V.S.: The Weil paring, and its eﬃcient implementation. J. Cryptology 17(1), 235–261 (2004) 22. Montgomery, P.L.: Speeding up the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48(177), 243–264 (1987) 23. Morain, F.: Primality proving using elliptic curves: An update. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 111–127. Springer, Heidelberg (1998) 24. Peeples Jr., W.D.: Elliptic curves and rational distance sets. Proc. Am. Math. Soc. 5, 29–33 (1954) 25. Silverman, J.H.: The Arithmetic of Elliptic Curves, ch III. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986)

Eﬃcient Pairing Computation with Theta Functions David Lubicz1,2 and Damien Robert3 1

2

DGA-MI, BP 7419, F-35174 Bruz IRMAR, Universt´e de Rennes 1, Campus de Beaulieu, F-35042 Rennes 3 LORIA, CARAMEL Project, Campus Scientiﬁque, BP 239, 54506 Vandoeuvre-l`es-Nancy Cedex

Abstract. In this paper, we present a new approach based on theta functions to compute Weil and Tate pairings. A beneﬁt of our method, which does not rely on the classical Miller’s algorithm, is its generality since it extends to all abelian varieties the classical Weil and Tate pairing formulas. In the case of dimension 1 and 2 abelian varieties our algorithms lead to implementations which are eﬃcient and naturally deterministic. We also introduce symmetric Weil and Tate pairings on Kummer varieties and explain how to compute them eﬃciently. We exhibit a nice algorithmic compatibility between some algebraic groups quotiented by the action of the automorphism −1, where the Z-action can be computed eﬃciently with a Montgomery ladder type algorithm.

1

Introduction

In recent years, many new and interesting cryptographic protocols have been proposed which use the existence of pairings on abelian varieties. In order to obtain eﬃcient and secure implementations of these protocols it is important to be able to compute quickly these pairings. Miller has proposed a method (see for instance [2]) to compute the function on an algebraic curve given up to a constant factor by the data of a principal divisor. This method is a key ingredient of all known algorithms to compute pairings. In this paper, we propose a diﬀerent approach based on theta functions. We ﬁrst make explicit the link between Weil and Tate pairings and the intersection pairing on the degree 1 homology of an abelian variety. Our method appears to be a very natural and straightforward way to compute the pairing associated to the Riemann form (or its arithmetic counterpart the commutator pairing) of an abelian variety. It is then easy to deduce practical formulas to compute Weil and Tate pairings. A ﬁrst beneﬁt of our approach is its generality: where Miller’s algorithm rely on the representation of an abelian variety as the Jacobian of an algebraic curve, our method works with any abelian varieties. The case of the Tate pairing is noticeable: while the original deﬁnition of Tate [8] deals with any abelian varieties, the formula of Lichtenbaum [9] used in cryptographic applications is restricted to Jacobian of curves. This restriction does not appear in our formulas. Our algorithm also G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 251–269, 2010. c Springer-Verlag Berlin Heidelberg 2010

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expand the algorithmic toolbox based on theta functions to compute with abelian varieties. For the complexity analysis of our algorithm we focus on the case of level 2 and 4 theta functions in order to obtain the best running time and memory consumption. The only diﬀerence between the two cases lies in the initialisation phase of the algorithm: in level 4 one can recover enough information from the data of two points to compute the pairings. This is not possible with the level 2 embedding since it does not distinguish a point and its opposite. Nonetheless it is possible to deﬁne a “symmetric pairing” on the quotient of an abelian variety by the action of the automorphism −1. These notions extend the deﬁnition of the trace pairing proposed in [3]. We have chosen to present all the formulas of this paper using the classical analytic theory of theta functions. In order to consider also rationality problems which are essential to the deﬁnition of the Tate pairing, we make the assumption that all the abelian varieties that we consider are deﬁned over a number ﬁeld K and we suppose given a ﬁxed embedding of K in its algebraic closure C. Nonetheless, it should be understood that all our algorithms apply to the case of abelian varieties deﬁned over any ﬁeld of characteristic not equal to 2. To see this one can invoke the Lefschetz’s principle or use Mumford’s theory of algebraic theta functions. We refer to [10] for proofs of the main formulas of this paper in the theory of Mumford. Our paper in organized as follows: in Section 2 we recall some basic deﬁnitions about theta functions. The Section 3 we give a method to compute the usual pairings by using a double and add algorithm based a theta addition formula. In Section 5 we make a precise assessment about the complexity of our algorithm. We also introduce symmetric pairings on Kummer varieties and explain how to adapt our algorithms to compute them eﬃciently. We end the paper with an example of computation in Section 6.

2

Some Notations and Basic Facts

In this section, in order to ﬁx the notations, we recall some well known facts on analytic theta functions (see for instance [14,6]). Let Hg be the g dimensional Siegel upper-half space which is the set of g × g symmetric matrices Ω whose imaginary part is positive deﬁnite. For Ω ∈ Hg , we denote by ΛΩ = ΩZg + Zg the lattice of Cg deﬁned by Ω. If A is an abelian variety of dimension g over the number ﬁeld K with a principal polarisation then A is analytically isomorphic to Cg /ΛΩ for a certain Ω ∈ Hg . In the rest of this paper, we denote by π : Cg → Cg /ΛΩ = A the canonical projection. The classical theory of theta functions gives a lot of functions on Cg that are pseudo-periodic with respect to ΛΩ and can be used as a projective coordinate system for A. More precisely, for a, b ∈ Qg , the theta function with rational characteristics (a, b) is an analytic function on Cg × Hg given by: θ [ ab ] (z, Ω) = exp πit (n + a).Ω.(n + a) + 2πit (n + a).(z + b) . (1) n∈Zg

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In order to write the pseudo-periodicity relations veriﬁed by the theta functions it is convenient to introduce a certain pairing on Cg . First we identify Cg to R2g via the isomorphism R2g → Cg , (x1 , x2 ) → Ωx1 + x2 . Then for α, β ∈ R2g with α = (α1 , α2 ) and β = (β1 , β2 ), we put eΩ (α, β) = exp(2πi(α1 β2 − α2 β1 )). The pseudo-periodicity of θ [ ab ] is given by t

θ [ ab ] (z +Ω.m+n, Ω) = eΩ (Ω.a+b, Ω.m+n)e−πi

m.Ω.m−2πit m.z

θ [ ab ] (z, Ω). (2)

We say that a function f on Cg is ΛΩ -quasi-periodic of level ∈ N if for all z ∈ Cg and m ∈ Zg , we have:f (z + m) = f (z), f (z + Ω.m) = exp(−πit m.Ω.m − 2πit z.m)f (z). For any ∈ N∗ , the set HΩ, of ΛΩ -quasi-periodic functions of level is a ﬁnite dimensional C-vector space basis can be given by the 0 whose theta functions with characteristics: (θ b/ (z, −1 .Ω))b∈[0,...,−1]g . If = k 2 , g then an alternative basis of HΩ, is (θ a/k b/k (kz, Ω))a,b∈[0,...,k−1] . A theorem of Lefschetz tells that if ≥ 3, the functions in HΩ, give a projective embedding g of A in P −1 , the projective space over C of dimension g − 1. For = 2, the functions in HΩ,2 do not give a projective embedding of A. It is easy to check that for all f ∈ HΩ,2 , we have f (−z) = f (z). Under some well known general conditions [7, cor 4.5.2], the image of the embedding deﬁned by HΩ,2 in 2 P −1 is the Kummer variety associated to A, which is the quotient of A by the automorphism −1. Once we have chosen a level ∈ N, for the rest of this paper, we adopt g g the following conventions: we let Z() 0 = (Z/Z) and for 2a point zP ∈ C and i ∈ Z() we put θi (zP ) = θ i/ (zP , Ω/). If = k , for i, j ∈ Z(k), g we let θi,j (zP ) = θ i/k (k.zP , Ω). We denote by P the element of A (C) j/k

with coordinates Pi = θi (zP ) and let P be the associated point of A that g we consider depending on the situation as embedded in P −1 or as a point on the analytic variety Cg /ΛΩ . In this paper, for n, ∈ N, such that n divides we will implicitly consider Z(n) as a subgroup of Z() via the morphism x → (/n).x. We denote by Ξ the theta divisor of level on A which is the divisor of zero of θ [ 00 ] (z, −1 .Ω). There is an isogeny ϕ : A → Aˆ = Pic0A , deﬁned by x → τx∗ Ξ − Ξ where τx is the translation by x morphism on A. The kernel of ϕ is A[]. For = 1 we let Ξ1 = Ξ . We denote by K(A) the function ﬁeld of A and if f ∈ K(A), we denote (f ) the divisor of the function f . Let Z 0 (A) be the group of 0-cycles of A group over the set of closed that is the free commutative points of A. If D = ni Pi is an element of Z 0 (A) and f ∈ K(A) then we put f (D) = i f (Pi )ni .

3

Weil and Tate Pairings and Theta Functions

In this section, we present formulas to compute Weil and Tate pairings from the knowledge of the theta coordinates of some points.

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The Weil Pairing

For Ω ∈ Hg , let A = Cg /ΛΩ be the associated complex abelian variety and denote by π : Cg → A the natural projection. Let be a positive integer, we denote by μ the subgroup of C∗ of th roots of unity. For zP , zQ ∈ Cg , let P, Q be the associated points of A, we consider the pairing: eW : A[] × A[] → μ , (P, Q) → eΩ (zP , zQ ) . It is clear that eW does not depend on the choice of zP and zQ representing P and Q respectively and that eW is a non-degenerate skew linear form. The following proposition gives an expression of this pairing in term of the values of certain theta functions. Lemma 1. Let Ω ∈ Hg . Let a, b ∈ Qg , let be a positive integer and let zP , zQ ∈ Cg be such that .zP = .zQ = 0 mod ΛΩ . Set zP = Ω.zP 1 + zP 2 and zQ = Ω.zQ1 + zQ2 with for i = 1, 2, zP i , zQi ∈ Rg . Let P = π(zP ) and Q = π(zQ ). For all z ∈ Cg , we have: a+zQ1 (z, Ω) θ b+zQ2 θ [ ab ] (z + .zP , Ω) . (3) eW (P, Q) = a+z Q1 θ [ ab ] (z, Ω) (z + .zP , Ω) θ b+zQ2

Proof. By (2), we have: a+zQ1 θ b+zQ2 (z + .zP , Ω) = eΩ (Ω.(a + zQ1 ) + (b + zQ2 ), Ω.zP 1 + zP 2 ) a+zQ1 exp[(πi2 (t zP 1 .Ω.zP 1 ) − 2πit zP 1 .z]θ b+zQ2 (z, Ω), θ [ ab ] (z + .zP , Ω) = eΩ (Ω.a + b, Ω.zP 1 + zP 2 ) exp[−πi2 (t zP 1 .Ω.zP 1 ) − 2πit zP 1 .z]θ [ ab ] (z, Ω). The lemma follows immediately. Let eW : A[]×A[] → μ be the usual Weil pairing. We recall a possible deﬁnition ∗ for eW [13, p. 184]. Let P, Q ∈ A[]. Let D = τQ Ξ − Ξ , then D represents a 0 ˆ point of A[] = PicA []. As a consequence, there exists a function fQ ∈ K(A) such that (fQ ) = .D. In the same way, there exists a function gQ ∈ K(A) such that (gQ ) = []∗ (D). As []∗ (fQ ) = .[]∗ D = (gQ ) there exists a constant c ∈ C∗ such that []∗ fQ = c.gQ . Thus for X a general point of A, of μ which is equal to eW (P, Q).

gQ (X) gQ (X+P )

is an element

Proposition 1. Keeping the notations from above, let zP = Ω.zP 1 + zP 2 and zQ = Ω.zQ1 + zQ2 be elements of Cg such that P = π(zP ) and Q = π(zQ ). For z ∈ Cg , we have the following equalities, up to a multiplication by a constant: z

0 θ zQ1 (.z, Ω) θ [ 0 ](z + zQ ) Q2 −1 , fQ (z) = μQ (z) gQ (z) = , (4) θ [ 00 ] (.z, Ω) θ [ 00 ](z) where μQ (z) : Cg → C is given by μQ (z) =

θ [ 0 ](z+zQ ) 0 . θ[ 0 0 ](z)

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Remark 1. In the preceding equations, the domain of the functions gQ and fQ is Cg but we will see in the course of the proof that gQ and fQ are periodic with respect to ΛΩ and are in fact well deﬁned functions on A. Proof. As π ∗ Ξ is the divisor of zero of θ [ 00 ] (z, Ω), π ∗ D is the divisor of zero of t g (z) = θ [ 00 ] (z + zQ , Ω)/θ [ 00 ] (z, Ω). But g(z) = exp[πitzQ1 Ωz Q1 + 2πi zQ1 (z + zQ1 zQ2 )]g (z) has the same zero divisor as g (z) and g(z) = θ zQ2 (z, Ω)/θ [ 00 ] (z, Ω). : Cg → Cg , z → z. It is clear from its deﬁnition that up to a multiplication Let [l] which gives the left hand of (4). It is easily seen using by a constant gQ = g ◦ [l] (2) that gQ (z) is periodic with respect to ΛΩ and as a consequence descends to a function on A. We turn to the proof of the second equality. As μQ (z) is a non vanishing function, the zero divisor of the function μQ (z)−1 (θ [ 00 ](z + zQ )/θ [ 00 ](z)) is ∗ π (D). Moreover, it is easily seen using (2) that this function is periodic with respect to ΛΩ , and descends to a function on A which up to a multiplication by a constant is fQ (z). Corollary 1. The pairing eW is the Weil pairing. Proof. This is an immediate consequence of Lemma 1 with a = b = 0, ProposigQ (X) tion 1 and the deﬁnition of the Weil pairing as eW (P, Q) = gQ (X+P ). Corollary 2. Let Ω ∈ Hg . Let a, b ∈ Qg , let be a positive integer and let zP , zQ ∈ Cg be such that .zP = .zQ = 0 mod ΛΩ . Let P, Q ∈ A be such that P = π(zP ) and Q = π(zQ ) and let: θ [ ab ] (.zP + zQ , Ω) θ [ ab ] (0, Ω) , θ [ ab ] (zQ , Ω) θ [ ab ] (.zP , Ω) θ [ ab ] (.zQ + zP , Ω) θ [ ab ] (0, Ω) R(zP , zQ ) = . θ [ ab ] (zP , Ω) θ [ ab ] (.zQ , Ω) L(zP , zQ ) =

(5)

If L(zP , zQ ) and R(zP , zQ ) are well deﬁned and non null, we have: eΩ (zP , zQ ) = eW (P, Q) = L(zP , zQ )−1 .R(zP , zQ ).

(6)

Proof. Since Q + P = Q and P = 0, L(zP , zQ ) does not depend on [ ab ] so we can assume that a = b = 0. The corollary can then be proved by a direct computation. But it also follows immediately from Proposition 1 and the formula eW (P, Q) = fP (Q − 0)/fQ (P − 0). In fact, using the notations of Proposition 1, we have fP (Q − 0) μP (zQ )μQ (0) = . fQ (P − 0) μP (0)μQ (zP ) The result follows an immediate computation. Remark 2. One can recognize in (6) a classical formula to compute the ﬁrst Chern class of a line bundle from the knowledge of its factors of automorphy, see for instance [1, Th. 2.1.2].

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The Tate Pairing

Let K be a number ﬁeld and we suppose that A is deﬁned over K. In this section, we suppose that μ ⊂ K and that A[] is rational over K. Let K be the algebraic closure of K and let G = Gal(K/K). Let δ1 : K ∗ /K ∗ → Hom(G, μ ) (resp. δ2 : A(K)/[]A(K) → Hom(G, A[])) be the connecting morphism of the Galois cohomology long exact sequence associated to the Kummer exact sequence (resp. to the exact sequence 0 → A[] → A(K) → A(K) → 0). There exists a bilinear application often referred to as the Tate pairing eT : A(K)/[]A(K) × A[] → K ∗ /K ∗ such that for (P, Q) ∈ A(K)/[]A(K) × A[], eW (δ2 (P ), Q) = δ1 (eT (P, Q)). In the statement of the next proposition, we suppose that the principal polarization L of A deﬁned by the matrix period is deﬁned over K. Thus for any X ∈ A(K) there exits zX ∈ Cg such that π(zX ) = X and θ(zX )/θ(0) ∈ K. In general this rationality condition on L is not veriﬁed but we will see later on in Remark 4 how to adapt the formulas of the next proposition to cover the general case. Proposition 2. Let K be a number ﬁeld and let A be a dimension g abelian variety over K. Let Ω ∈ Hg be such that A is analytically isomorphic to Cg /ΛΩ . Let a, b ∈ Qg , and let be a positive integer. Let P ∈ A(K)/[]A(K) and Q ∈ A[](K) and let zP , zQ ∈ Cg be such that π(zP ) = P and π(zQ ) = Q where π : Cg → A is the natural projection (by abuse of notation we use P, Q to denote the corresponding points of an algebraic and analytic model of A). Suppose that we have chosen zP , zQ and zP +Q such that θ [ 00 ](zP + zQ ) θ [ 00 ](0) ∈ K ∗, θ [ 00 ](zP ) θ [ 00 ](zQ ) then we have eT (P, Q) =

(7)

θ [ 00 ](.zQ + zP ) θ [ 00 ](0) . θ [ 00 ](zP ) θ [ 00 ](.zQ )

(8)

Proof. By Proposition 1, we have θ [ 00 ](.zQ ) θ [ 00 ](zP ) fQ (P − 0) = 0 θ [ 0 ](.zQ + zP ) θ [ 00 ](0)

θ [ 00 ](zP + zQ ) θ [ 00 ](0) θ [ 00 ](zP ) θ [ 00 ](zQ )

.

(9)

Taking care of the fact that eT (P, Q) has value in K ∗ /K ∗ we just have to prove that eT (P, Q) = fQ (0 − P ). The proof follows exactly the same computations as [16, p. 280].

4

Pairing Computations

In this section, we describe a general method to compute Weil or Tate pairings which does not rely on the usual Miller’s loop and prove its correctness. We postpone to the next section the analysis of the running time of these algorithms.

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257

Let n, ∈ N. We suppose that 2 divides n and that and n are relatively prime. Let A be an abelian variety over C with period matrix Ω. We represent g A as a closed subvariety of Pn −1 by the way of level n theta functions and we the pullback of suppose that this embedding is deﬁned over K. Denote by A ng ng −1 A via the natural projection κ : A → P . In the following, we adopt the following convention: if P is a point of A, we denote by P an aﬃne lift of P that g is a point P of An such that κ(P ) = P . An important ingredient of our algorithm is the Riemann addition formulas. The usual form of these formulas works for theta functions of level divisible by 4 (see for instance [6, p. 139]). In this paper we need a slight generalisation of these formulas for working also with level 2 theta functions. We recall that following the convention for the notation of theta functions described 0 at the end of the introduction, we let for all i ∈ Z(n), z ∈ Cg , θi (z) = θ i/n (z, Ω/n). Moreover, we recall that in the following we consider Z(n) (resp. Z(2)) as a subgroup of Z(2n) via the map x → 2x (resp. x → nx). Theorem 1. Let i, j, k, l ∈ Z(2n). We suppose that i + j, i + k and i + l ∈ Z(n). ˆ ˆ Let Z(2) be the dual group of Z(2). For all χ ∈ Z(2) and z1 , z2 ∈ Cg we have ⎛ ⎞⎛ ⎞ ⎝ χ(η)θi+j+η (z1 + z2 )θi−j+η (z1 − z2 )⎠ ⎝ χ(η)θk+l+η (0)θk−l+η (0)⎠ η∈Z(2)

⎛

=⎝

⎞⎛ χ(η)θi+k+η (z1 )θi−k+η (z1 )⎠ ⎝

η∈Z(2)

η∈Z(2)

⎞

χ(η)θj+l+η (z2 )θj−l+η (z2 )⎠

η∈Z(2)

0 (z, Ω/(2n)). Let Proof. For i ∈ Z(2n) and z ∈ Cg , we let θi (z) = θ i/(2n) g i, j ∈ Z(2n) be such that i + j ∈ Z(n) and let z1 , z2 ∈ C . The usual duplication formula [6, p. 139] gives θi+j (z1 +z2 )θi−j (z1 −z2 ) = 21g η∈Z(2) θi+η (z1 )θj+η (z2 ). ˆ For χ ∈ Z(2), using this formula, we compute

1 χ(η1 + η2 )θi+η (z1 )θj+η (z2 ) 1 2 2g η1 ,η2 ∈Z(2) ⎞⎛ ⎞ ⎛ 1 ⎝ = g χ(η)θi+η (z1 )⎠ ⎝ χ(η)θj+η (z2 )⎠ . (10) 2

χ(η)θi+j+η (z1 + z2 )θi−j+η (z1 − z2 ) =

η∈Z(2)

η∈Z(2)

η∈Z(2)

Using this last equation to compute the left and right hand sides of the preceding equation we obtain the result. We suppose that the theta null point 0 = (θi (0))i∈Z(n) is known. We deduce im= mediately from Theorem 1 an algorithm that takes as inputs P = (Pi )i∈Z(n) , Q − Q = ((P − Q)i )i∈Z(n) and outputs P + Q = ((P + Q)i )i∈Z(n) . (Qi )i∈Z(n) and P P We write P + Q = PseudoAdd(P , Q, − Q). Indeed we will see later (Proposition 3) that if n = 4, we can recover the projective point P + Q from P and Q

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using the Riemann addition formulas. It is then easy to see that if we moreover and P know P, Q − Q, then there is a unique aﬃne point P + Q above P + Q that satisfy the addition formulas from Theorem 1. If n = 2, the point P +Q is also unique provided the abelian variety satisﬁes the generic condition from Theorem 3. Chaining the algorithm PseudoAdd in a classical Montgomery ladder [2, alg. = (Q i )i∈Z(n) , P 9.5 p. 148] yields an algorithm that takes as inputs Q +Q = 0 = ( 0i )i∈Z(n) and an integer and outputs ((P + Q)i )i∈Z(n) , P = (Pi )i∈Z(n) , P , P + Q. We write P + Q = ScalarMult(P + Q, Q, 0, ). In particular, we have P = ScalarMult(P , P , 0, 0, ). The following lemma tells that the output of ScalarMult does not depend on the particular chain of PseudoAdd calls it uses. Lemma 2. Let L = {0, 1, . . . , } be a Lucas sequence. Let A0 = P , B0 = 0, A1 = For m ∈ L, m 2, write m = j + k with j, k, j − k ∈ L. Let P + Q and B1 = Q. Bm = PseudoAdd(Bj , Bk , Bj−k ) and Am = PseudoAdd(Aj , Bk , Aj−k ). Then + Q. In other words P + Q does not depend on the Lucas sequence A = P used to compute it. = (θi (zQ ))i∈Z(n) Proof. If there exist zP , zQ ∈ Cg such that P = (θi (zP ))i∈Z(n) , Q and P + Q = (θi (zP + zQ ))i∈Z(n) then by Theorem 1 and a recursion we see that Aj = (θi (zP + jzQ ))i∈Z(n) and Bj = (θi (jzQ ))i∈Z(n) . Hence A = (θi (zP + + Q. zQ )) = P Otherwise there exist λP , λQ and λP +Q in C∗ such that P = λP (θi (zP ))i∈Z(n) , = λQ (θi (zQ ))i∈Z(n) and P + Q = λP +Q (θi (zP + zQ ))i∈Z(n) . Since we have Q λ2 λ2Q P), an PseudoAdd(λP +Q P + Q, λQ Q, λP P) = P +Q PseudoAdd(P + Q, Q, λP 2

j(j−1)

easy recursion shows that Bj = λjQ (θi (jzQ ))i∈Z(n) and Aj = λjP +Q λQ (−1)

(θi (zP + jzQ ))i∈Z(n) . Hence A = λP +Q λQ P + Q. ∗

/λj−1 P ·

/λ−1 · (θj (zP + zQ ))j∈Z(n) = P g

Remark 3. There is a natural action of K on An − {0} by multiplication of g ∗ the coordinates of a point that we denote by α ∗ P for α ∈ K and P ∈ An (K). In the proof of the preceding lemma we have seen the eﬀect of this action on P the output of the algorithm ScalarMult: let P, Q ∈ A(K) and let P, Q, +Q be aﬃne lifts of P , Q and P + Q. Let R = ScalarMult(P + Q, Q, P , 0, ). Let α, β, γ, δ ∈ K, we have γ ∗ P , δ ∗ (11) ScalarMult(α ∗ P + Q, β ∗ Q, 0, ) = (α β (−1) /γ −1 δ (−1) ) ∗ R, 2

α 0, 0, ). (12) ScalarMult(α ∗ P , α ∗ P, δ ∗ 0, δ ∗ 0, ) = 2 −1 ∗ ScalarMult(P, P , δ

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Given P and Q with projective coordinates (θi (zP ))i∈Z(n) and (θi (zQ ))i∈Z(n) for zP , zQ ∈ Cg , we would like to compute eW (P, Q) and eT (P, Q). We can state the main theorem of this section Theorem 2. We suppose that n and are relatively prime. For X, Y ∈ A(K), Y , X denote by X, + Y any aﬃne lifts of X, Y and X + Y . Recall that for i ∈ i the coordinate i of the point X. For ∈ N and i ∈ Z(n), Z(n), we denote by X Y , X let fT (X, + Y , 0, , i) = ScalarMult(X+Y ,X,Y ,0,)i 0i . Then for P, Q ∈ A[] X, 0,0,)i Yi ScalarMult(X, and i ∈ Z(n), we have: P P , P + Q, 0, , i)−1 fT (Q, + Q, 0, , i), eW (P, Q)n = fT (P , Q,

(13)

whenever the right hand side is well deﬁned. and Moreover, for P ∈ A(K)/[]A(K), Q ∈ A[], if we suppose that 0, P , Q P + Q are aﬃne lifts of 0, P , Q and P + Q with coordinates in K, then we have for i ∈ Z(n), P, P + Q, 0, , i), eT (P, Q)n = fT (Q,

(14)

whenever the right hand side is well deﬁned. Proof. Let zP , zQ ∈ Cg such that π(zP ) = P and π(zQ ) = Q (recall that π : = Cg → A = Cg /ΛΩ is the natural projection). Let P = (θi (zP ))i∈Z(n) , Q + Q = (θi (zP + zQ ))i∈Z(n) . Then applying Corollary 2, if (θi (zQ ))i∈Z(n) and P P, Q ∈ A[], we obtain that P P, P eΩ/n (zP , zQ ) = eW (P, Q)n = fT (P , Q, + Q, 0, , i)−1 fT (Q, + Q, 0, , i). In the same way, by Proposition 2 (which apply for i = 0, but it is easy to see that the same result is true for any i ∈ Z(n)), we have for P ∈ A(K)/[]A(K) P, P + Q, 0, , i). Next, let α, β, γ, δ ∈ K. By and Q ∈ A[], eT (P, Q)n = fT (Q, Remark 3, we have γ δ β ∗ Y , γ ∗ X Y , X fT (α ∗ X, + Y,δ ∗ 0, , i) = .fT (X, + Y , 0, , i). αβ

(15)

This shows that the expressions (13) and (14) for the Weil and Tate pairing do not depend on the choice of aﬃne liftings (rational over K in the case of the Tate pairing) of P , Q and P + Q. Remark 4. In this remark we keep the notations of the previous theorem. Let L be a polarization of A associated to Ξn for n ∈ N∗ which is rational over K. Let (θi )i∈Z(n) be a basis of global sections of a trivialisation of π ∗ (L ) (and we rigidify this basis by setting θ0 (0) = 1). In general, it is not true that the polarization deﬁned by the level n classical theta functions is rational over K. Nonetheless we know that there exits a non vanishing function ζ of Cg such that θi = ζθi for i ∈ Z(n) (up to a renumbering of the basis θi ).

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alg = zX ∈ {0, zP , zQ , zP +Q }, if we denote by X ∗ constant factors cX ∈ C such that for X ∈ alg = X. {0, P, Q, P + Q} we have cX ∗ X alg for X ∈ {0, P, Q, P + As we can suppose that the coordinates of the points X Q} are deﬁned over K, we can rewrite (9) as: −

θi (.zQ + zP ) θi (0) cP +Q c0 , eT (P, Q) = cP cQ θi (zP ) θi (.zQ )

Let 0, zP , zQ , zP +Q ∈ Cg . For (θi (zX ))i∈Z(n) , then there exist

alg

alg , P alg , P for i ∈ Z(n). But by (15) we have the equation: fT (Q + Q , 0alg , , i) − − cP +Q c0 cP +Q c0 θi (.zQ +zP ) θi (0) P , P = .fT (Q, + Q, 0, , i) = . ComcP cQ

cP cQ

θi (zP )

θi (.zQ )

paring these formulas, we obtain that we can compute the Tate pairing by taking aﬃne lifts of 0, P , Q and P + Q provided by the coordinates θi . Now using (15) again, we obtain that to compute the Tate pairing we only have to choose aﬃne lifts of 0, P , Q, and P + Q which are rational over K. As we have shown that the formulas of Theorem 2 do not depend on a choice of the aﬃne lifts of the input points of the algorithm (as long as the choices are the same for the computation of the two functions fT in the case of the Weil pairing), from now on we only consider projective points. In order to have a working algorithm to compute Weil and Tate pairings, it remains to explain how to compute P +Q from the knowledge of P and Q. As the formulas to compute the pairings only involve one of the level n theta functions, and since the number of the coordinates used in the computation of ScalarMult is ng , for the sake of eﬃciency it is important to have a small n. As 2 divides n, from now on, we focus on the two interesting cases: n = 2 and n = 4. We ﬁrst treat the case n = 4. Let zP , zQ ∈ Cg and let P = (Pi )i∈Z(n) = (θi (zP ))i∈Z(n) and Q = (Qi )i∈Z(n) = (θi (zQ ))i∈Z(n) . From the knowledge of P and Q, with the addition formula (Theorem 1), one can compute the products: χ(η)θi+j+η (zP + zQ )θi−j+η (zP − zQ ) χ(η)θk+l+η (0)θk−l+η (0) , (16) η∈Z(2)

η∈Z(2)

ˆ for χ ∈ Z(2) and i, j, k, l ∈ Z(2n) such that i + j, i + k, and i + l ∈ Z(n). If we ˆ can prove that for any such choice of i, j, k, l ∈ Z(2n) and χ ∈ Z(2) there exist k ∈ k +Z(n) and l ∈ l+Z(n) such that η∈Z(2) χ(η)θk +l +η (0)θk −l +η (0) = 0, then by summing over the characters the left bracket of (16) one can compute all the products θi (zP + zQ )θj (zP − zQ ), for i, j ∈ Z(n) from which it is easy to recover by taking quotients the projective point (θi (zP + zQ ))i∈Z(n) . Now, using equation (10), we have

χ(η)θk+l+η (0)θk−l+η (0) =

η∈Z(2)

where for k ∈ Z(8), θk (z) = θ

1 χ(η)θk+η (0) χ(η)θl+η (0) , (17) g 2 η∈Z(2)

0 k/8

η∈Z(2)

(z, Ω/8). We have the

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Proposition 3. Let δ ∈ N be such that 4 divides δ. For any a ∈ K(2δ) there exˆ ists an element b 0∈ a + K(δ) such that for all χ ∈ Z(2) we have that η∈Z(2) χ(η)θ (b+η)/(2δ) (0, 1/(2δ).Ω) = 0. Proof. This is just a rephrasing of [11, equation (*) p. 339]. Applying the preceding proposition to the factors of the right hand of equation (17), we obtain that there exists k ∈ k + Z(n) and l ∈ l + Z(n) such that χ(η)θ (0)θ (0) = 0 and we are done. k +l +η k −l +η η∈Z(2) 0 (z, 1/2.Ω). In the case n = 2, as usual, for all i ∈ Z(2), we put θi (z) = θ i/2 ˆ Then by Theorem 1, we have for any χ ∈ Z(2) and for well chosen pairs of quadruples (i, j, k, l), (i , j , k , l ) ∈ Z(2)4 an equation

χ(η)θi+η (zP + zQ )θj+η (zP − zQ ) χ(η)θk+η (0)θl+η (0)

η∈Z(2)

=

η∈Z(2)

χ(η)θi +η (zP )θj +η (zP ) χ(η)θk +η (zQ )θl +η (zQ ) .

η∈Z(2)

(18)

η∈Z(2)

If the kernel of χ does not contain the subgroup of Z(2) generated by k + l then we have η∈Z(2) χ(η)θk+η (0)θl+η (0) = 0, so it is not possible to recover θi+η (zP +zQ ) as before. This is consistent with the fact that for i ∈ Z(2) and z ∈ Cg , θi (z) = θi (−z), the right hand side of (18) is invariant for the transformation zQ → −zQ while it is not the case of the left hand side. The best we can hope is that for almost all period matrices Ω ∈ Hg there exists a k ∈ Z(2) such that ˆ such that k + l is in the kernel of χ, we have for all l ∈ Z(2) and χ ∈ Z(2) the content of Theorem 3. In η∈Z(2) χ(η)θk+η (0)θl+η (0) = 0. This is exactly order to prove this theorem, we let Tk,l,χ = η∈Z(2) χ(η)θk+η (0)θl+η (0) and we state the following lemma: Lemma 3. For Ω ∈ Hg , the two following properties are equivalent: ˆ such that 1. There exists a k ∈ Z(2) such that for all ∈ Z(2) and χ ∈ Z(2) k + l is in the kernel of χ, we have Tk,l,χ = 0. 2. For all i, j ∈ Z(2) such that t i.j = 0, θi,j (0) = 0. t ˆ Proof. For χ ∈ Z(2), let μ ∈ Z(2) be such that χ(η) = (−1) η.μ . Let ρ : Z(4) → Z(2), x → x mod Z(2) be the canonical projection. Then we have (see [14, prop 1.3 p. 124]), for all i ∈ Z(4) η∈Z(2) χ(η)θi+η (0) = 2g .θμ,ρ(i) (0), where 0 θk (z) = θ k/4 (z, 1/4.Ω). Combining this relation together with (17), for all i, j ∈ Z(4) such that i + j ∈ Z(2), let k = i + j, l = i − j, we obtain the equality

Tk,l,χ = Ti+j,i−j,χ = 2g .θμ,ρ(i) (0)θμ,ρ(j) (0) = 2g .θμ,k+l (0)2 . t

Since χ(k + l) = (−1)

(k+l).μ

the lemma follows immediately from (19).

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It is well known that for z ∈ Cg , and k, l ∈ Z(2), we have θk,l (−z) = (−1) k.l θk,l (z). As a consequence, for all k, l ∈ Z(2) such that t k.l = 1 (the odd characteristics), we have θk,l (0) = 0. Denote by M4 the quasi-projective variety over C deﬁned as the locus of zeros of θi,j (0) considered as functions of Ω. It is clear that M4 parametrizes the set of principally polarized abelian varieties together with a level 4 structure since from the knowledge of a point in M4 one can recover the projective embedding of the corresponding abelian variety provided by the Riemann equations. Theorem 3. For all k, l ∈ Z(2) such that t k.l = 0, the function θk,l (0) on M4 is non-trivial and as consequence, its zero locus is a proper subvariety of M4 of codimension 1. Proof. We sketch the proof of the theorem. Suppose on the contrary that for k, l ∈ Z(2) such that t k.l = 0, θk,l (0) is a constant function of Ω. This is a degree 1 relation level 4 theta constants, call it Rk,l . We have for all k ∈ Z(4), 0 for θk (0) = θ (2k)/8 (0, (2Ω)/8). Thus, the level 4 degree 1 relations Rk,l induce degree 1 relations for level 8 theta constants. The hypothesis t k.l = 0 means that these level 8 relations are not a linear combination of the symmetry relations θk (0) = θ−k (0) for all k ∈ Z(8). This is a contradiction with the description of M8 the modular space of level 8 marked abelian varieties given by Mumford in [12, main th. p. 83] as an open subset of the reduced projective variety given by the symmetry relations and the Riemann relations. Remark 5. The preceding theorem shows that the symmetric pairing computation algorithms that we describe in the next section works for a general abelian variety. However, one can ask if the closed proper subset of M4 , given by the cancellation of some even level 4 theta constants contains noticeable abelian varieties. Actually, this is the case since a theorem of Frobenius [15, cor. 6.7 p. 3.102] tells us that the locus of Jacobian of hyperelliptic curves inside M4 can be given by equations of the form θk,l (0) = 0 where (k, l) is an even characteristic. As a consequence, the algorithms of Section 5.2 to compute symmetric pairings don’t apply to Jacobian of hyperelliptic of genus g when g 3. It should be noted however that following [7, cor 4.5.2 and remark (2)], the condition that for all k, l ∈ Z(2) such that t k.l = 0, θk,l (0) = 0 is equivalent to the fact the level 2 theta functions give a projectively normal embedding. Considering this result, the condition of Theorem 3 should be considered as natural.

5

Complexity Analysis

In this section, we explain how to use the results of the preceding section to compute eﬃciently pairings on abelian and Kummer varieties with a special focus on dimension 1 and 2 since these cases are particularly interesting for cryptographic applications.

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5.1

263

Abelian Varieties

We begin with the case of abelian varieties since the main loop of the algorithm can also be used for the computation of symmetric pairings on Kummer varieties. Initialisation phase. The initialisation phase depends on the representation of the points P and Q on the abelian variety A. If P and Q are given by theta coordinates of level 4 we can apply the procedure described in Section 4 to compute the homogeneous coordinates of (θi (P + Q))i∈Z(4) . Suppose that another coordinate system is used to represent P and Q that we denote by (Xi )i∈I where Xi are rational functions on a Zariski open subset of A. Then by deﬁnition there exist formulas to compute θi (P ) and θi (Q) from the knowledge of Xi (P ) and Xi (Q). In practise, the dictionary between some useful coordinate system and the theta coordinates can easily be deduced from well known properties of theta functions. It should be remarked that in order to carry out these computations we might have to do a base ﬁeld extension since in the projective embedding of A provided by the level 4 theta functions the 4-torsion of A is rational over the base ﬁeld, whereas this may not the case with other models of A. The advantage of the level 4 is that no square root extraction is needed for the computation of P + Q,contrarily to the level 2 case as we will see. 0 From the knowledge of θ i/4 (zX , 1/4.Ω), i ∈ Z(4) for X = P, Q, P +Q we can 0 then compute the level 2 coordinates given by ( j∈Z(2) θ i+2j (zX , Ω4 ))i∈Z(2) 4

for the coordinates of the (isogeneous) points X = P, Q, P + Q. Pairing computation phase. As we have seen before, we can carry out the computations of the main loop of the algorithm with level 2 theta functions since at the end we only need one theta coordinate to compute the pairings. This is more eﬃcient because we only need 2g coordinates to represent a point and we can do the computation on the ﬁeld of deﬁnition of the 2-torsion of A. We suppose that we are given the level 2 coordinates of P , Q, P + Q. Rather than considering the formulas of Theorem 1 for the double and add algorithm, we use the level 2 formulas given in [4] for the genus 2 case, and in [5] for the genus 1 case. For instance, let E be an elliptic curve deﬁned by Ω ∈ H1 , let Ω = Ω/2 and put 0 (0, Ω ); A = ϑ [ 00 ] (0, 2Ω ); B = ϑ 1/2 (0, 2Ω ). a = ϑ [ 00 ] (0, Ω ); b = ϑ 1/2 0 The duplication formulas are given by the equalities: 1/2 2 0 0 (z, 2Ω )2 , aϑ 0 0[ 0 ] (z, Ω ) = ϑ [ 00 ] (z, 2Ω )2 + ϑ 1/2 bϑ 1/2 (z, Ω ) = ϑ [ 0 ] (z, 2Ω ) − ϑ 0 (z, 2Ω )2 . 0 2 2Aϑ [ 00] (2z, 2Ω ) = ϑ [ 00 ] (z, Ω )2 + ϑ 1/2 (z, Ω )2 , 1/2 0 2 0 2Bϑ 0 (2z, 2Ω ) = ϑ [ 0 ] (z, Ω ) − ϑ 1/2 (z, Ω ) . 0 Let x = θ [ 00 ] (z, Ω ) and z = θ 1/2 (z, Ω ) using the above formulas yield the following algorithms:

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Doubling Algorithm: Input: A point P = (x : z). Output: The double 2.P = (x : z ). 1. 2. 3. 4. 5.

x0 = (x2 + z 2 )2 ; A2 2 2 2 z0 = B 2 (x − z ) ; x = (x0 + z0 ); z = ab (x0 − z0 ); Return (x : z ).

Diﬀerential Addition Algorithm: Input: Two points P = (x : z) and Q = ( x : z) on E, and R = (x : z) = P −Q, with xz = 0. Output: The point P + Q = (x : z ). 1. 2. 3. 4. 5.

x0 = (x2 + z 2 )( x2 + z2 ); 2 A 2 2 z0 = B x2 − z2 ); 2 (x − z )( x = (x0 + z0 )/x; z = (x0 − z0 )/z; Return (x : z ).

Recall that in order to compute the pairing eT (P, Q), we have to compute P , = ScalarMult(Q, Q, P + Q = ScalarMult(P + Q, Q, 0, ) and Q 0, 0, ). It should be remarked that in the computation of P + Q, we need exactly the Since same values of j.Q for some j ∈ {1, . . . , } as those required to obtain Q. we want to avoid a division in each step, we use a Montgomery ladder so that the diﬀerences in the adding step are always the same points. To speed up the diﬀerential additions, we have renormalised the theta null point (a, b) to (1, b/a). It is easy to see by doing the same computation as in Remark 3 that this does not change the value of the Tate pairing eT (P, Q). Moreover we also have renormalised the theta null point (A, B). Looking back at the proof of 1, we see that this change each aﬃne addition by the constant factor B −2 . This also does not aﬀect the ﬁnal value of the Tate pairing eT (P, Q), since we use the same Lucas and P sequence for computing Q + Q. This give the following steps for the pairing: from (j − 1)Q, jQ and P + jQ we compute 2(j − 1)Q, (2j − 1)Q, P + (2j − 1)Q or (2j − 1)Q, 2jQ and P + 2jQ depending on the binary decomposition of . We remark that at each step we do a doubling and two adding, and that we add the same point to the triple A2 2 2 (j−1)Q, jQ, P +jQ. For instance in genus 1, we only have to compute B 2 (x −z ) once, where (x : z) are the coordinates of the doubled point. The ﬁgure below summarises the cost per bit of computation of the Tate pairing with our algorithm in genus 1 and 2 with the following notations: S is for squaring, M is for general multiplication, m is for multiplication by a constant. Tate pairing First pairing e(P, Q) Following pairings e(P , Q) Dimension 1 8S+4m+4M 2S+1m+2M Dimension 2 13S+12m+11M 4S+3m+4M The algorithms that we have presented in this section are deterministic and generalize immediately to the higher dimension case. Usually when computing a pairing, the ﬁeld of deﬁnition of Q has a smaller degree than the ﬁeld of deﬁnition of P , so that at each step one adding and one doubling is done with points in the smaller ﬁeld. We also remark that if we have to compute several pairings e(P1 , Q), e(P2 , Q), . . . with the same Q, it makes sense to store the

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results of the computations of the jQ so that for the next pairings we only have to compute the Pi + jQ. For instance when g = 1 if we store the log2 () A2 2 2 coordinates (x2 + z 2 , B 2 (x − z )) of each doubling step, we can compute the subsequent pairings with only ﬁve multiplications at each step. 5.2

Kummer Varieties

Let A be a principally polarized abelian variety of dimension g deﬁned by Ω ∈ Hg . As we have seen in the introduction, the level 2 theta functions deﬁned by Ω give a projective embedding of the Kummer variety associated to a A. We recall that the Kummer variety K A of A is the quotient of A by the action of the automorphism −1 of A. Let ζ : A → K A be the natural projection. In the following, if P ∈ A(K) we denote by P its image by ζ. The construction of K A does not preserve the group structure of A. Nonetheless, we remark that from the data of P ∈ K A (K) one can compute 2P without ambiguity, and from the data of P , Q and P − Q one can compute P + Q. As a consequence, K A inherits from A of an action of Z on its points which can be computed by a Montgomery ladder like algorithm. ∗ ∗ Let e be a pairing on A, and let K 0 be the quotient of K by the action of ∗ ∗ the automorphism −1. Let ζ0 : K → K 0 be the natural projection. The pairing ∗ e gives a well deﬁned application e : K A (K) × K A (K) → K 0 , (P , Q) → ∗ ζ0 (e(P, Q)). It is easily seen that the elements of K 0 are in bijection with the ∗ ∗ set S = {x + 1/x, x ∈ K }. Identifying K 0 with S, the application ζ0 is given by ∗ ζ0 (x) = x + 1/x, x ∈ K from which we deduce the expression of e : (P , Q) → e(P, Q) + e(−P, Q). This pairing has been introduced in [3]. In the following, if e is a pairing, we say that e is the symmetric pairing associated to e. The symmetric pairing e can be seen as a version of e for compressed coordinates as it takes as input points with 2g coordinates rather than 4g . Its cryptographic relevance comes from the compatibility of e with the Z-set ∗ structures of K A and K 0 : for all λ, μ ∈ Z, P , Q ∈ K A , we have e(λ.P , μ.Q) = (λμ).e(P , Q). In [3], the authors give an algorithm based on Lucas sequences to compute the action of Z on K 0 for certain ﬁnite ﬁelds. Here we would like to emphasize that the compatibility of the Z-structure of K A and K 0 is also algorithmic. It comes from the fact and on any quotient of an algebraic group by the automorphism −1 there exists a natural Montgomery ladder algorithm to compute the resulting Z-action. In the case of K 0 we obtain very simple and general formulas. For x ∈ K, and i, j ∈ Z, we have 1 2 1 1 1 1 1 ) = (x2i + 2i +2); (xi + i )(xj + j ) = (xi+j + i+j )+(xi−j + i−j ). xi x x x x x We have seen that the codomain of the Tate pairing eT is the multiplicative group K ∗ /K ∗ . Again, we can take the quotient of this group by the action of (−1) on it, denote it by (K ∗ /K ∗ )0 . It is clear that there is a bijection between the set (K ∗ /K ∗ )0 and the set ST = {x + 1/x, x ∈ KT } where KT is a set of representatives of K ∗ /K ∗ . Moreover, one can compute the Z-action on such representatives using the preceding algorithm. (xi +

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Initialisation phase. We suppose that we know the level 2 coordinates θi (zP ) and θi (zQ ), i ∈ Z(2) of P and Q. We may assume (by multiplying by a projective factor) that the values of the projective coordinates (θi (zP ))i∈Z(2) and (θi (zQ ))i∈Z(2) are in K. Using Theorem 1 and Theorem 3, we obtain that for a ˆ it is possible to compute for all i, j ∈ Z(2) and χ ∈ Z(2) general choice of K A , such that χ(i − j) = 1, η∈Z(2) χ(η)θi+η (zP + zQ )θj+η (zP + zQ ) from the inputs. By summing over the characters, we obtain for all i, j ∈ Z(2) κij = θi (zP + zQ )θj (zP − zQ ) + θj (zP + zQ )θi (zP − zQ ).

(20)

We suppose that θ0 (zP +zQ )θ0 (zP −zQ ) = 0, if necessary by replacing the index 0 by another one. By rescaling the projective coordinates, we do our computations as if θ0 (zP − zQ ) = 1 hence we know θ0 (zP + zQ ). θ (z +z ) i0 ii For i ∈ Z(2), let Pi (X) = X 2 −2 κκ00 X + κκ00 . The roots of Pi (X) are θ0i (zPP +zQ , Q) θi (zP −zQ ) θ0 (zP −zQ ) .

If P or Q is a point of 2-torsion, P + Q = P − Q ∈ K A so each Pi (X) has a double root. Otherwise, exist α ∈ Z(2), α = 0

we may suppose that there θ0 (zP + zQ ) θ0 (zP − zQ ) is invertible. such that the matrix M = θα (zP + zQ ) θα (zP − zQ ) We can compute {θα (zP + zQ ), θα (zP − zQ )} by ﬁnding the roots of Pα (X). As by hypothesis, P +Q, P −Q ∈ A(K), we deduce that these roots are in K. We ﬁx an arbitrary ordering (θα (zP + zQ ), θα (zP − zQ )) of these roots (depending on the ordering, we will compute P − Q or P + Q). We can then ﬁnd {θi (zP + zQ ), θi (zP − zQ )} by solving the system

θi (zP − zQ ) κi0 θ0 (zP + zQ ) θ0 (zP − zQ ) = . (21) θα (zP + zQ ) θα (zP − zQ ) θi (zP + zQ ) κiα This method requires one square root. Pairing computation phase. Let P ∈ A(K)/[]A(K) and Q ∈ A[] and denote by P , Q the corresponding points on K A . Denote by θi (z), i ∈ Z(2), the level 2 theta functions associated to Ω. We present two methods to compute the symmetric Tate pairing. A ﬁrst method is to consider the formula eT (P , Q) = eT (P, Q) + eT (P, −Q). We have explained in the last paragraph how to compute the set S = {P + Q, P − Q} at the expence of a square root extraction. By choosing a point in S, we can use the algorithm from Section 5.1 to compute e(P, Q) (resp e(P, −Q)). We can then compute eT (P, Q) = e(P, Q) + e(P, −Q) with a simple division. Another approach is to work in the algebra A = K[X]/(Pα (X)) for α ∈ Z(2) as before. We denote by g the unique automorphism of the algebra of A leaving K invariant and diﬀerent from the identity. For each i ∈ Z(2) by using equation (21) we can express θi (zP + zQ ) = γi X + δi . (We can always compute an inverse of γX + δ except when −δ/γ is a root of Pα . But in this case we have found a root of Pα and we can use the ﬁrst method.) Now, consider the vector (Tj )j∈Z(2) where T0 = 1, Tα = X and Tj = γj X + δj . We compute R = ScalarMult(T, Q, P, 0, )i . Then it is easily seen that

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R + g.R = ScalarMult(P + Q, Q, P, 0, )i + ScalarMult(P − Q, Q, P, 0, )i . By Proposition 2, and using the fact that θi (−zQ ) = θi (zQ ) we have for [θ (.z +z )+θi (−.zQ +zP )]θi (0) i ∈ Z(2) eT (P , Q) = i Q θPi (zP )θ . We can now compute i (.zQ ) eT (P, Q) =

0, )i ]θi (0) [ScalarMult(P +Q, Q, P, 0, )i +ScalarMult(P −Q, Q, P, , θi (zP )ScalarMult(Q, Q, 0, 0, )i

By an application of Lemma 3, the result of the preceding equation is a well deﬁned element of (K ∗ /K ∗ )0 . With this method, we have to compute 1 ScalarMult with value in A and 1 ScalarMult with value in K. It is interesing to note that it avoids the non determinism of the square root computation of the ﬁrst method. In some cryptographic applications, it is important to have a unique value as the result of the Tate pairing. In order to have this property, it is common to compose the Tate pairing with a th root extraction on K which can be done in the case that K is a ﬁnite ﬁeld by an exponentiation in K0∗ . This operation can be performed using the Montgomery ladder type algorithm presented above. The symmetric Weil pairing computation. Since we compute P + Q with the ﬁrst method, we can compute the Weil pairing as in the level 4 case. We explain how to compute it with the second method: let P, Q ∈ A[] and denote by P , Q the corresponding points in K A . Denote by θi (z), i ∈ Z(2) the level 2 theta functions associated to Ω. By Corollary 2, we have: eW (P , Q) =

θi (zQ )θi (.zP ) × θi (zP )θi (.zQ )θi (zQ + .zP )θi (zQ − .zP ) [θi (.zQ + zP )θi (zQ − zP ) + θi (.zQ − zP )θi (zQ + zP )] . (22)

The denominator of this expression can be easily computed from the knowledge of θi (zQ ), θi (.zQ ), θi (zP ) and θi (.zP ) by using the addition formula (1). The numerator can be computed in the algebra A in the following way: keeping the notations from above, we compute R = ScalarMult(T, Q, P, 0, )i .ScalarMult (gT, P, Q, 0, )i . We obtain that R + g.R = ScalarMult(P + Q, Q, P, 0, )i . ScalarMult(P − Q, P, Q, 0, )i + ScalarMult(P − Q, Q, P, 0, )i .ScalarMult(P + Q, P, Q, 0, )i , which gives the numerator of (22).

6

An Example in Dimension 2

In this section we give an example of compution of the pairings on a dimension 2 Jacobian. Let H be the hyperelliptic curve over the prime ﬁeld Fp , p = 331, given by the equation: Y 2 = X 5 + 204X 4 + 198X 3 + 80X 2 + 179X. Let J be the Jacobian of H. The cardinal of J(Fp ) is 26 · 1889 (since we are in level 2, all the 2-torsion points of J are rational), so that we let = 1889, and

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the embedding degree k corresponding to is 4. A theta null point of level 2 associated to J is given by (328 : 213 : 75 : 1). Let P = (255 : 89 : 30 : 1), we have P ∈ J[](Fp ). Let Fpk Fp (t)/(t4 + 3t2 + 290t + 3). We let Q be the Fpk -point of -torsion whose coordinates are: (158t3 + 67t2 + 9t + 293 : 290t3 + 25t2 + 235t + 280 : 155t3 + 84t2 + 15t + 170 : 1).

We compute (and ﬁx an arbitrary ordering): P + Q = (217t3 + 271t2 + 33t + 303 : 308t3 + 140t2 + 216t + 312 : 274t3 + 263t2 + 284t + 302 : 1), P − Q = (62t3 + 16t2 + 255t + 129 : 172t3 + 157t2 + 43t + 222 : 258t3 + 39t2 + 313t + 150 : 1). k

Finally, we let r = p −1 = 6354480 and ζ = tr be a primitive th -root of unity. We then compute using the doubling and diﬀerential addition algorithms: = (12, 141, 31, 327) = 327.0, P = (21t + 280t + 101t + 180, 164t3 + 311t2 + 111t + 129, Q 3

2

0, 137t3 + 282t2 + 123t + 134, 324t3 + 17t2 + 187t + 271) = (324t3 + 17t2 + 187t + 271).

, P, 0, ) = (45t3 + 118t2 + 219t + 308, 152t3 + 97t2 + 166t + 40, ScalarMult(P + Q, Q

, 200t3 + 267t2 + 201t + 192, 117t3 + 42t2 + 106t + 205) = (117t3 + 42t2 + 106t + 205).P

, Q , 0, ) = (50t3 + 31t2 + 84t + 309, 168t3 + 196t2 + 275t + 234, ScalarMult(P + Q, P

. 67t + 186t + 159t + 102, 243t + 320t + 222t + 200) = (243t + 320t + 222t + 200).Q 3

2

3

2

3

2

We then compute (following the previous ordering): eW (P, Q) =

243t3 + 320t2 + 222t + 200 324t3 + 17t2 + 187t + 271 . = ζ −1 , 327 117t3 + 42t2 + 106t + 205

eT (P, Q) =

eT (Q, P ) =

117t3 + 42t2 + 106t + 205 324t3 + 17t2 + 187t + 271

r

243t3 + 320t2 + 222t + 200 327

=ζ

r

1068

,

= ζ 1184 .

Here the Tate pairings are normalized by taking their r = (pk − 1)/-power. The symmetric pairings are then given by eW (P, Q) = 61t3 + 285t2 + 196t + 257 and eT (P, Q) = 194t3 + 163t2 + 97t + 164.

7

Conclusion

In this paper, we have presented an algorithm based on theta functions to compute Weil and Tate pairings. It would be interesting to carry out a ﬁne grained study of the eﬃciency of our algorithm depending on the target implementation (software, hardware etc.) and to compare it with existing implementations based on Miller’s algorithm.

Acknowledgement The authors of this paper would like to thank anonymous referees for their careful reading and helpful comments on an earlier version of the paper.

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References 1. Birkenhake, C., Lange, H.: Complex abelian varieties, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 302. Springer, Berlin (2004) 2. Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications. Chapman & Hall/CRC (2006) 3. Galbraith, S., Lin, X.: Computing pairings using x-coordinates only. Designs, Codes and Cryptography (2008) 4. Gaudry, P.: Fast genus 2 arithmetic based on Theta functions. J. of Mathematical Cryptology 1, 243–265 (2007) 5. Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields Appl. 15(2), 246–260 (2009) 6. Igusa, J.-i.: Theta functions. Springer, New York (1972); Die Grundlehren der mathematischen Wissenschaften, Band 194 7. Koizumi, S.: Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98(4), 865–889 (1976) 8. Lang, S.: Reciprocity and correspondences. Amer. J. Math. 80, 431–440 (1958) 9. Lichtenbaum, S.: Duality theorems for curves over p-adic ﬁelds. Invent. Math. 7, 120–136 (1969) 10. Lubicz, D., Robert, D.: Computing isogenies between abelian varieties (2010), http://arxiv.org/abs/1001.2016 11. Mumford, D.: On the equations deﬁning abelian varieties. I. Invent. Math. 1, 287– 354 (1966) 12. Mumford, D.: On the equations deﬁning abelian varieties. II. Invent. Math. 3, 75– 135 (1967) 13. Mumford, D.: Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay (1970) 14. Mumford, D.: Tata lectures on theta I. Progress in Mathematics, vol. 28. Birkh¨ auser Boston Inc., Boston (1983); With the assistance of Musili, C., Nori, M., Previato E., Stillman, M. 15. Mumford, D.: Tata lectures on theta II. Progress in Mathematics, vol. 43. Birkh¨ auser Boston Inc., Boston (1984); Jacobian theta functions and diﬀerential equations, With the collaboration of Musili, C., Nori, M., Previato, E., Stillman, M., Umemura, H. 16. Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986); Corrected reprint of the 1986 original (1986)

Small-Span Characteristic Polynomials of Integer Symmetric Matrices James McKee Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, England, UK [email protected]

Abstract. Let f (x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd . The span of f (x) is deﬁned to be αd − α1 . Monic irreducible f (x) of span less than 4 are special. In this paper we give a complete classiﬁcation of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we ﬁnd some low-degree polynomials that do not arise as the minimal polynomial of any integer symmetric matrix: these provide lowdegree counterexamples to a conjecture of Estes and Guralnick [6].

1 1.1

Introduction History of the Small Span Problem

Let f (x) ∈ Z[x] be a monic polynomial having only real roots. If these roots are α1 ≤ . . . ≤ αd then we say that f (x) has span αd − α1 . In the case where f (x) is irreducible, the roots are (Galois) conjugates of each other and we then refer to {α1 , . . . , αd } as a conjugate set. If a real interval I has length strictly less than 4, then it is known [19] that I contains only ﬁnitely many conjugate sets of algebraic integers. If I has length greater than 4 then it contains inﬁnitely many such conjugate sets [17]. The problem remains open for intervals of length exactly 4, unless the endpoints are integers, in which case there are inﬁnitely many such sets [11]. Monic f (x) ∈ Z[x] of span less than 4 have therefore attracted some interest: for convenience we shall call these small-span polynomials. The span is unchanged if we replace f (x) by εdeg f f (εx + c) for any choice of ε ∈ {−1, 1} and any integer c: two polynomials related in this way are deemed to be equivalent. The number of equivalence classes of small-span polynomials of any given degree is ﬁnite. Robinson [18] produced a complete list of representatives for degrees up to 6, with conjectured lists for degrees 7 and 8 that were later veriﬁed as complete. Recently Capparelli, Del Fra and Sci` o [2] extended this computation (using new techniques) up to degree 14. For any natural number m, the totally real algebraic integer 2 cos(2π/m) has its conjugate set lying in the interval [−2, 2]; we call the minimal polyomial of such a number a cosine polynomial. Examples of irreducible small-span f (x) not equivalent to one of these cosine polynomials are of special interest. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 270–284, 2010. c Springer-Verlag Berlin Heidelberg 2010

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1.2

271

Characteristic Polynomials of Integer Symmetric Matrices

For any n-by-n integer symmetric matrix A we deﬁne its characteristic polynomial, χA (x), by χA (x) = det(xI − A), where I is the n-by-n identity matrix. Clearly χA (x) is a monic polynomial with integer coeﬃcients; moreover all its roots are real since A is a real symmetric matrix. We deﬁne the span of A to be the span of its characteristic polynomial, and we say that A is a small-span integer symmetric matrix if it has span less than 4. A more usual measure of the size of the eigenvalues of A is its spectral radius, deﬁned to be the largest modulus of any eigenvalue. Plainly the span of A is bounded above by twice its spectral radius. If the spectral radius is at most 2, then the characteristic polynomial is a small-span cosine polynomial (or a product of such polynomials). See [14] for a classiﬁcation of all integer symmetric matrices of spectral radius below 2.019: there are no non-cosine small-span examples. There is a similar list in [14] of all f (x) arising as characteristic polynomials of integer symmetric matrices for which the Mahler measure of xdeg f f (x + 1/x) is below 1.3: if the Mahler measure is 1, then one has a cosine example, and amongst those for which the Mahler measure is close to 1 one ﬁnds some, but not all, non-cosine small-span examples. Petrovi´c [16] classiﬁed all graphs whose characteristic polynomial has span at most 4. From this one can easily deduce which cases give span less than 4. The adjacency matrices of such graphs are special cases of integer symmetric matrices, with the entries restricted to {0, 1}, and with only zero entries on the main diagonal. If f (x) ∈ Z[x] is monic and totally real, then one can sensibly ask whether or not it arises as the characteristic polynomial of an integer symmetric matrix. Not every such f (x) arises in this way: we shall see some examples that do not, below. On the other hand, it is known (see [5], or [1]) that every totally real algebraic integer α is the eigenvalue of some integer symmetric matrix A, so that the minimal polynomial of α divides χA (x). 1.3

Minimal Polynomials of Integer Symmetric Matrices: A Conjecture of Estes and Guralnick

With mystery surrounding the question of which polynomials f (x) arise as χA (x) for some integer symmetric matrix A, Estes and Guralnick [6] turned their attention to the minimal polynomial mA (x), deﬁned as the monic polynomial in Z[x] of minimal degree such that mA (A) = 0. One has that mA (x) divides χA (x), and that every root of χA is a root of mA [9, §11.6]. For an integer symmetric matrix A, the minimal polynomial mA (x) must be separable (i.e., its roots are distinct) since A is diagonalisable. Estes and Guralnick showed [6, Corollary C] that if f (x) ∈ Z[x] has degree n ≤ 4, has all roots real, and is monic and separable, then f (x) is the minimal polynomial of a 2n-by-2n integer symmetric matrix. For example, one can easily show that x2 − 3 is not the characteristic polynomial of an integer symmetric matrix, but it satisﬁes all the hypotheses of the Estes-Guralnick theorem, and sure enough we ﬁnd that

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⎛

−1 ⎜ 1 ⎜ ⎝ 1 0

⎞ 1 1 0 1 0 1⎟ ⎟ 0 1 −1 ⎠ 1 −1 −1

has minimal polynomial x2 − 3. For a less trivial example, we shall see below in §3 that x3 − 4x − 1 is not the characteristic polynomial of any integer symmetric matrix. Yet it is the minimal polynomial of ⎞ ⎛ 1 0 1 1 1 0 ⎜ 0 1 1 −1 0 1 ⎟ ⎟ ⎜ ⎜ 1 1 −1 0 0 0 ⎟ ⎟ ⎜ ⎜ 1 −1 0 −1 0 0 ⎟ . ⎟ ⎜ ⎝1 0 0 0 0 0⎠ 0 1 0 0 0 0 At the end of their paper [6], Estes and Guralnick ask whether or not every monic, separable, totally real f (x) ∈ Z[x] is the minimal polynomial of an integer symmetric matrix: they conjecture that the answer is ‘yes’ (p. 84). This question was answered in the negative by Dobrowolski [4]. He showed that any degree-n irreducible minimal polynomial of an integer symmetric matrix has discriminant at least nn , and then observed that inﬁnitely many cosine polynomials have smaller discriminant than this (for a precise formula for the discriminant of a cosine polynomial see [18, p. 554], derived from a formula in [12]). The smallest degree of any of Dobrowolski’s counterexamples to the conjecture of Estes and Guralnick is 2880; we shall give below some counterexamples of degree 6, for which the discriminant is too large for Dobrowolski’s argument to apply. It remains an open problem as to whether or not there are any counterexamples of degree 5. 1.4

The Contributions of This Paper

In this paper we ask which monic, irreducible, totally real polynomials in Z[x] of span less than 4 arise as characteristic polynomials of integer symmetric matrices. For this restricted class of polynomials, we are able to give a complete classiﬁcation (Theorem 3; more precisely, Theorem 3 classiﬁes the integer symmetric matrices that give rise to small-span characteristic polynomials). As a byproduct of this, we are able to address the conjecture of Estes and Guralnick about minimal polynomials [6, p. 84], and produce some counterexamples with degree as small as 6. In §2 we describe the algorithm for computing the complete list of representatives of equivalence classes of small-span integer symmetric matrices up to any desired degree. This builds on similar algorithms in [13] and [14]. In §3 we detail the results. In §4, we prove a classiﬁcation theorem for the small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. The paper concludes by applying this to the conjecture of Estes and Guralnick.

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2 2.1

273

The Growing Algorithm Equivalence

Let On (Z) be the orthogonal group of n-by-n signed permutation matrices. If A is an n-by-n integer symmetric matrix, and P ∈ On (Z), then we call A and P −1 AP = P T AP strongly equivalent. Strongly equivalent matrices have the same characteristic polynomial. Let A be an n-by-n integer symmetric matrix, and let c be any integer. Then χA+cI (x) = χA (x − c). Also χ−A (x) = (−1)n χA (−x). Thus if f (x) is the characteristic polynomial of an integer symmetric matrix, then so is any polynomial equivalent to f (x) in the sense of §1.1. We deﬁne integer symmetric matrices A and B to be equivalent if A is strongly equivalent to ±B + cI for some integer c. Thus equivalent matrices have equivalent characteristic polynomials. If A has span less than 4, then by adding cI for suitable c we can move to an equivalent matrix B with all eigenvalues in the interval [−2, 3); if B has an eigenvalue greater than 2.5, then it has no eigenvalue smaller than −1.5, and we replace B by the equivalent matrix −B + I. We see that any small-span integer symmetric matrix is equivalent to one with all eigenvalues in the interval [−2, 2.5). Our conclusion is that in order to ﬁnd which monic, totally real polynomials in Z[x] of degree n and span less than 4 arise as characteristic polynomials of integer symmetric matrices, it is enough to ﬁnd all n-by-n integer symmetric matrices up to strong equivalence that satisfy both: (i) the span is less than 4; and (ii) all eigenvalues lie in the interval [−2, 2.5). 2.2

Indecomposable Matrices

An integer symmetric matrix will be called decomposable if one can apply a permutation to the rows, and the same permutation to the columns, to produce a matrix in block diagonal form with more than one block. A matrix that is not decomposable is indecomposable. The characteristic polynomial of a decomposable matrix is the product of the characteristic polynomials of its blocks. In attempting to understand which polynomials arise as characteristic polynomials, it is therefore enough to restrict to indecomposable matrices. There is a nice graph-theoretic description of the property of being indecomposable. The underlying graph of an integer symmetric matrix has vertices labelled by the rows, with an edge between vertex i and vertex j precisely when the (i, j)-entry in the matrix is non-zero. Then a matrix is indecomposable if and only if the underlying graph is connected. We record a standard lemma whose proof is obvious given this interpretation. Lemma 1. Let A be an n-by-n indecomposable matrix, with n ≥ 2. Then there is a choice of i between 1 and n such that deleting row i and column i from A leaves an indecomposable submatrix. When convenient, we shall use the language of graphs to talk about our matrices. We speak of vertices to indicate rows, edges to indicate non-zero matrix entries,

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with natural interpretations of paths, cycles, connectedness, and so on. The distance between two vertices will mean the minimal number of edges on a path from one to the other. If our matrix has a non-zero entry on the diagonal, then we refer to the corresponding vertex as being charged. Lemma 1 is a corollary of the following slightly more precise result, which we shall exploit later. Lemma 2. Let G be a connected graph with at least 2 vertices, and let i and j be vertices for which the distance between i and j is maximal. Then deleting vertex i (and all incident edges) does not disconnect the graph. Proof. Suppose that after deleting i there was a vertex k not in the same component as j. Then every path from k to j in G would have to pass through i, and so the distance from k to j would be strictly greater than that from i to j, giving a contradiction. 2.3

Interlacing

We shall make much use of Cauchy’s interlacing theorem [3] (for more accessible proofs, see [8], [10] or [7]). Theorem 1 (Cauchy, 1829). Let A be an n-by-n integer symmetric matrix, with n ≥ 2, and let B be an (n − 1)-by-(n − 1) submatrix formed by deleting row i and column i from A (for some choice of i between 1 and n). Let λ1 ≤ λ2 ≤ . . . ≤ λn be the eigenvalues of A, and let μ1 ≤ . . . ≤ μn−1 be those of B. Then these two sets of eigenvalues interlace: λ1 ≤ μ1 ≤ λ2 ≤ μ2 ≤ . . . ≤ μn−1 ≤ λn . From this we have an immediate corollary which will be of use in our algorithm for computing small-degree small-span integer symmetric matrices. Corollary 1. Let A be an n-by-n integer symmetric matrix, with n ≥ 2, and let B be an (n − 1)-by-(n − 1) submatrix formed by deleting row i and column i from A (for some choice of i between 1 and n). Then the span of A is at least as large as the span of B. Moreover, if A has all its eigenvalues in the interval [−2, 2.5), then so does B. 2.4

Reduction

Our situation would be considerably more pleasant if for any integer symmetric matrix A we could quickly ﬁnd a canonical representative of its strong equivalence class. Unfortunately this is not the case, and we content ourselves with a quick ‘reduction’ process that gives us a semi-canonical representative, but with the possibility that there are several diﬀerent ‘reduced’ elements in the same strong equivalence class. Some balance must be struck between the speed of reduction and the possible number of strongly-equivalent reduced matrices.

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In practice we used two complementary reduction processes, which for convenience we call fast reduction and slow reduction. The ﬁrst of these is generally much faster and was used to identify quickly many cases of strong equivalence. The slower reduction process was then used to produce further weeding of our lists of matrices. This double reduction was then repeated until no further weeding was achieved. Any matrices in the ﬁnal list having the same characteristic polynomial (and sharing a few other invariants of strong equivalence) were ﬂagged for further inspection: in all such cases, either an equivalence between the two examples was found, or some simple argument established that the two were not equivalent. The principle of fast reduction is to give a ‘score’ to each row of the matrix, such that the multiset of scores is invariant under strong equivalence. The rows and columns would then be ordered according to this score. Finally, if the ﬁrst non-zero entry of any row was negative (and not on the diagonal) then that row (and the corresponding column) would have its sign changed. A more complicated scoring system would take longer to compute but would reduce the number of rows having equal score and thereby reduce the risk of having more than one possible reduced matrix in the same strong equivalence class. The scoring system that we used was to compute the ﬁrst three powers of the matrix A and then rank rows by a linear combination of: (i) the sum of the moduli of the entries in the row; (ii) the same for A2 ; (iii) the same for A3 ; (iv) the size of the diagonal entry. The aim of slow reduction was to attempt to ﬁnd the lexicographically smallest element of a strong equivalence class. If always successful then this would provide a perfect reduction process, but to achieve this perfection would be painfully slow. Instead one deemed a matrix to be reduced if it was ‘locally minimal’ with respect to lexicographical ordering in the sense that: (i) changing the sign of any row (and column) would give a larger matrix (in the sense of the ordering); (ii) swapping any two rows (and the corresponding columns) would give a larger matrix; (iii) cyclically permuting any three rows (and the corresponding columns) would produce a larger matrix. There is no claim that the combination of fast and slow reduction detailed above is optimally eﬃcient, but both reduction methods signiﬁcantly reduced the number of matrices needing to be considered, and enabled the computations to proceed smoothly up to the sizes detailed below. 2.5

Bounds on Entries and Valencies

Using interlacing (Theorem 1 to bound the size of diagonal entries, and Corollary 1 to deal with oﬀ-diagonal entries) we can rapidly restrict the possible entries for integer symmetric matrices that are of interest to us. Lemma 3. Let A be a small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). Then all entries of A have absolute value at most 2, and all oﬀ-diagonal entries have absolute value at most 1.

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Proof. Let a be a diagonal entry in A. Then since (a) has a as an eigenvalue, repeated use of Theorem 1 shows that A has an eigenvalue with modulus at least as large as |a|. Our restriction on the eigenvalues of A shows that |a| ≤ 2. Let b be an oﬀ-diagonal entryof A. Then deleting other rows and columns ab . By repeated use of Corollary 1, this gives a submatrix of the shape bc

submatrix must have span less than 4, giving (a − c)2 + 4b2 < 4. This implies |b| ≤ 1. The cases where there is an entry that has absolute value 2 are extremely restricted. The following Lemma describes the complete list. Lemma 4. Up to strong equivalence, the only indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5) and containing an entry of modulus greater than 1 are: (−2), (2),

⎞ ⎛ ⎞ ⎛ 2100 210 ⎜1 0 1 0⎟ 21 ⎝ 2 1 ⎟ , 1 0 1⎠,⎜ , ⎝0 1 0 1⎠ . 10 1 −1 010 0010

(The ﬁrst two matrices listed in Lemma 4 are equivalent, but not strongly equivalent.) Proof. Each of the ﬁve 1-by-1 matrices (−2), (−1), (0), (1), (2) was grown in all possible ways to larger indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5), allowing entries from {−2, −1, 0, 1, 2} in accordance with Lemma 3. After producing a provisional list of 2-by-2 matrices, this list was weeded by reduction, as described in §2.4. Repeating this growing process three more times revealed that there are no 5-by-5 examples containing an entry having modulus greater than 1, and by interlacing the same must be true for all larger indecomposable integer symmetric matrices. The output of this computation also established the advertised list. Having reduced to the problem of considering matrices that have absolute value at most 1, we now further restrict the possible entries in each row. Lemma 5. Let A be an indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). Then each row of A has at most 4 non-zero entries. Proof. After Lemma 4, we can suppose that all entries in A are from the set {−1, 0, 1}. If Lemma 5 were false, then by interlacing (and making use of strong equivalence) there would be a small-span integer symmetric matrix M with all eigenvalues in the interval [−2, 2.5) and with M being one of

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⎞ 0111 11 11111 −1 1 1 1 1 ⎟ ⎜ ⎜ 1 a b c d⎟ ⎜1 a b c d⎟ ⎜1 a b c d e⎟ ⎟ ⎜1 b f g h i⎟ ⎟ ⎜ ⎜ ⎟, ⎜ 1 b e f g⎟,⎜1 b e f g⎟,⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ 1 c f h i⎠ ⎝1 c f h i⎠ ⎜1 c g j k l⎟ ⎝1 d h k m n⎠ 1d g ij 1d g ij 1e i l no ⎛

⎞ ⎛

⎞

⎛

where the unspeciﬁed entries are all from {−1, 0, 1}. A computer search showed that no such matrix M exists. 2.6

The Algorithm

Lemma 1 and Corollary 1 suggest a means of ‘growing’ indecomposable smallspan integer symmetric matrices with all eigenvalues in [−2, 2.5) from smaller matrices. This idea has been used before for computing integer symmetric matrices with small spectral radius or small Mahler measure ([13] and [14]). Having established Lemmas 4 and 5, we grow indecomposable matrices with all entries coming from the set {−1, 0, 1}, and with the extra restriction that each row can contain no more than four non-zero entries. After producing a provisional list of r-by-r matrices, this list is weeded by reduction, as described in §2.4, before growing to produce a list of (r + 1)-by-(r + 1) matrices. The complete search up to 13-by-13 matrices was completed in under ﬁve hours on a single processor. This was enough to provide the computational element of the proof of Theorem 3 below. The computation was pushed up to 20-by-20 matrices in under six days; perfect agreement of the results with Theorems 2 and 3 for larger matrices provided conﬁdence in the correctness of the output for smaller matrices. The PARI code for all of this is freely available from the author on request. After each growing of a list of (n − 1)-by-(n − 1) matrices to a list of n-by-n matrices, any examples from the ﬁrst list that had not been grown to one or more examples in the second were recorded in a list of maximal examples. Some of these maximal examples ﬁtted into inﬁnite families, described in Theorem 2; others did not, and these we call sporadic.

3

Results

We shall call an indecomposable small-span integer symmetric matrix that has all eigenvalues in the interval [−2, 2.5) maximal if it cannot be obtained by deleting rows (and corresponding columns) from any larger indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). It turns out that every indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5) can be grown to a maximal one (part of Theorem 3). In view of Corollary 1, it is enough to describe all the maximal matrices. Up to strong equivalence there are 197 sporadic examples and 10 inﬁnite families. In this section we tabulate the number of sporadic examples of

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each size, found by computation as outlined above. The inﬁnite families and the proof of completeness of the classiﬁcation will follow in §4 (Theorems 2 and 3). Members of the inﬁnite families all in fact have eigenvalues in the smaller interval [−2, 2]. The following table includes the three maximal examples from Lemma 4. Maximal examples that are members of the inﬁnite families of Theorem 2 are excluded: only the sporadic cases are counted. The computations had been done up to size 20-by-20, but the only maximal cases that were not covered by the inﬁnite families of Theorem 2 were 12-by-12 or smaller. That no further sporadic maximal examples arise is the point of Theorem 3. Sporadic maximal indecomposable small-span integer symmetric matrices with all eigenvalues in [−2, 2.5), up to strong equivalence n n-by-n cosine examples n-by-n non-cosine examples total 1 1 0 1 2 0 1 1 3 0 1 1 4 10 9 19 5 0 19 19 6 0 43 43 7 0 28 28 8 11 39 50 9 0 15 15 10 0 15 15 11 0 2 2 12 0 3 3 total 22 175 197 For degrees up to 8, most small-span irreducible polynomials arise as characteristic polynomials of integer symmetric matrices: it is simpler to record which of Robinson’s polynomials from [18] do not arise. It is interesting to note that all examples of degrees 4 and 5 appear. The missing examples for degrees 2 and 3 are those mentioned in §1.3 above, namely x2 − 3 and x3 − 4x − 1. The other missing polynomials are numbers 6g, 6i, 6k, 7j, 7k, 7l, 8a, 8c, 8l, 8m, 8t, 8u, 8y in Robinson’s list [18]. For degree 9, both of the inequivalent cosine polynomials arise as characteristic polynomials, and three other irreducibles: x9 − x8 − 9x7 + 7x6 + 28x5 − 15x4 − 34x3 + 10x2 + 12x − 1, x9 − 4x8 − 2x7 + 21x6 − 5x5 − 37x4 + 12x3 + 24x2 − 5x − 4, x9 − 3x8 − 5x7 + 18x6 + 7x5 − 34x4 − x3 + 20x2 − 3x − 1. For degree 10, the only irreducible small-span characteristic polynomial is the non-cosine example x10 − 5x9 + x8 + 26x7 − 21x6 − 49x5 + 40x4 + 42x3 − 20x2 − 15x − 1. For degree 11, the only one (up to equivalence) is the cosine case. For degree 13 and above, Theorem 3 (below) gives a complete description of which characteristic polynomials arise. All degree-13 examples that have span below 4 and all eigenvalues in the interval [−2, 2.5) in fact have all eigenvalues in the subinterval [−2, 2] (this is the content of Theorem 3), and hence are described in Theorem 2.

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The following table compares the complete lists of [18] and [2] with the results of the computations for characteristic polynomials, restricting to irreducible polynomials. Degree

1 2 3 4 5 6 7 8 9 10 11 12 13

4

Number of irreducible Number that arise as small-span polynomials characteristic polynomials up to equivalence: of integer symmetric matrices: cosine + non-cosine = total cosine + non-cosine = total 1+0=1 1+0=1 3+1=4 2+1=3 2+3=5 2+2=4 4 + 10 = 14 4 + 10 = 14 1 + 14 = 15 1 + 14 = 15 4 + 13 = 17 1 + 13 = 14 0 + 15 = 15 0 + 12 = 12 5 + 21 = 26 5 + 14 = 19 2 + 19 = 21 2+3=5 3 + 15 = 18 0+1=1 1 + 10 = 11 1+0=1 7 + 9 = 16 0+0=0 0+4=4 0+0=0

Classification of Small-Span Integer Symmetric Matrices

One result of our computations is that any indecomposable small-span 13-by13 integer symmetric matrix with all its eigenvalues in [−2, 2.5) in fact has all its eigenvalues in [−2, 2]. We shall now prove that this holds for all larger indecomposable matrices too. As a ﬁrst step, we classify those indecomposable small-span integer symmetric matrices that have all their eigenvalues in the interval [−2, 2]. After Lemma 4, we are reduced to considering matrices that have entries 0, 1 or −1. These are conveniently represented by charged signed graphs. Vertices are labelled with their charges (corresponding to diagonal entries of the matrix); oﬀ-diagonal entries 1 and −1 are represented respectively by solid and dotted edges. Zero charges can be omitted to reduce clutter. For example, the matrix ⎞ ⎛ t t t 1 1 0 0 @ 1 qq 1 ⎜1 0 1 1⎟ ⎟ ⎜ @ qq . ⎝ 0 1 1 −1 ⎠ is drawn as @qt 0 1 −1 0 In the graphs below, the symbol k t t

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denotes a path with k solid edges (and all vertices uncharged) between the displayed end vertices (if k = 0 then these end vertices are identiﬁed as a single vertex). Deﬁne + − + − graphs O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , Yn+ , Yn− , Zk,l , Zk,l as shown. 2k − 3 t t

2k − 1 t t J

J

Jt

tq q q q q q q t O2k (k ≥ 2)

+ − (k ≥ 1) O2k+1 (k ≥ 1) O2k+1

k t t ± Pk+3 (k ≥ 0)

1t

2k − 1 t t q J q J qq Jt

t−1

t @

@ tq q q [email protected] t1

1t

t @ @ k tq q q [email protected] t qt + Xk+5 (k ≥ 0)

1t

X4+ t @

@ tq q q [email protected] t−1 X4−

−1 t

t @ @ k tq q q [email protected] q t t − Xk+5 (k ≥ 0)

−1 t

1t

1t

@ 1 tq q q [email protected] q t1

1 tq q q [email protected] qt

@

@

@

Y3+ −1 t

−1 t

@

−1 t

@

@

@t−1

−1 t

Y3−

t1

−1 t

+ Yk+4

k t

k t

t @

t + Zk,l

k

t (k ≥ 0)

k @t t − Yk+4 (k ≥ 0)

@

@ l tq q q [email protected] q t t (k ≥ 0, l ≥ 0)

t1

t−1

t1

t−1

t1

t @

t − Zk,l

@ l tq q q [email protected] t qt (k ≥ 0, l ≥ 0)

t−1

Theorem 2. Every indecomposable small-span integer symmetric matrix M1 that has all its eigenvalues in the interval [−2, 2] is a submatrix of an indecomposable

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281

small-span integer symmetric matrix M2 that is maximal subject to being smallspan and having all its eigenvalues in [−2, 2.5). Up to strong equivalence, the possibilities for M2 are the sporadic maximal examples tabulated in Section 3 and the + adjacency matrices of the charged signed graphs O2k (k ≥ 4), O2k+1 (k ≥ 3), − ± + − + O2k+1 (k ≥ 2), Pn (n ≥ 6), Xn (n ≥ 7), Xn (n ≥ 4), Yn (n ≥ 6), Yn− (n ≥ 3), + − Zk,l (k ≥ l ≥ 0, except for (k, l) ∈ {(0, 0), (1, 0), (1, 1), (2, 1)}), Zk,l (k ≥ l ≥ 0) pictured above. Proof. This is a tedious but easy extension of the work in [13, §12] where all examples with eigenvalues in the open interval (−2, 2) were described; here we relax this to consider the intervals (−2, 2] and [−2, 2). A convenient technique is that of Gram vectors. If an integer symmetric matrix A has all its eigenvalues in [−2, 2], then both B = A + 2I and C = −A + 2I have all eigenvalues at least 0. Thus there are lists of Gram vectors v1 , . . . , vn and w1 , . . . , wn contained in Rn such that the (i, j)-entry of B (respectively C) is given by vi · vj (respectively wi · wj ). Now −2 is an eigenvalue of A if and only if v1 , . . . , vn are linearly dependent, and 2 is an eigenvalue of A if and only if w1 , . . . , wn are linearly dependent. We start by noting that the following charged signed graphs have span 4: in each case one readily writes down linearly dependent sets of Gram vectors as above, showing that both −2 and 2 are eigenvalues, following the ideas in [13]. t t t t t qt qt qt qq q q qq @ q @ q q q t t q qq @ q [email protected] t t tq q q [email protected] qt t t t @t t t k k k 1t −1 t t t @ @ @ @ @ @ @ @ t tq q q [email protected] qt qt t @t t tq q q [email protected] t 1 tq q q [email protected] −1 k k 1t 1 1 1 −1 t t t t−1 q qt @ q q qq @ @ [email protected] qq @ @ 1 tqq q q [email protected] 1 tq q q [email protected] q t1 q t tq q q q q t1 −1 t t−1 @t t k k −1 t 1t t t t t−1 1 t t t−1 @ −1 @ @ k @ @ @ −1 t −1 1 @t tq q q [email protected] t t t @t−1 1 tq q q q q t −1 k t t @ @ @ @ 1t 1t tq q q [email protected] t t t−1 t t tq q q [email protected] t−1 k k 1t t1 t t −1 @ @ @ @ @ @ 1 tq q q [email protected] 1t q t−1 @t−1 t t tq q q [email protected] q t t t−1 k k

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+ − Next we note that the charged signed graphs O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , + − + − Yn , Yn , Zk,l , Zk,l have all eigenvalues between −2 and 2 (they are equivalent to subgraphs of those listed in [13, §4]), and have span less than 4 (writing down Gram vector representations for each graph and its negative, one ﬁnds that in every case exactly one of the sets of Gram vectors is linearly independent). Finally we check readily that any connected subgraph of one of those in [13, §4] that does not contain any subgraph equivalent to one of the span-4 examples + − listed above must be a subgraph of one of O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , + − + − Yn , Yn , Zk,l , Zk,l . The restrictions on n, k and l require a trawl through the sporadic examples to see which of them contain any of the members of these 10 inﬁnite families as subgraphs. For example, P5± is a subgraph of the maximal sporadic example t−1 t1 t t t t−1 .

Theorem 3. Up to strong equivalence, the indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5) are precisely the indecomposable submatrices of the 197 sporadic cases accounted for in §3 and the 10 inﬁnite families of Theorem 2. In particular, every such matrix with more than 12 rows has all its eigenvalues in the interval [−2, 2]. Proof. In view of Theorem 2 and the computational results of §3, it is enough to show that every indecomposable integer symmetric matrix with more than 12 rows and all its eigenvalues in the interval [−2, 2.5) in fact has all its eigenvalues in the interval [−2, 2]. Suppose for a contradiction that this is not the case. Let A be a counterexample that has as few rows as possible. We know from our computations that A has at least 14 rows, and this minimal counterexample would then have the property that any proper submatrix has all its eigenvalues in the interval [−2, 2]. The result now follows from the classiﬁcation of all integer symmetric matrices minimal subject to not all eigenvalues being in the interval [−2, 2]: there are no such matrices with more than 10 rows [14]. But the current case is much easier, so we outline a direct proof. The key idea in the proof is that the property of having all eigenvalues in the interval [−2, 2] is essentially described by local structure. In the general case treated in [14] this local structure is much more complicated than in the small-span case treated here. Let G be the charged signed graph with adjacency matrix A (using Lemma 4). Pick vertices u and v as far apart as possible in G. Deleting either u or v leaves a connected (Lemma 2) charged signed graph with all eigenvalues in [−2, 2] and with at least 13 vertices, and hence a connected subgraph of one of the inﬁnite families of Theorem 2. Deleting u leaves an underlying graph that is either a cycle or not. Suppose ﬁrst that the underlying graph of G with u deleted is a cycle. Since u and v are maximally distant in G, we deduce that u is joined to vertices as far (or almost as far) as possible from v on this cycle, and since deleting v from G must give a connected subgraph of one of the inﬁnite families of Theorem 2, the only possibility for G (up to strong equivalence) is a charged signed graph of the

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283

+ shape formed by identifying the end vertices of Zk,l , with the charges removed. But then A has all eigenvalues in the interval [−2, 2] (see [13]) and in fact also has span 4, giving two contradictions. Now suppose that deleting u does not leave a cycle. Then it leaves a structure that is up to strong equivalence either an uncharged path (perhaps with one + − negative edge) or is one of Pn± , Xn+ , Xn− , Yn+ , Yn− , Zk,l , Zk,l , perhaps with one or more vertices removed in a way that does not disconnect the graph. Then either v is near the middle and u is adjacent to vertices at or near both ends of this structure, or v is at one end and u is adjacent to vertices at or near the other end. Again one sees (on considering deleting v, and using the classiﬁcation in [13]) that A must have all eigenvalues in [−2, 2], giving a contradiction.

5

Low-Degree Counterexamples to a Conjecture of Estes and Guralnick

Let f (x) be a monic, irreducible, totally real, small-span polynomial of degree n > 6 that has all its eigenvalues in the interval [−2, 2.5) but is not the characteristic polynomial of an integer symmetric matrix. Suppose further that f (x) is not a cosine polynomial. Then f (x) cannot be the minimal polynomial of any integer symmetric matrix. For if it were, then the smallest such matrix would be indecomposable and have characteristic polynomial f (x)r for some r > 1. But Theorem 3 precludes the existence of such characteristic polynomials, since the degree rn would be greater than 12. In particular, none of the polynomials x7 − x6 − 7x5 + 5x4 + 15x3 − 5x2 − 10x − 1, x7 − 8x5 + 19x3 − 12x − 1 or x7 − 2x6 − 6x5 + 11x4 + 11x3 − 17x2 − 6x + 7 is the minimal polynomial of an integer symmetric matrix. These provide degree-7 counterexamples to the conjecture of Estes and Guralnick [6]. Finally we remark that none of the three degree-6 cosine polynomials x6 − 5 x − 6x4 + 6x3 + 8x2 − 8x + 1, x6 − 7x4 + 14x2 − 7 and x6 − 6x4 + 9x2 − 3 is the minimal polynomial of any integer symmetric matrix. Our computations revealed that these three do not arise as characteristic polynomials, nor as minimal polynomials for any 12-by-12 or 18-by-18 matrix. Moreover the smallest span of an indecomposable 19-by-19 matrix is already larger than the spans of all three of these degree-6 polynomials, so by interlacing they cannot appear as the minimal polynomial of any larger matrix. It remains an open problem as to whether or not there exists a degree-5, monic, separable, totally real polynomial that does not arise as the minimal polynomial of an integer symmetric matrix. All the small-span cases are covered, so the techniques of this paper cannot be applied.

Acknowledgments This work was prompted by conversations with Georges Rhin and Chris Smyth at a workshop on Discovery and Experimentation in Number Theory, at the

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Fields Institute, Toronto in September 2009: I am grateful to the organisers of that workshop. I have also beneﬁted from conversations with Gary Greaves. Finally, I thank the referees for their numerous helpful suggestions.

References 1. Bass, H., Guralnick, R., Estes, D.: Eigenvalues of symmetric matrices and graphs. J. Algebra 168, 536–567 (1994) 2. Capparelli, S., Del Fra, A., Sci` o, C.: On the span of polynomials with integer coeﬃcients. Math. Comp. 79, 967–981 (2010) 3. Cauchy, A.: Sur l’´equation a l’aide de laquelle on determine les in´egalit´es s´eculaires des mouvements des plan`etes. In: Oeuvres Compl`etes d’ Augustin Cauchy Seconde S´erie IX, pp. 174–195. Gauthier-Villars, Berkeley (1891) 4. Dobrowolski, E.: A note on integer symmetric matrices and Mahler’s measure. Canadian Mathematical Bulletin 51(1), 57–59 (2008) 5. Estes, D.: Eigenvalues of symmetric integer matrices. J. Number Theory 42, 292– 296 (1992) 6. Estes, D.R., Guralnick, R.M.: Minimal polynomials of integral symmetric matrices. Linear Algebra and its Applications 192, 83–99 (1993) 7. Fisk, S.: A very short proof of Cauchy’s interlace theorem. Amer. Math. Monthly 112, 118 (2005) 8. Godsil, C., Royle, G.: Algebraic Graph Theory. In: Graduate Texts in Mathematics, vol. 207. Springer, New York (2000) 9. Hartley, B., Hawkes, T.O.: Rings, modules and linear algebra. Chapman and Hall, Boca Raton (1970) 10. Hwang, S.-G.: Cauchy’s interlace theorem for eigenvalues of Hermitian matrices. Amer. Math. Monthly 112, 157–159 (2004) 11. Kronecker, L.: Zwei s¨ atse u ¨ber gleichungen mit ganzzahligen coeﬃcienten. J. Reine Angew. Math. 53, 173–175 (1857) 12. Lehmer, E.: A numerical function applied to cyclotomy. Bull. Amer. Math, Soc. 36, 291–298 (1930) 13. McKee, J.F., Smyth, C.J.: Integer symmetric matrices having all their eigenvalues in the interval [−2, 2]. J. Algebra 317, 260–290 (2007) 14. McKee, J.F., Smyth, C.J.: Integer symmetric matrices of small spectral radius and small Mahler measure, arXiv:0907.0371v1 15. Batut, C., Belebas, K., Bernardi, D., Cohen, H., Olivier, M.: PARI/GP version 2.3.4, http://pari.math.u-bordeaux.fr/ 16. Petrovi´c, M.M.: On graphs whose spectral spread does not exceed 4. Publ. Inst. Math. Beograd 34(48), 169–174 (1983) 17. Robinson, R.M.: Intervals containing inﬁnitely many sets of conjugate algebraic integers. In: Mathematical Analysis and Related Topics: Essays in Honor of George P´ olya, Stanford, pp. 305–315 (1962) 18. Robinson, R.M.: Algebraic equations with span less than 4. Math. Comp. 18(88), 547–559 (1964) ¨ 19. Schur, I.: Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeﬃzienten. Math. Z. 1, 377–402 (1918)

Decomposition Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field Koh-ichi Nagao Dept. of Engineering, Kanto Gakuin Univ., 1-50-1 Mutsuura Higashi Kanazawa-ku Yokohama 236-8501, Japan [email protected]

Abstract. We propose some kind of new attack which gives the solution of the discrete logarithm problem for the Jacobian of a curve defined over an extension field Fqn , considering the set of the union of factor basis and large primes B0 given by points of the curve whose x-coordinates lie in Fq . In this attack, an element of the divisor group which is written by a sum of some elements of factor basis and large primes is called (potentially) decomposed and the set of the factors that appear in the sum, is called decomposed factors. So, it will be called decomposition attack. In order to analyze the running of the decomposition attack, a test for the (potential) decomposedness and the computation of the decomposed factors are needed. Here, we show that the test to determine if an element of the Jacobian (i.e., reduced divisor) is written by an ng sum of the elements of the decomposed factors and the computation of decomposed factors are reduced to the problem of solving some multivariable polynomial system of equations by using the Riemann-Roch theorem. In particular, in the case of hyperelliptic curves of genus g, we construct a concrete system of equations, which satisfies these properties and consists of (n2 − n)g quadratic equations. Moreover, in the case of (g, n) = (1, 3), (2, 2) and (3, 2), we give examples of the concrete computation of the decomposed factors by using the computer algebra system Magma. Keywords: Decomposition Attack, Hyperelliptic curve, Discrete logarithm problem, Weil descent attack.

1

Introduction

In this work, we treat the solution of the discrete logarithm problem of the Jacobian of a curve C of genus g deﬁned over an extension ﬁeld Fqn (n ≥ 2) by decomposition attack. In particular, when C is a hyperelliptic curve and ng(≥ 3) is a small integer, we give the concrete algorithm for computing what is called decomposed factors. In [6], Gaudry proposes the decomposition attack for the Jacobian of a hyperelliptic curve deﬁned over a general ﬁnite ﬁeld Fq considering a set of factor basis given by the Fq -rational points of the curve. This attack is usually called ’Index Calculus’ and such variations are widely used [3], [11]. However, the behavior of this attack, when it is used for solving the discrete G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 285–300, 2010. c Springer-Verlag Berlin Heidelberg 2010

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logarithm of algebraic curve, is quite diﬀerent to the original index calculus, which is a method to compute indices, that is, discrete logarithms in multiplicative groups of ﬁnite prime ﬁelds. Because of this, we use the name decomposition attack to refer to the attack. By recent works on the decomposition attack, which are the improvements of [6], it is known that the techniques of 1) using rebalancing [5] and 2) using large primes [15], [13], [7] are available. On the contrary, the techniques of large prime variations of normal index calculus associated to number ﬁeld sieve are known as no contribution and do not lead to a decrease of the complexity. In [8](also c.f. [4]), Gaudry also presents the decomposition attack for an elliptic curve deﬁned over an extension ﬁeld Fqn considering the set of factor basis given by points of the curve whose x-coordinates lie in Fq . Actually, Gaudry proposes also the rebalancing and the large prime variations. In these variations, the set of factor basis B is taken by some subset of B0 which is given by points of the curve whose x-coordinates lie in Fq and an element in B0 \B is called large prime. In these methods, the test for the potential decomposedness of P ∈ E(Fqn ) (i.e., for being a sum of n elements of the B0 ) and the computation of the decomposed factors (i.e., n elements of B0 whose summation equals to P ) are reduced to the problem of solving some system of multivariable polynomial equations of degree 2n−1 , n variables, and n equations, using Semaev’s summation polynomials [14]. Moreover, Gaudry generalizes this decomposition attack to the case of the abelian varieties deﬁned over an extension ﬁeld, including the case of Jacobians of curves. However, in the case of non-elliptic curves, Semaev’s summation polynomials are not available. It is, in principle, possible to derive a similar system of equations using the group law. Unfortunately, such is cumbersome. In fact, in the case of the Jacobian of a hyperelliptic curve of genus g, the sum of ng generic points is needed. Assuming that an element of Jacobian is written by the Mumford representation and that the group law is done by the Cantor algorithm [2], since the Cantor algorithm needs g − 1 times reduction steps, explosions of the degree and terms occur in this computation. In this work, we show that instead of using the group law, another system of equations is obtained from the theory of Riemann-Roch spaces (only in the case of Jacobians of curves). With this tool, the system of the equations is now simple to compute, and its parameters are easily controlled. In particular, in the case of Jacobians of hyperelliptic curves, this system of the equations consists of (n2 − n)g quadratic equations in (n2 − n)g indeterminates. So, under the heuristic assumption that this system of the equations is (essentially) projectively 0-dimensional, the computational amount for solving this 2 system of equations is estimated by O(2(n −n)g·C ) where C is some constant less than 3. In the case of an elliptic curve (i.e., g = 1), this computational amount heuristically equals to that of Gaudry’s original equations system using Semaev’s summation polynomials.

Decomposition Attack for the Jacobian of a Hyperelliptic Curve

2

287

Decomposition Attack for the Jacobian of a General Plane Curve

This section adapts the idea of [8] to the setting of a smooth plane curve with a single missing point at inﬁnity, and presents an overview of the decomposition attack for the Jacobian of a general plane curve using the Riemann-Roch theorem. Let Ca be the aﬃne curve of genus g deﬁned over an extension ﬁeld Fqn (i.e., n ≥ 2) given by the equation f (x, y) = 0, and let C be the corresponding non-singular complete curve. Assume that Ca is non-singular. From this, we have a canonical embedding ι : Ca → C. It is also assumed that C\ι(Ca ) only consists of a single Fqn -valued point, which is denoted by ∞ and is called the point at inﬁnity. These assumptions are true for hyperelliptic curves so there is no problem for the main results of this work. Let D0 be a divisor of the form D0 = Q1 + .. + Qg − (g)∞

(1)

where Q1 , .., Qg ∈ C(Fqn ) and the multiset {Q1 , .., Qg } is stable under the action of galois group Gal(Fqn /Fqn ). Put φ1 (x) :=

g (x − x(Qi ))

(2)

i=1

and note that it is in Fqn [x]. Also put B0 := {P ∈ C |P = (x, y) ∈ C(Fqn ), x ∈ Fq }, as a set of factor basis and large primes. (Strictly saying, B0 must be a subset of JacC (Fqn ), and it is the set of the elements of the divisors P − ∞ where P has the above properties. Here, the term “−∞” is omitted for simplicity.) Assumption 1. Let n be a ﬁxed positive integer. Then the number of the mulng tisets P = {P , .., P } with P ∈ B , which satisfy the relation 1 ng i 0 i=1 Pi ∼ ng P for some diﬀerent (P = P ) multiset P = {P , .., P } with P 1 ng i ∈ B0 , i=1 i ng−ε is less than q , where ε is some positive constant. Here, we shortly state the validity of this assumption in the case of hyperelliptic curve. Let C : y 2 = f (x) be the equation of hyperelliptic curve. For any P = (x, y) ∈ C, put P¯ = (x, −y) ∈ C. So, there are series of trivial relations P + P¯ ∼ P + P¯ for any P, P ∈ B0 . The number of the multisets satisfying the condition of Assumption 1 and coming from these trivial relations is only O(q ng−1 ) and it seems to be no series including many trivial relations. So, Assumption 1 seems to be valid. Assumption 2. |B0 | ≈ q. Here, we also state the validity of this assumption in the case of hyperelliptic curve. Let C : y 2 = f (x) be the equation of hyperelliptic curve. If f (x) is chosen

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randomly, the probability that f (x) (x ∈ Fq ) is square in Fqn is around 1/2 and this assumption seems to hold. In the following, we assume Assumption 1 and Assumption 2. From these assumptions, we see easily that since “the number of the divisors of the form(1)”≈ q gn , the probability that there are some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed) such that + P2 + ... + Png − (ng)∞ D0 + P1 g = i=1 Qi + P1 + P2 + ... + Png − (ng + g)∞ ∼ 0,

(3)

is approximately 1/(gn)!, when q ng. Definition 1. If a divisor D0 is written by the form (3) for some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed), D0 is called potentially decomposed and in this case, the elements P1 , P2 , .., Png are called decomposed factors and the multiset {Pi }ng i=1 is called decomposed divisor. We now ﬁx D0 and discuss how it can be tested that D0 is potentially decomposed and the decomposed factors can be computed. So, Q1 , ..., Qg and φ1 (x), which are dependent on D0 , are also ﬁxed. Let D = P ∈C(Fqn ) np P , np ∈ Z be a divisor of C/Fqn . Assume that D is sta ble under the action of galois group Gal(Fqn /Fqn ). Put deg(D) := P ∈C(Fqn ) np , and L(D) := {f ∈ Fqn (C) | (f )+D ≥ 0}∪{0}. From the Riemann-Roch theorem (cf [10] Corollary A.4.2.3), we have the following lemma. Lemma 1. (Riemann-Roch) 1) L(D) is an Fqn vector space. 2) If deg(D) ≥ 2g − 1, dim L(D) = deg(D) − g + 1. g From this Lemma, dim L((ng)∞ − D0 ) = dim L((ng + g)∞ − i=1 Qi ) = ng − g + 1. Let {f0 (x, y), f1 (x, y), ..., fng−g (x, y)} be a base of L((ng)∞ − D0 )) and an element h ∈ L((ng)∞ − D0 ) is written by a0 f0 (x, y) + a1 f1 (x, y) + ... + ang−g fng−g (x, y)

(4)

where ai are values in Fqn . From Hess [9], we have the following lemma. Lemma 2. A base of L((ng)∞ − D0 ) is computable within Poly(ng log q) time. Let h(x, y) := A0 f0 (x, y) + A1 f1 (x, y) + ... + Ang−g fng−g (x, y)

(5)

be a multivariable polynomial in Fqn [A0 , ..., Ang−g , x, y]. For aaﬀ = (a0 , a1 , ..., ang−g ) ∈ Ang−g+1 (Fqn ) and some polynomial p(x) ∈ Fqn [A0 , ..., Ang−g , x], let paaff (x) be the polynomial obtained from p(x) by substituting ai for Ai .

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Definition 2. A multivariable polynomial p(x) in Fqn [A0 , ..., Ang−g , x] is called A-homogenous, when paaﬀ (x) = Const × pkaaﬀ (x) holds for all aaﬀ = (a0 , a1 , ..., ang−g ) ∈ Ang−g+1 (Fqn ) and k ∈ F∗qn . For

apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn )

and some A-homogenous polynomial p(x) ∈ Fqn [A0 , ..., Ang−g , x], let monic(paaff (x)) be the polynomial obtained from p(x) by substituting ai for Ai and dividing by the leading coeﬃcient. Now, we compute the intersections of hapro (x, y) = 0 on C. Remember that the equation of Ca is f (x, y) = 0. Put S(x) := Resultanty (f (x, y), h(x, y)). From this construction, we then have the following lemma. Lemma 3. 1) S(x) is a multivariable A-homogeneous polynomial in Fqn [A0 , .., Ang−g , x]. 2) degx S(x) = ng + g. 3) φ1 (x) | S(x). Proof. 1) is trivial. For any apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn ), since hapro (x, y) has only poles (ng + g)∞ on points at inﬁnity, we have 2) and since hapro (x, y) have zeros at each Qi ’s, we have 3). Put g(x) := S(x)/φ1 (x). Since φ1 (x) ∈ Fqn [x], g(x) is also a multivariable Ahomogeneous polynomial in Fqn [A0 , .., Ang−g , x]. Thus, g(x) is written in the form g(x) = Cng xng + Cng−1 xng−1 + ... + C0 where each Ci ∈ Fqn [A0 , .., Ang−g ] has the same multi degree of Ai . Note that if the indeterminates Ai s are replaced by values ai and the obtained polynomial is divided by the leading coeﬃcient, then one obtains a polynomial monic(gapro (x)) in Fqn [x]. The solutions of monic(gapro (x)) = 0 mean the x-coordinates of the intersections hapro (x, y) = 0 on C except Q1 , ..., Qg . So, we have the following lemma. Lemma 4. The condition that D0 is potentially decomposed is equivalent to the following: There is some apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn ) such that monic(gapro (x)) ∈ Fq [x] and monic(gapro (x)) ∈ Fq [x] factors completely in Fq [x]. Now, we ﬁnd such ai ’s. Let [α0 (= 1), α1 , .., αn−1 ] be a base of Fqn /Fq . We ﬁx this base. Let Ai,j (1 ≤ i ≤ ng, 0 ≤ j ≤ n − 1) be new indeterminates over Fq , and let us consider the polynomials obtained by substituting A0 by 1 and Ai by n−1 j=0 Ai,j αj (1 ≤ i ≤ ng − g) in g(x). Let us denote the coeﬃcients obtained in this way again by Ci . Then the coeﬃcients can be written in the form Ci =

n−1

Ci,j αj ,

Ci,j ∈ Fq [∪1≤i≤ng, 0≤j≤n−1 {Ai,j }].

j=0

Then, the condition that there is some apro ∈ Png−g (Fqn ) satisfying 1) monic(gapro (x)) ∈ Fq [x] and

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2) First coordinate of apro is non-zero, is equivalent to the condition that the system of the equations Ci,j = Ti Cng,j

(0 ≤ i ≤ ng − 1, 0 ≤ j ≤ n − 1)

(6)

of (n2 + n)g indeterminates ∪{Ai,j } and T0 , ..., Tng−1 deﬁned over Fq has some solutions Ai,j = ai,j , Ti = ti in Fq . In this case, monic(gapro (x)) is written by xng−g + tng−g−1 xng−g−1 + ... + t1 x + t0 .

(7)

Thus, the test of the decomposedness of D0 and the computation of the decomposed factors are reduced to ﬁnd the solutions of the system of the equations (6) and factorizations of the polynomials (7). In the next section, we will investigate the case of the hyperelliptic curve. In this case, there is a concrete representation of the Riemann-Roch space, and so we have a more concrete system of equations.

3

Decomposition Attack for the Jacobian of a Hyperelliptic Curve

Now, we discuss the special case of Jacobians of hyperelliptic curves. In this case, there are concrete representations of the Riemann-Roch space and some techniques that g(x) can be taken as a monic polynomial, and from this, a simple system of equations is derived. Let C be a hyperelliptic curve (including an elliptic curve) of genus g of the form C : y 2 = f (x), where f (x) = x2g+1 + a2g x2g + ... + a0 over Fqn where the characteristic of Fq is not 2 and n ≥ 2. Put ∞ by the unique point at inﬁnity on C. Let D0 be a reduced divisor (i.e.,Fqn -rational point of the Jacobian) of C. To represent D0 , we use the so-called Mumford representation: D0 = (φ1 (x), φ2 (x)), where φ1 (x) ∈ Fqn [x] is a monic polynomial with deg(φ1 (x)) ≤ g and φ2 (x) ∈ Fqn [x] satisﬁes deg(φ2 (x)) < deg(φ1 (x)) and f (x) − φ2 (x)2 ≡ 0 mod φ1 (x). In the following, we will assume deg(φ1 (x)) = g. This assumption holds for all but a negligible fraction of divisor classes D0 . Note that there are Q1 , .., Qg ∈ C(Fqn )\{∞} satisfying the equation (1) and the multiset {Q1 , .., Qg } is stable under the action of galois group Gal(Fqn /Fqn ). Similarly, put B0 := {P ∈ C |P = (x, y) ∈ C(Fqn ), x ∈ Fq } as a set of factor basis and large primes. Then, from the Assumption 1 and Assumption 2, we can see easily that the probability, that there are some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed) satisfying the equation (3), is approximately 1/(gn)!, when q ng. In the following, we ﬁx a reduced divisor D0 . So, φ1 (x), φ2 (x), and Q1 , ..., Qg , which are dependent on D0 , are also ﬁxed. In this work, we show the following theorem.

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Theorem 1. Let V1 , V2 , ..., V(n2 −n)g be indeterminates and let D0 be a reduced divisor of C/Fqn . Then there are some computable degree 2 polynomials Ci,j ∈ Fq [V1 , V2 , ..., V(n2 −n)g ]

(0 ≤ i ≤ ng − 1, 0 ≤ j ≤ n − 1)

satisfying the following: The condition that D0 is potentially decomposed is equivalent to the following 1) and 2): 1) The system of equations {Ci,j = 0 | 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1} has some 2 solution v = (v1 , .., v(n2 −n)g ) ∈ A(n −n)g (Fq ). 2) Put ci = Ci,0 (v1 , .., v(n2 −n)g ) for 0 ≤ i ≤ ng − 1. Then G(x) = xng + cng−1 xng−1 + ... + c0 ∈ Fq [x] factors completely. Moreover, if D0 is potentially decomposed, the x-coordinates of the decomposed factors are the solutions of G(x) = 0. From this theorem, the test, whether D0 is potentially decomposed and the computation of the decomposed factors (if possible), is reduced to solving the system of the equations {Ci,j = 0 | 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1} and factorizing the polynomials G(x) obtained form the solutions of the system of these equations. In the following, we construct such multivariable polynomials {Ci,j } and show Theorem 1. From the equation of C, we see ord∞ x = 2, and ord∞ y = 2g + 1. Put N1 := and N2 := ng−g−1 .

(n+1)g 2 2 Lemma 5. 1) N1 + N2 = ng − 1. 2) N2 + g − 1 < N1 . Proof. Trivial. Lemma 6. {1, x, x2 , .., xN1 , y, xy, ...xN2 y} is a base of L((ng + g)∞). Proof. From ord∞ x = 2, ord∞ y = 2g + 1, each element in the above list is in L( (ng +g)∞). The independence is from the deﬁnition of the hyperelliptic curve. Thus, since the number of the elements of the list N1 + N2 + 2 = ng + 1 is the same as the dim L((ng + g)∞) (from Lemma 1), we ﬁnish the proof. Lemma 7 {φ1 (x), φ1 (x)x, ..., φ1 (x)xN1 −g , (y − φ2 (x)), (y − φ2 (x))x, ..., (y − φ2 (x))xN2 } is a base of L((ng)∞ − D0 ) = L((ng + g)∞ − gi=1 Qi ). Proof. From the deﬁnition of φ1 (x) and φ2 (x), each element in the list has a zero at each Qi . Since deg(φ1 (x)) = g, deg(φ2 (x)) ≤ g − 1, and N2 + g − 1 < N1 (from Lemma 5), each element in the list has at most (ng + g) poles at ∞. Then they are in L((ng)∞ − D0 ). Now, we show the independence. Assume they are not independent, and there are some non zero f1 (x), f2 (x) ∈ Fqn [x] such that φ1 (x)f1 (x) + (y − φ2 (x))f2 (x) = 0. However, the relation φ1 (x)f1 (x) + (y − φ2 (x))f2 (x) = 0 induces yf2 (x) ∈ Fqn [x] and f1 (x) = f2 (x) = 0. As this is a contradiction, they are independent. On the other hand, the number of the elements in the list is N1 + N2 + 2 − g = ng − g + 1 from Lemma 5, which is the same as the dim L((ng)∞ − D0 ). So we ﬁnish the proof.

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From Lemma 7, an element h ∈ L((ng)∞ − D0 ) is written by h(x, y) = φ1 (x)(a0 + a1 x+ ...+ aN1 −g xN1 −g )+ (y − φ2 (x))(b0 + b1 x+ ...+ bN2 xN2 ) (8) where ai ,bi are values in Fqn . Lemma 8. Let h(x, y) ∈ L((ng)∞ − D0 ). Assume div(h(x,y)) is written in the form P1 + P2 + ... + Png + gi=1 Qi − (ng + g)∞ for Pi ∈ C(Fqn )\{∞}. Then we have the following: 1) aN1 −g = 0 when ng + g is even. 2) bN2 = 0 when ng + g is odd. Proof. When ng + g is even, assume aN1 −g = 0, thus we have the order of the pole of h(x, y) at ∞ being truly less than ng + g and div(h(x, y)) is not written by the form of (3). Similarly, when ng + g is odd, assume bN2 = 0. Thus we have the order of the pole of h(x, y) at ∞ being truly less than ng + g and div(h(x, y)) is not written by the form of (3). So, we can assume that aN1 −g = 0, if ng + g is even, and bN2 = 0, if ng + g is odd. Now, we compute the intersections of h(x, y) = 0 on C. For this purpose, y must be eliminated. Note that the point (x, y) fulﬁlls h(x, y) = 0, if and only if the equation −φ1 (x)(a0 + a1 x + ... + aN1 −g xN1 −g ) + φ2 (x)(b0 + b1 x + ... + bN2 xN2 ) . b0 + b1 x + ... + bN2 xN2 (9) holds. By this y’s representation, the number of the parameters must be decreased. So, put aN1 −g = 1 when ng + g is even and put bN2 = 1 when ng + g is odd (this can be done from the above lemma). Also put M1 = when ng + g is even N2 N1 − g − 1 when ng + g is even , and M2 = . N1 − g N2 − 1 when ng + g is odd when ng + g is odd Note that M1 + M2 = ng − g − 2 from Lemma 5. Put −(denominator of (9))2 f (x) + (numerator of (9))2 , if ng + g is even s(x) := . (denominator of (9))2 f (x) − (numerator of (9))2 , if ng + g is odd y=

and let S(x) be the multivariable polynomial obtained from the deﬁnition of s(x) replacing the values ai and bi by the indeterminates Ai and Bi . From the construction, S(x) is a monic polynomial of the degree ng + g, whose coeﬃcients are degree 2 polynomials in Fqn [A0 , .., AM1 , B0 , .., BM2 ], and φ1 (x)|S(x). Put g(x) := S(x)/φ1 (x). Since φ1 (x) is a monic polynomial in Fqn [x], g(x) is also a monic polynomial of degree ng, whose coeﬃcients are degree 2 polynomials in Fqn [A0 , .., AM1 , B0 , .., BM2 ]. Put Ci ∈ Fqn [A0 , .., AM1 , B0 , .., BM2 ] by i-th coeﬃcient of g(x), i.e., g(x) = xng + Cng−1 xng−1 + ... + C0 .

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Similarly, for v = (a0 , ...aM1 , b0 , ..., bM2 ) ∈ AM1 +M2 +2 (Fqn ) and some polynomial p(x) in Fqn [A0 , ..., AM1 , BM0 , ..., BM2 , x], let pv (x) be the polynomial obtained from p(x) by substituting ai and bi for Ai and Bi . Then, the zeros of gv (x) = 0 are the x-coordinate of the intersections of h(x, y) = 0 on C except Q1 , ..., Qg . Thus, we have the following lemma. Lemma 9. The condition that D0 is a potentially decomposed reduced divisor is equivalent to the following: There is some v = (a0 , .., aM1 , b0 , ...bM2 ) ∈ AM1 +M2 +2 (Fqn ) such that gv (x) ∈ Fq [x] and gv (x) ∈ Fq [x] factors completely in Fq [x]. We now show how to ﬁnd ai in Fqn (0 ≤ i ≤ M1 ) and bi in Fqn (0 ≤ i ≤ M2 ) such that gv (x) in Fq [x]. Let [α0 (= 1), α1 , .., αn−1 ] be a base of Fqn /Fq and ﬁx this base. Let Ai,j (0 ≤ i ≤ M1 , 0 ≤ j ≤ n − 1) and Bi,j (0 ≤ i ≤ M2 , 0 ≤ j ≤ n − 1) be new indeterminates over Fq . Note that the number of the indeterminates {Ai,j } ∪ {Bi,j } is (M1 + M2 + 2)n = (N1 + N2 − g + 1)n = (n2 − n)g. For simplicity, substitute the variables Ai,j (0 ≤ i ≤ M1 , 0 ≤ j ≤ n − 1) and Bi,j (0 ≤ i ≤ M2 , 0 ≤ j ≤ n − 1) by {V1 , V2 , ..., V(n2 −n)g }. Let us consider the n−1 n−1 polynomials obtained by substituting Ai by j=0 Ai,j αj and Bi by j=0 Bi,j αj in g(x). Also let us denote the coeﬃcients obtained in this way again by Ci . Then the coeﬃcients can be written in the form Ci =

n−1

Ci,j αj ,

Ci,j ∈ Fq [V1 , V2 , ..., V(n2 −n)g ].

j=0

Thus from Lemma 9, the condition gv (x) ∈ Fq [x] is equivalent to the condition that there are some v1 , v2 , ..., v(n2 −n)g ∈ Fq such that Ci,j (v1 , v2 , ..., v(n2 −n)g ) = 0 for 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1. Moreover, when gv (x) ∈ Fq [x], g(x) = xng + Cng−1,0 xng−1 + ... + C0,0 . The condition that gv (x) factors completely in Fq [x] is equivalent to the above condition, and G(x) := xng + cng−1 xng−1 + ... + c0 factors completely in Fq [x] where ci = Ci,0 (v1 , v2 , ..., v(n2 −n)g ). In this case, the solutions of G(x) = 0 are the xcoordinates of the decomposed factor. Then, we ﬁnish the proof of proposition 1 and construct the equation system {Ci,j = 0}.

4

Example

In this section, we examine three computational experiments of the decomposed factors of Jacobian. The computations are done by using the computer algebra

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system magma on a Windows XP preinstalled PC (CPU:Pentium M 2GHz, RAM:1GB). (In order to solve equation system, the function “variety” prepared in magma is used.) We compute three cases 1) (g, n) = (1, 3), 2) (g, n) = (2, 2), and 3) (g, n) = (3, 2) where g and n are the genus and the extension degree of the deﬁnition ﬁeld of the chosen hyperelliptic/elliptic curve, respectively. In all cases, one trial, which means the judge as to whether a given element of Jacobian is decomposed or not and compute its decomposed factor, if it is decomposed, is done within 1 second. Since the probability that an element of Jacobian is decomposed is approximately 1/(gn)!, the amount of the time for obtaining one potentially decomposed reduced divisor is within 6 sec, 24 sec, and 720 sec, respectively. Further, we will give the following three examples. Case 1. Let q = 1073741789(prime number), Fq3 := Fq [t]/(t3 + 456725524t2 + 251245663t + 746495860), and let E/Fq3 be an elliptic curve deﬁned by y 2 = x3 + (1073741788t2 + t)x + (126t + 3969) and P0 := (t, t + 63) ∈ E. We investigate whether nP0 : n = 1, 2, ..30 are decomposed and ﬁnd the following 7 decompositions. (24P0 is written by 2 forms.) 2P0 = (1050861583, 6509843t2 + 387051565t + 920296030) + (742900894, 362262801t2 + 6480079t + 886701711) + (571975376, 938916909t2 + 910769097t + 139897863) 5P0 = (806296922, 113931706t2 + 863383473t + 133427995) + (797256157, 360646567t2 + 663390692t + 1012046566) + (389333914, 986077188t2 + 829314065t + 687783827) 8P0 = (1063441336, 113661172t2 + 942865616t + 744283566) + (894045278, 863335768t2 + 637284565t + 937810737) + (694935460, 740353309t2 + 505910431t + 597402219) 20P0 = (996570058, 341336613t2 + 450680674t + 72874200) + (141768271, 589122734t2 + 930205049t + 713557032) + (73505168, 432994198t2 + 405986289t + 233154172) 24P0 = (529735815, 20343700t2 + 780030904t + 490121669) + (515960254, 269821984t2 + 561547517t + 348990487) + (207183771, 712543643t2 + 356522343t + 895634732) = (818683055, 1034251164t2 + 705927333t + 1062879754) + (754504105, 23461217t2 + 961620879t + 1015889110) + (489159707, 271295793t2 + 600348670t + 1022482426) 26P0 = (628174301, 138296704t2 + 104824480t + 858118320) + (371888603, 417445284t2 + 850151153t + 126970733) + (55411433, 560274594t2 + 609956706t + 821692494) Case 2. Let q = 1073741789(prime number), Fq2 := Fq [t]/(t2 + 746495860t + 206240189), and let C/Fq2 be a hyperelliptic curve deﬁned by y 2 = x5 + (673573223t + 771820244)x + 6t + 9 and let D0 := (x2 + 1073741787tx + 327245929t + 867501600, (1023168391t + 350252228)x + 658555356t + 446913597)

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be a reduced divisor of C. We investigate whether nD0 : n = 1, 2, ..100 are decomposed and ﬁnd the following 9 decompositions. (71D0 is written by 2 forms.) 6D0 ∼ (1025731975, 776505688t + 911495013) + (728060789, 648475468t + 1067025179) + (341799975, 145077925t + 187604034) + (61964999, 227570631t + 639782700) − 4∞ 19D0 ∼ (1039361498, 15180988t + 396695374) + (828360115, 179412594t + 719919461) + (483171045, 677645208t + 604714840) + (34566209, 753841024t + 14375633) − 4∞ 33D0 ∼ (970690833, 608141084t + 889165804) + (260086243, 894605411t + 261264640) + (208957980, 43330622t + 581461318) + (190782894, 124873649t + 510328990) − 4∞ 35D0 ∼ (699447787, 267523741t + 562899544) + (559470007, 197827114t + 99971197) +(472594781, 579187919t+266558458) +(453661772, 449424806t+977318920)− 4∞ 48D0 ∼ (1009979214, 959734525t + 990871450) + (995813251, 44186049t + 288496638) +(521299995, 556594200t+468424666) +(17946008, 977064852t+1071618742)− 4∞ 71D0 ∼ (1019155056, 573896856t + 103042116) + (944470217, 829781939t + 184620624) + (727156004, 462612591t + 582877732) + (281900623, 553507533t + 42660552) − 4∞ ∼ (502979299, 412632304t + 1036827718) + (74527656, 927651409t + 452588110) + (50078888, 801072540t + 888737005) + (2986754, 556402789t + 236723678) − 4∞ 73D0 ∼ (843747137, 682161676t + 600252618) + (829302257, 145878028t + 853397395) +(290487906, 645896278t+279001181) +(184873704, 567002729t+620354511)− 4∞ 80D0 ∼ (907811987, 216534804t + 936839244) + (808513243, 873487475t + 273845273) +(520893378, 757248670t+381150138) +(486203744, 494475019t+791571132)− 4∞

Case 3. Let q = 1073741789(prime number), Fq2 := Fq [t]/(t2 + 746495860t + 206240189), and let C/Fq2 be a hyperelliptic curve deﬁned by y 2 = x7 + (111912375t + 1046743132)x + 6t + 9 and let D0 := (x2 + 1073741787tx + 327245929t + 867501600, (473621736t + 256126568)x + 145989647t + 687383736) be a reduced divisor of C. We investigate whether nD0 : n = 1, 2, ..3000 are decomposed and find the following 6 decompositions. 414D0 ∼ (1001437837, 752632260t+700158497)+(747112084, 656073918t+400137619) + (620249588, 127943213t + 635474623) + (614180498, 206297635t + 445250468) +(515769009, 607297126t+554290493) +(488549466, 627952783t+854182612)− 6∞ 657D0 ∼ (939617127, 695261735t + 239531611) + (933351280, 935312661t + 961494096) + (799612924, 341923983t + 677495100) + (294787599, 279723229t + 760003067) + (273118782053704103t + 577497766) + (153381525, 983211238t + 517037777) − 6∞ 921D0 ∼ (1034634787, 400751409t+829801342)+(763888873, 757155774t+829936954) + (619620874, 800641683t + 200272230) + (603032615, 115219564t + 655011145) +(436423191, 285214454t+450812747) +(125198811, 884750621t+123305741)− 6∞ 1026D0 ∼ (1024020017, 267457905t+41452942)+(794174628, 615676821t+723336407) + (738567269, 433647609t + 128304659) + (629287731, 465842490t + 789390318) + (435082408, 878213106t + 603353206) + (79621979, 479459622t + 672937516) − 6∞ 1121D0 ∼ (764081031, 812350603t+347878564)+(673426715, 687737442t+381588704) + (6102522082007139t + 99219637) + (467560104, 619342780t + 228756808) + (179787786, 333322906t + 75482151) + (59221667, 860686653t + 625301206) − 6∞ 2289D0 ∼ (729358563, 482925408t + 170057124) + (529840657, 42328987t + 857983002)

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+ (514618236, 436901100t + 416530686) + (350106356, 183495333t + 950710579) + (175898979, 411808870t + 427518366) + (96240558, 703780413t + 461022225) − 6∞

5

Conclusion

In this manuscript, we have proposed an algorithm which checks whether a reduced divisor is potentially decomposed or not, and we have computed the decomposed factors, if it is potentially decomposed. From this algorithm, concrete computations of decomposed factors are done by computer experiments when the pairs of the genus of the hyperelliptic curve and the degree of extension ﬁeld are (1, 3), (2, 2), and (3, 2).

Acknowledgment The author would like to thank Professor Kazuto Matsuo in the Institute of Information Security for useful comments and fruitful discussions and Professor Lisa Bond in Kanto Gakuin University for English writing. Also, the author would like to thank the anonymous reviewers who pointed out many mistakes and suggested a revisal plan.

References 1. Adleman, M., DeMarrais, J., Huang, M.-D.: A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 28–40. Springer, Heidelberg (1994) 2. Cantor, D.G.: Computing in the Jacobian of hyperelliptic curve. Math. Comp. 48, 95–101 (1987) 3. Diem, C.: An Index Calculus Algorithm for Plane Curves of Small Degree. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 543–557. Springer, Heidelberg (2006) 4. Diem, C.: On the discrete logarithm problem in class groups (2009) (preprint), http://www.math.uni-leipzig.de/~ diem/preprints/small-genus.pdf 5. Enge, A., Gaudry, P.: A general framework for subexponential discrete logarithm algorithms. Acta Arith. 102(1), 83–103 (2002) 6. Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000) 7. Gaudry, P., Thom´e, E., Th´eriault, N., Diem, C.: A double large prime variation for small genus hyperelliptic decomposed attack. Math. Comp. 76, 475–492 (2007) Preprint Version, http://eprint.iacr.org/2004/153/ 8. Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. Journal of Symbolic Computation 44(12), 1690– 1702 (2009), Preprint version http://eprint.iacr.org/2004/073 9. Hess, F.: Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symb. Comp. 11, 1–22 (2001) 10. Hindry, M., Silverman, J.H.: Diophantine Geometry An introduction. In: Graduate Texts in Math., vol. 201. Springer, Heidelberg (2000)

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11. Granger, R., Vercauteren, F.: On the Discrete Logarithm Problem on Algebraic Tori. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 66–85. Springer, Heidelberg (2005) 12. LaMacchia, B.A., Odlyzko, A.M.: Solving large sparse linear systems over finite fields. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 109–133. Springer, Heidelberg (1991) 13. Nagao, K.: Index calculus for Jacobian of hyperelliptic curve of small genus using two large primes. Japan Journal of Industrial and Applied Mathematics 24(3) (2007); Preprint version entitled by Improvement of Th´eriault Algorithm of decomposed attack for Jacobian of Hyperelliptic Curves of Small Genus, http://eprint.iacr.org/2004/161 14. Semaev, I.: Summation polynomials and the discrete logarithm problem on elliptic curves (2004) (preprint) 15. Th´eriault, N.: Index calculus for hyperelliptic curves of small genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003) 16. Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory IT-32(1), 54–62 (1986)

6

Appendix

In the appendix, we estimate the complexity of the decomposition attack, as a function of q, for ﬁxed g, n (i.e., g, n are considered as constants) under the Assumption 1 and Assumption 2. Here, we apply the ideas of the “Rebalancing method” [5],“One large prime method” [15], and “Two large prime method” [13] [7], which are the techniques of solving discrete logarithm of the Jacobian of a hyperelliptic curve over a general ﬁnite ﬁeld, to our cost estimation for the case of an extension ﬁeld. Note that as g and n are ﬁxed, the input length is linear in log q. These techniques are very complicated, and we only give the outline of the algorithm and estimation of the complexity. In this estimation, since n, g are ﬁxed, the cost for solving the system of the equations is considered as Poly(log q). For simplicity, the terms of Poly(log q)˜ part of the complexity is omitted. For this purpose, we denote the symbol O ˜ where the complexity O(N (q)) is estimated by ˜ (q)) < x(log q)y N (q) for some constants x, y ∈ R>0 , O(N and the symbol ≈ that the relation N1 (q) ≈ N2 (q) is deﬁned by N2 (q) < N1 (q) < x2 (log q)y2 N2 (q) for some constants x1 , x2 , y1 , y2 ∈ R>0 , x1 (log q)y1 where N (q), N1 (q) and N2 (q) are functions of input size q. Now, let G be a general ﬁnite abelian group whose group law is written additively and we consider the general decomposition attack over G. In the following, we also assume that i) The group order is known, and ii) G has a prime order. The assumption ii) is not an essential assumption, but make here for simplicity. Let us now ﬁx a set B0 subset of G.

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Definition 3. Let N be a ﬁxed positive integer(ﬁxed constant). 1) An element of g ∈ G written by g = g1 + .. + gN for g1 , ...gN ∈ B0 is called potentially decomposed. 2) g1 , ..., gN are then called decomposed factors and the multiset {g1 , ..., gN } is called decomposed divisor. Further, we also assume the following iii), iv), v), and vi): iii) The probability that g ∈ G is potentially decomposed is O(1). iv) For a g ∈ G, the cost for checking whether g is potentially decomposed or ˜ not is O(1). v) For the potentially decomposed g ∈ G the cost of computing decomposed ˜ divisor {g1 , ..., gN } from g is O(1). (If there are several decomposed divisors, the computation of all decomposed divisors is needed.) vi) |B0 |2 |G|. ˜ from iv). (Otherwise, the Note that o(|G|) < |B0 |N from ii) and |B0 |N < O(|G|) ˜ ε ) for some ε > 0 expected number of decomposed divisors is bigger than O(q and iv) does not hold.) In the normal index calculus, the number of B0 which are used for the decomposition is basically large (i.e.,N 1). So, the randomly chosen element is basically written by some linear sum of B0 in many ways. However, it is diﬃcult to compute such linear sums, so, by the use of the lifting to integer or number ﬁeld ring and by the use of the sieving method, one can ﬁnd some decomposition of randomly chosen element. So, remark carefully that the prerequisite condition of the normal index calculus for number ﬁeld sieve and that of the decomposition attack for the Jacobian of algebraic curve is quite diﬀerent. In our case (i.e., G being the Jacobian of a hyperelliptic curve of genus g over extension ﬁeld Fqn , B0 being the set of Fqn -rational point of the curve whose x-coordinate lie in Fq , N = ng), iii) is from Assumption 1 and Assumption 2, iv) and v) are from Theorem 1, and vi) is from the notations. Let us now ﬁx a set B subset of B0 . The set B is called the factor base and an element in B0 \B is called a large prime. Definition 4. 1) An element of g ∈ G written by g = g1 + .. + gN for g1 , ...gN ∈ B is called decomposed. 2) An element of g ∈ G written by g = g1 + .. + gN for one gi ∈ B0 \B, and the other gj ∈ B (1 ≤ j ≤ N, j = i) is called almost decomposed. 3) An element of g ∈ G written by g = g1 + .. + gN for two gi1 , gi2 ∈ B0 \B, and the other gj ∈ B (1 ≤ j ≤ N, j = i1 , i2 )is called 2-almost decomposed. 4)In every case, g1 , ..., gN are also called decomposed factors and the multiset {g1 , ..., gN } is called decomposed divisor. Now, we give the outlines of the algorithms named ’rebalancing method’, ’one large prime method’ and ’two large prime method’, which are the variants of the decomposition attack [5], [15], [13], and [7], by Algorithm 1 and Algorithm 2. Note that Algorithm 1 and Algorithm 2 are probabilistic, since they need random numbers. Also note that the probability that r1 a + r2 b is potentially

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Algorithm 1. The outline of the Rebalancing method Input: a, b ∈ G s.t. a = nb for some unknown n ∈ Z/|G|Z. Output: find n. 1: Initializing the list of the relations L = {} 2: while |L| < suitable number N0 do 3: For a pair of random numbers (r1 , r2 ), computing r1 a + r2 b. 4: if r1 a + r2 b being decomposed then 5: adding the informations of (r1 , r2 ) and the decomposed factor to L. 6: (If there are several decomposed factors, choosing one decomposed factor randomly.) 7: Solving the linear algebraic computation of roughly |B| × |B| size, modulo |G| 8: Computing n

Algorithm 2. The outlines of the One (resp. Two)large prime method Input: a, b ∈ G s.t. a = nb for some unknown n ∈ Z/|G|Z. Output: find n. 1: Initializing the list of the relations L = {} 2: while |L| < suitable numberN1 (resp. N2 ) do 3: For a pair of random numbers (r1 , r2 ), computing r1 a + r2 b. 4: if r1 a + r2 b being almost-decomposed (resp. 2-almost decomposed) then 5: adding the informations of (r1 , r2 ) and the decomposed factor to L. 6: (If there are several decomposed factors, choosing one decomposed factor randomly.) 7: Updating L by the elimination of the terms of external elements. 8: Solving the linear algebraic computation of roughly |B| × |B| size, modulo |G| 9: Computing n

decomposed is O(1), since |G| is a prime number and r1 a+ r2 b can be considered as a random element of G. In Algorithm 1 and Algorithm 2, N0 (resp. N1 , resp N2 ) be the number of decomposed (resp. almost decomposed , resp. 2-almost decomposed ) elements of G which are required in the rebalancing method (resp. one large prime method, resp. two large prime method). From the ideas of [5], [15], [13], and [7], the estimations of the following conjecture is expected. Conjecture . 1) N0 is estimated by Const × |B|, i.e., N0 = O(|B|). 2) N12 /|B0 | is estimated by Const × |B|, i.e., N1 = O(|B|1/2 |B0 |1/2 ). 3) N2 is estimated by Const × |B0 |, i.e., N2 = O(|B0 |). Further, we have the following estimations of the complexity. Lemma 10. Under the assumptions of i),ii),iii), iv) v), vi), and Conjecture, we have the following: 1) The complexity of the general decomposition attack taking B as a set of factor basis by the rebalancing method is minimized at |B| ≈ |B0 |N/(N −1) , and it is ˜ 0 |(2N )/(N +1) ). estimated by O(|B 2) The complexity of the general decomposition attack taking B as a set of factor basis and taking B0 \B as a set of large primes by the one large prime method is min˜ 0 |(4N −2)/(2N +1) ). imized at |B| ≈ |B0 |(2N −1)/(2N +1) , and it is estimated by O(|B

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3) The complexity of the general decomposition attack taking B as a set of factor basis and taking B0 \B as a set of large primes by the two large prime method is ˜ 0 |(2N −2)/N ). minimized at |B| ≈ |B0 |(N −1)/N , and it is estimated by O(|B Proof. (Sketch of the proof) In every case, the cost of the part of linear algebra 2 ˜ is O(|B| ), and for the rebalance, which is needed for minimizing the complexity, it is the same as the cost of the collecting divisors. So, we only need to estimate the optimized size |B|. 1)In the case of rebalancing method: The probability that the randomly chosen g ∈ G is a decomposed is O(|B/B0 |N ). So, the cost to obtain one decomposed g ˜ 0 /B|N ). From Conjecture , we must have O(|B|) number of such g. So is O(|B |B0 /B|N · |B| ≈ |B|2 where the left hand side is the cost for collecting enough decomposed group elements, and the right hand side is the cost for the linear algebra. Thus we have |B| ≈ |B0 |N/(N +1) . 2) In the case of one large prime method: The probability that the randomly chosen g ∈ G is an almost decomposed is O(|B/B0 |N −1 ). From Conjecture , we must have O(|B|1/2 |B0 |1/2 ) number of such g. Similarly, we have |B0 /B|N −1 · |B|1/2 |B0 |1/2 ≈ |B|2 and |B| ≈ |B0 |(2N −1)/(2N +1) is obtained. 3) In the case of two large prime method: The probability that the randomly chosen g ∈ G is a 2-almost is O(|B/B0 |N −2 ). From Conjecture , we must have O(|B0 |) number of such g. Similarly, we have |B0 /B|N −2 · |B0 | ≈ |B|2 and |B| ≈ |B0 |(N −1)/N is obtained. Now, we apply this lemma for the decomposition attack for the Jacobian of a curve over an extension ﬁeld. Note that B0 = {P − ∞ |x(P ) ∈ Fq }, |B0 | ≈ q, N = ng and thus, we have the following claim, which is based on the assumptions i),ii),iii),iv),v),vi),and Conjecture. Claim . 1) The complexity of the decomposition attack with the rebalancing ˜ (2ng)/(ng+1) ). method is estimated by O(q 2) The complexity of the decomposition attack with the one large prime method ˜ (4ng−2)/(2ng+1) ). is estimated by O(q 3) The complexity of the decomposition attack with the two large prime method ˜ (2ng−1)/(ng) ). is estimated by O(q

Factoring Polynomials over Local Fields II Sebastian Pauli Department of Mathematics and Statistics University of North Carolina at Greensboro, Greensboro, NC 27412, USA s [email protected]

Abstract. We present an algorithm for factoring polynomials over local fields, in which the Montes algorithm is combined with elements from Zassenhaus Round Four algorithm. This algorithm avoids the computation of characteristic polynomials and the resulting precision problems that occur in the Round Four algorithm.

1

Introduction

Polynomial factorization is fundamental in working with local ﬁelds. In addition to the irreducible factors of a given polynomial, computer algebra systems that support extensions of local ﬁelds (e.g., Magma [1], Sage [16]) require explicit representations of the unramiﬁed and totally ramiﬁed parts of the extensions generated by arbitrary irreducible polynomials, as these systems represent such extensions as a tower of unramiﬁed and totally ramiﬁed extensions. Moreover, there are many applications of global ﬁelds that include the construction of integral bases, decomposition of ideals, and the computation of completions. The algorithms [2,4,7,14] for factoring a polynomial Φ(x) over a local ﬁeld ﬁnd successively better approximations to the irreducible factors of Φ(x) until gaining suﬃcient precision to apply Hensel lifting. The algorithms diﬀer in how the approximations are computed. Algorithms based on the Zassenhaus Round Four algorithm (e.g. [3,4,14]) suﬀer from loss of precision in computing characteristic polynomials and approximating greatest common divisors. The Montes algorithm [10,11,7,8] avoids the computation of characteristic polynomials by exploiting Newton polygons of higher order. Here the most expensive operations are division with remainder and polynomial factorization over ﬁnite ﬁelds. We present the algorithm of Montes in the terminology of [14] and use the techniques of the Round Four algorithm to derive a factorization when a breaking element is found. We also give a complexity analysis. Notation Let K be a ﬁeld complete with respect to a non-archimedian exponential valuation ν with ﬁnite residue class ﬁeld K ∼ = Fq of characteristic p; we call K a local ﬁeld. Assume ν is normalized with ν(π) = 1 for the uniformizing element G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 301–315, 2010. c Springer-Verlag Berlin Heidelberg 2010

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π in the valuation ring OK of K. For γ ∈ OK denote by γ the class γ + (π) in K. The unique extension of ν to an algebraic closure K of K (or to any intermediate ﬁeld) is also denoted ν. In our algorithm we will be concerned with the ﬁrst non-zero coeﬃcient of the expansion of an element in a ﬁnite subextension of K/K. We introduce an equivalence relation on the elements of K which reﬂects this (also see [9]). ∗

∗

Definition 1. For γ ∈ K and δ ∈ K we write γ ∼ δ if ν(γ − δ) > ν(γ) and make nthe supplementary assumption 0 ∼ 0. For ϕ(x) = ϑ(x) = i=0 ϑi xi in K[x] we write ϕ(x) ∼ ϑ(x) if

n

i=0

ϕi xi and

min 0≤i≤n ν(ϕi − ϑi ) > min 0≤i≤n ν(ϕi ). Let L be a ﬁnite extension of K with uniformizing element πL . Two elements γ = γ0 πLv ∈ L and δ = δ0 πLw ∈ L with ν(γ0 ) = ν(δ0 ) = 0 are equivalent with respect to ∼ if and only if v = w and γ0 ≡ δ0 mod (πL ). It follows immediately that the relation ∼ is symmetric, transitive, and reﬂexive.

2

Reducibility

Assume we want to factor a polynomial Φ ∈ OK [x] of degree N . If Φ(x) splits into the product of two co-prime factors over the residue class ﬁeld K of K, say Φ(x) = Φ1 (x) · Φ2 (x), then Hensel lifting yields a factorization of Φ(x) to any given precision. In addition to this classic situation we give two further situations that we can exploit to obtain a factorization of Φ(x). We consider a polynomial ϑ(x) ∈ OK [x] as a representative of an element in the algebra K[x]/(Φ(x)) and determine a polynomial χϑ (x) ∈ K[x] from ϑ(x) such that χϑ (ϑ(ξ)) = 0 for all roots ξ of Φ(x). Definition 2. Let Φ(x) = N j=1 (x − ξj ) ∈ OK [x], where ξj ∈ K for 1 ≤ j ≤ N and ϑ(x) ∈ K[x]. Then we set χϑ (y) :=

N (y − ϑ(ξi )) = resx (Φ(y), y − ϑ(x)). i=1

Assume we ﬁnd ϑ ∈ K[x] such that χϑ (y) = χ1 (y)χ2 (y) with gcd(χ1 , χ2 ) = 1. Reordering the roots ξi (1 ≤ i ≤ N ) of Φ(x) if necessary, we may write χ1 (y) = (y − ϑ(ξ1 )) · · · (y − ϑ(ξr )) and χ2 (y) = (y − ϑ(ξr+1 )) · · · (y − ϑ(ξN )), where 1 ≤ r < N and obtain a proper factorization of Φ(x): Φ(x) = gcd(Φ(x), χ1 (ϑ(x))) · gcd(Φ(x), χ2 (ϑ(x))).

(1)

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Definition 3. We say a polynomial ϑ(x) ∈ K[x] with χϑ (t) ∈ OK [t] passes the Hensel test if χϑ (t) = ρ(t)g for some irreducible polynomial ρ(t) ∈ K[t]. If ϑ(x) ∈ K[x] fails the Hensel test, that is, χϑ (y) splits into two co-prime factors over K, say χϑ (y) = χ1 (y)χ2 (y), then Hensel lifting yields a factorization χϑ (y) = χ1 (y)χ2 (y) and equation (1) gives a proper factorization of Φ(x). ∗ Definition 4. For ϑ ∈ K[x] we set vΦ (ϑ) := minΦ(ξ)=0 ν(ϑ(ξ)) and say the polynomial ϑ(x) passes the Newton test if ν(ϑ(ξ)) = ν(ϑ(ξ )) for all roots ξ and ξ of Φ(x).

If ϕ(x) ∈ K[x] fails the Newton test, the Newton polygon of χϕ (y) consists of at ∗ least two segments. Let h/e = vΦ (ϕ) be the minimum of the valuations ν(ϕ(ξi )) (1 ≤ i ≤ N ) in lowest terms. Then −h/e is the gentlest slope of the segments of the Newton polygon of χϕ (y). We set ϑ(x) := ϕ(x)e /π h and obtain ν(ϑ(ξ)) = 0 for all roots ξ of Φ(x) with ν(ϕ(ξ)) = h/e and ν(ϑ(ξ)) > 0 for all roots ξ of Φ(x) with ν(ϕ(ξ)) > h/e. Thus χϑ (t) splits into two co-prime factors and the considerations above yield a proper factorization of Φ(x).

3

Irreducibility and the Sequence ϕt(x) t

In the polynomial factorization algorithm we construct a sequence of polynomials ϕt (x) ∈ OK [x] such that ν(ϕt+1 (ξ)) > ν(ϕt (ξ)) for all roots ξ of Φ(x) until we either ﬁnd a polynomial that fails the Newton test, which leads to a factorization of Φ(x) or we have established the irreducibility of Φ(x). If we assure that the degrees of the polynomials ϕt (x) are less than or equal to the degree of all irreducible factors of Φ(x), we either obtain a factorization of Φ(x) or we establish the irreducibility of Φ(x) in ﬁnitely many steps [14]: Theorem 5. Let ξ1 , . . . , ξN be elements of an algebraic closure of a local ﬁeld K and assume the following hypotheses hold. N – Φ(x) = j=1 (x − ξj ) is a square-free polynomial in OK [x]. – ϕ(x) ∈ K[x]. – N ν(ϕ(ξj )) > 2ν(disc Φ) for 1 ≤ j ≤ N . – The degree of any irreducible factor of Φ(x) is greater than or equal to deg ϕ. Then N = deg ϕ and Φ(x) is irreducible over K. While we construct the sequence of polynomials ϕt (x) we gather information about the extensions generated by the irreducible factors of Φ(x). In particular we will at all times know divisors Et and Ft of the ramiﬁcation index and inertia degree of these extensions respectively. If we ﬁnd that not all of these extensions have the same inertia degree and ramiﬁcation index, we will have encountered a polynomial that fails the Hensel or the Newton test. On the other hand if Et · Ft = deg Φ we know that Φ(x) is irreducible.

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Definition 6. Let Φ(x) ∈ OK [x] be irreducible and let ξ be a root of Φ(x). We call a pair of Π(x) ∈ K[x] and Γ (x) ∈ K[x] with ν(Π(ξ)) = 1/E polynomials and F = K Γ (ξ) : K such that E ·F = deg Φ a two element certiﬁcate for the irreducibility of Φ(x). Remark 7. If a two element certiﬁcate exists then Φ(x) is irreducible and an integral basis of the extension of K(ξ)/K generated by a root ξ of Φ(x) is given by the elements Γ (ξ)i Π(ξ)j with 0 ≤ i ≤ F − 1 and 0 ≤ j ≤ E − 1. In the polynomial factorization algorithm we construct a sequence of polynomials (ϕt (x))t∈N where ϕt ∈ OK [x] such that 1. ν(ϕt+1 (ξ)) > ν(ϕt (ξ)) for all roots ξ of Φ(x), 2. ν(ϕt (ξ)) = ν(ϕt (ξ )) for all roots ξ and ξ of Φ(x), and 3. the degree of ϕt (x) is less than or equal to the degree of any irreducible factor of Φ(x). In the following we assume that all polynomials that occur in our constructions pass the Hensel and Newton tests, as we can otherwise derive a factorization of Φ(x). For convenience of notation we deﬁne: ∗ ∗ Definition 8. If vΦ (ϕ−ϑ) > vΦ (ϕ) for polynomials ϕ(x) ∈ K[x] and ϑ(x) ∈ K[x] n we write ϕ ∼ ϑ. For polynomials χ(y) = i=0 ai (x)y i ∈ K[x][y] and τ (y) = Φ n i i=0 bi (x)y ∈ K[x][y] we write χ(y) ∼ τ (y) if Φ

∗ (ai min 0≤i≤n vΦ

4

∗ − bi ) > min 0≤i≤n vΦ (ai ).

The First Iteration

N Let Φ(x) = i=0 ci xi and ϕ1 (x) := x ∈ OK [x]. Assume the Newton polygon of Φ(x) consists of one segment and let −h1 /E1 be its slope in lowest terms. Then ν(ϕ1 (ξ)) = ν(ξ) = h1 /E1 for all roots ξ of Φ(x). This implies that the ramiﬁcation index of all extension generated by irreducible factors of Φ(x) is divisible by E1 . Let β ∈ K with β E1 = π h1 where π is the uniformizing element of K. We ﬂatten the Newton polygon of Φ(x) so that it lies on the x-axis: Φ (y) :=

N

Φ(βy) = ci β i−N y i . βN i=0

Because we can only have ν(ci β i−N ) = 0 when E1 | i, we have

Φ (y) ∼

N/E1

cj·E1 π h1 (j−N/E1 ) y j·E1 .

j=0

Replacing y

E1

by z yields N/E1

A1 (z) :=

j=0

cj·E1 π h1 (j−N/E1 ) z j .

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The polynomial A1 (z) ∈ K[z] is called the associated polynomial [11,10] or residual polynomial [7,8] of Φ(x) with respect to ϕ1 (x). Assume that A1 (z) = ρ1 (z)r for some irreducible polynomial ρ1 ∈ K. Otherwise ϕ1 (x)E1 /π h1 = xE1 /π h1 would fail the Hensel test and (1) would yield a factorization of Φ(x). All ﬁelds K(ξ), where ξ is a root of Φ(x), contain an element ξ E1 /π h1 , whose minimal polynomial is a power of ρ1 (z) over K[z]; therefore their ramiﬁcation indices are divisible by F1 := deg ρ1 . Let γ1 ∈ K be a root of a lift ρ1 (z) ∈ OK [z] of ρ1 (z). In the unramiﬁed extension K1 := K(γ1 ) we have the relation xE1 ∼ π h1 · γ1 . Since Φ ν ρ1 (ϕ1 (ξ)E1 /π h1 ) > 0 for all roots ξ of Φ(x), we get

1 ϕ1 (ξ)E1 h1 F1 > ν(π h1 ) = ν ϕE ρ1 ν π 1 (ξ) > ν ϕ1 (ξ) = ν(ξ). h 1 π We set ϕ2 (x) := π h1 F1 ρ1 (ϕ1 (x)E1 /π h1 ) and continue the construction of our sequence of polynomials (ϕt )t . Obviously deg ϕ2 = E1 F1 , which divides the degree of every irreducible factor of Φ(x). Remark 9. Because the Newton polygon of ϕ2 (x) consists of one segment of slope −h1 /E1 with gcd(h1 , E1 ) = 1 and its associated polynomial with respect to x is ρ1 (z) of degree F1 , the extensions K(α), where α is a root of ϕ2 (x), have inertia degree F1 and ramiﬁcation index E1 . Hence ϕ2 (x) with deg ϕ2 = E1 F1 is irreducible.

5

The Second Iteration

Definition 10. Let Φ(x) ∈ OK [x] of degree N and ϕ(x) ∈ OK [x] of degree n be monic polynomials and assume n | N . We call N/n

Φ(x) =

ai (x)ϕi (x)

i=0

with deg(ai ) < deg(ϕ) the ϕ-expansion of Φ(x). We use the ϕ2 -expansion of Φ(x) to ﬁnd the valuations ν(ϕ2 (ξ)). Set n2 := deg ϕ2 N/n and let Φ(x) = i=0 2 ai (x)ϕi2 (x) be the ϕ2 -expansion of Φ(x). For each root ξ of Φ(x) we have N/n2 0 = Φ(ξ) = ai (ξ)ϕi2 (ξ). i=0

Hence χ2,ξ (y) =

m

ai (ξ)y i ∈ OK(ξ) [y]

i=0

with m = N/n2 = deg(Φ)/ deg(ϕ2 ) is a polynomial with root ϕ2 (ξ). Assume n2 −1 ai,j xj . As the valuations that ai (x) = j=0 ∗ (ϕ1 ) = vΦ

h1 (E1 − 1)h1 ∗ , . . . , vΦ (ϕ1E1 −1 ) = E1 E1

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are distinct (and not in Z) and ϕ1 (x)E1 1, ∼ γ1 , . . . , Φ π h1

ϕ1 (x)E1 π h1

F1 −1

∼ γ1F1 −1 Φ

are linearly independent over K, we have ∗ vΦ (ai ) =

min

0≤j≤n2 −1

ν(ai,j )(h1 /E1 )j.

If the Newton polygon of χ2,ξ (y) consists of more than one segment then ϕ2 (x) fails the Newton test and we can derive a factorization of Φ(x). Otherwise let −h2 /e2 be the slope of the Newton polygon of χ2,ξ (y) in lowest terms. Then ν(ϕ2 (ξ)) = h2 /e2 for all roots ξ of Φ(x). We set E2+ := e2 / gcd(E1 , e2 ). For all roots ξ of Φ(x) the ramiﬁcation index of K(ξ) is divisible by E2 := E1 · E2+ . Because the denominator of E2+ h2 /e2 is a divisor of E1 there is ψ2 (x) := π sπ ϕ1 (x)s1 = π sπ xs1 ∈ K[x] ∗ with s1 ∈ {0, . . . , E1 − 1} and sπ ∈ Z such that vΦ (ψ2 ) = E2+ h2 /e2 . + We ﬂatten the Newton polygon of χ2,ξ (y). Let β ∈ K with β E2 = ψ2 (x) and consider the polynomial χ2,ξ (y) := χ2,ξ (βy)/β m . As only the valuations of the +

coeﬃcients of y i·E2 (0 ≤ i ≤ m/E2+ ) can be zero we get χ2,ξ (y)

m/E2+

=

i=0 m/E2+

=

i=0 E1

+

2

+

+

ai·E + (ξ)ψ2 (ξ)i−m/E2 y i·E2 ∈ K2 [y]. 2

∼ π h1 · γ1 , which is independent of ξ, we ﬁnd coeﬃcients

Using the relation x

Φ

i−m/E2+

ai ∼ ai·E + (x)ψ2 ai ∈ K1 with Φ

+

ai·E + (ξ)β i·E2 −m y i·E2

2

m/E2+

A2 (z) :=

(x). We set

ai z i ∼ Φ

i=0

m/E2+

i=0

i−m/E2+

ai·E + (x)ψ2 2

(x)z i

and obtain the associated polynomial A2 (z) ∈ K1 [z] of Φ(x) with respect to ϕ2 (x). If A2 (y) splits into two or more co-prime factors over K1 = K(γ1 ), we can derive a factorization of Φ(x): Since deg ψ2 (x) is less than the degree of any irreducible factor of Φ(x) we have gcd(ψ2 (x), Φ(x)) = 1 and the extended Euclidean algorithm yields ψ2−1 (x) ∈ OK1 [x] such that ψ2 (x) · ψ2−1 (x) ≡ 1 mod Φ(x). The E+

polynomial ϕ2 2 (x) · ψ2−1 (x) fails the Hensel test. Otherwise A2 (z) = ρ2 (z)r2 for some irreducible polynomial ρ2 (z) ∈ K1 [z]. We set K2 := K(γ2 ) where γ2 is a root of a lift ρ2 (z) ∈ OK1 [z] of ρ2 (z) ∈ K1 [z], let +

F2+ := deg ρ2 , and obtain ϕ2 (x)E2 ∼ γ2 ψ2 (x). Φ

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307

∗ ∗ Next we construct ϕ3 (x) ∈ OK [x] with vΦ (ϕ3 ) > vΦ (ϕ2 ) and deg ϕ3 = E2 F2 . The coeﬃcients of ρ2 (z) ∈ OK1 can be written as polynomials in γ1 ∼ xE1 /π h1 , Φ say +

F2 F1 −1

ρ2 (z) =

ri,j γ1j z i

i=0 j=0

where ri,j ∈ OK . We are looking for

F2+

ϕ3 (x) ∼ ψ2 (x) Φ

ρ2

+

ϕ2 (x)E2 ψ2 (x)

+

=

F2 F1 −1

ri,j

i=0 j=0

xE1 π h1

j

+

ψ2 (x)F2

−i

+

ϕ2 (x)iE2

∗ ρ1 (xE1 /π h1 ) > 0. If we write with deg ϕ3 = E2 F2 = E2+ F2+ E1 F1 . We have vΦ ρ1 (z) = z F1 + ρ∗1 (z) with deg(ρ∗1 ) < F1 this implies 1 F1 ϕE ∼ −(π h1 )F1 ρ∗1 1

Φ

xE1 π h1

.

It follows that we can ﬁnd a polynomial Ri,j (x) with deg Ri,j < E1 F1 such that

Ri,j (x) ∼ ri,j Φ

xE1 π h1

j

F2+ −i

ψ2 (x)

= ri,j

xE1 π h1

j

+

(π sπ xs1 )F2

−i

.

Thus the polynomial E2+ F2+

ϕ3 (x) = ϕ2 (x)

F2+ −1 F1 −1

+

i=0

+

Ri,j (x)ϕ2 (x)iE2

j=0

∗ ∗ has the desired properties vΦ (ϕ3 ) > vΦ (ϕ2 ) and deg ϕ3 = E2 F2 .

Remark 11. ϕ3 (x) ∈ OK [x] is irreducible.

6

Data and Relations

In the algorithm we continue the construction of the sequence of polynomials (ϕt )t from the previous two sections. In the following steps the computation of ψt (x), the valuation of the coeﬃcients ai (x) of the ϕt -expansion of Φ(x), the coeﬃcients of the associated polynomial, and ϕt+1 becomes more involved and relies on the data computed in the previous iteration. We initially set K0 := K,

ϕ1 := x,

E0 := 1,

and compute the following data in every iteration:

F0 := 1

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S. Pauli

ϕt (x) ∈ OK [x]

∗ ∗ with vΦ (ϕt ) > vΦ (ϕt−1 ) and nt = deg(ϕt ) = Et−1 Ft−1 ; an approximation to an irreducible factor of Φ(x)

∗ ht /et = vΦ (ϕt ) et Et+ = gcd(Et−1 , et ) Et = Et+ ·Et−1 t−1 ψt (x) = π sπ i=1 ϕsi i

with gcd(ht , et ) = 1 the increase of the maximum known ramiﬁcation index the maximum known ramiﬁcation index E+

∗ ∗ with sπ ∈ Z and 0 ≤ si < Ei+ such that vΦ (ψt ) = vΦ (ϕt t )

At (y) ∈ Kt−1 [y]

the associated polynomial of Φ(x) with respect to ϕt (x)

ρt (y) ∈ Kt−1 [y]

irreducible with ρtrt (y) = At (y)

γt ∈ K t

such that ϕt t ∼ γt ψt

Kt = Kt−1 (γt )

the maximum known unramiﬁed subﬁeld

Ft+ = [Kt : Kt−1 ]

the increase of the maximum known inertia degree

Ft =

7

E+

Φ

Ft+ ·Ft−1

the maximum known inertia degree

The u-th Iteration

Assume we have computed the data and relations given above for t up to u − 1 and that ϕu (x) of degree nu = Eu Fu is the best approximation to an irreducible factor of Φ(x) found so far. We compute the ϕu -expansion Φ(x) = N/nu N/nu i i i=0 ai (x)ϕu (x) of Φ(x) and set χu (y) := i=0 ai (x)y . Definition 12. Let a(x) ∈ OK [x] with deg a < Et−1 Ft−1 . We call + + Et−1 Ft−1 −1

a(x) =

jt−1 ϕt−1 (x)

jt−1 =0

E2+ F2+ −1

···

ϕj22 (x)

j2 =0

E1 F1 −1

xj1 · aj1 ,...,jt−1 ,

j1 =0

where aj1 ,...,jt−1 ∈ OK (0 ≤ ji ≤ Ei , 0 ≤ i ≤ t), the (ϕ1 , . . . , ϕt−1 )-expansion of a(x). From the (ϕ1 , . . . , ϕu−1 )-expansion of ai (x) we obtain the valuations of ai (ξ) and see that they are independent of the choice of the root ξ of Φ(x). Since, by construction, the values E + −1

∗ ∗ ∗ ∗ vΦ (ϕ1 ), . . . , vΦ (ϕ1E1 −1 ), vΦ (ϕ2 ), . . . , vΦ (ϕ2 2

E+

u−1 ∗ ∗ ), vΦ (ϕ3 ), . . . . . . , vΦ (ϕu−1

−1

)

are distinct (and not in Z) and for 0 ≤ t ≤ u − 1 the elements +

F + −1

1, γt ∼ ϕt (x)Et /ψt (x), . . . , γt t Φ

F + −1 + ∼ ϕt (x)Et /ψt (x) t Φ

are linearly independent over Kt−1 = K(γ1 , . . . , γt−1 ) we have (see [7, Lemma 4.21]):

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309

Lemma 13. Let a(x) ∈ OK [x] with deg a < Et−1 Ft−1 and let aj1 ,...,jt−1 , with 0 ≤ ji < Ei+ Fi+ − 1, be the coeﬃcients of the (ϕ1 , . . . , ϕt−1 )-expansion of a(x). Then jt−1 ∗ ∗ vΦ ϕt−1 (x) · · · ϕj22 (x) · xj1 · aj1 ,...,jt−1 . (a) = min vΦ 1≤i≤t−1 + 1≤ji <E i

If the Newton polygon of χt (y) consists of one segment, say of slope −hu /eu , with u gcd(hu , eu ) = 1, then ϕt (x) passes the Newton test. We set Eu+ := gcd(Eeu−1 ,eu ) and construct u−1 sπ ψu (x) = π ϕt (x)st t=1 ∗ with sπ ∈ Z and 0 ≤ st < (1 ≤ t < u) such that vΦ (ψu ) = Eu+ hu /eu using the following algorithm. For q ∈ Q we denote by den(q) the denominator of q in lowest terms.

Et+

Algorithm 14 (Psi) ∗ Input: vΦ (ϕi ) and Ei+ for 0 ≤ i ≤ t, E = E0+ · · · Et+ , v ∈ Q with E |den(v). ∗ Output: sπ ∈ Z, 0 ≤ si ≤ Ei+ (1 ≤ i ≤ t) such that vΦ (π sπ ϕs00 · · · ϕst t ) = v. – d ← E, i ← t – for i from t to 1 by −1: ∗ • d ← d/Ei+ , v ← v · d, e ← vΦ (ϕi ) · d • Find si such that e · si ≡ v mod den(d · e) ∗ • v ← v − si vΦ (ϕi ) – sπ ← v – return sπ , s1 , . . . , st Next we determine the associated polynomial Au (y) of Φ(x) with respect to ϕu (x). Because we have representations of ai (x) (0 ≤ i ≤ N/ni ) and ψu (x) by + power products of π, ϕ1 , . . . , ϕu−1 we can use the relations ϕt (x)Et ∼ γt ψt (x) Φ

+

ai ∼ ai·Eu+ (x)ψu (x)i−m/Eu . We get to ﬁnd the coeﬃcients ai ∈ Ku−1 such that Φ

the associated polynomial + m/Eu

Au (z) =

ai z i

i=0

where m = N/nu . Assume that Au (z) = ρu (z)r for some irreducible polynomial +

ρu (z) ∈ Ku−1 (z). Otherwise we can ﬁnd ϑ(x) ∈ K[x] with ϑ(x) ∼ ϕu (x)Eu /ψu (x) Φ

that fails the Hensel test, which yields a factorization of Φ(x). Let ρu (z) ∈ Ku−1 be a lift of ρu (z), and set Fu+ := deg ρu . Finally we construct ϕu+1 (x) ∈ OK [x] of degree Eu Fu = Eu+ Fu+ Eu−1 Fu−1 such that +

ϕu+1 (x) ∼ Φ

Fu i=0

+

+

E+

ϑi (x)ϕu (x)iEu = ψu (x)Fu ρu (ϕu u (x)/ψu (x)),

(2)

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S. Pauli

where the ϑi (x) are sums of power products of π, ϕ1 , . . . , ϕu−1 . For t = u − 1, u − 2, . . . , 0 we recursively apply

E + ϕt t ∗ ρt >0 vΦ ψt to reduce the maximum exponent of ϕt (x) to Et+ Ft+ − 1, such that the de+ + gree of the ϕt (x) term is at most deg(ϕt (x)Et Ft −1 ) = (Et−1 Ft−1 )(Et+ Ft+ − 1). Thus we can ﬁnd a ϕu+1 (x) that fulﬁlls the degree condition deg ϕu+1 = Eu Fu . Furthermore

+ + ϕu (x)Eu Fu+ F ∗ ∗ ∗ ∗ vΦ (ϕu+1 ) = vΦ ψu ρu > vΦ ψu u ≥ vΦ (ϕu ). ψu (x) As a preparation for the next iteration we set Ku := Ku−1 (γu ) with γu a root of + ρu (z) and obtain the relation ϕEu (x) ∼ γu ψu (x). Φ

Remark 15. ϕu+1 (x) ∈ OK [x] is irreducible.

8

The Algorithm

We summarize the steps for the construction of the sequence (ϕt (x))t in an algorithm. Although we use the unramiﬁed extensions Kt /K above and in the algorithm, in practice the γi are represented as elements in the residue class ﬁeld Kt . Furthermore, many of the manipulations in the algorithm can be conducted on the representations of ψt (x) as power products of π, ϕ1 (x), . . . , ϕt−1 (x) and of ai (x) as sums of power products of π, ϕ1 (x), . . . , ϕt−1 (x) thus reducing these operations to operations of vectors of integers. Algorithm 16 (Polynomial Factorization) Input: a monic, separable, squarefree polynomial Φ(x) over a local ﬁeld K. Output: a proper factorization of Φ(x) if one exists, a two-element certiﬁcate for Φ(x) otherwise. (1) Initialize t ← 1, ϕ1 (x) ← x, E0 = 1, F0 = 1, K0 = K. (2) Repeat: deg ϕt (a) Find the ϕt expansion Φ(x) = N/ ai (x)ϕ(x)i of Φ(x). i=1 ∗ (b) Find vΦ (ai ) for 0 ≤ i ≤ N/ deg ϕt . (c) If ϕt (x) fails the Newton test: return a proper factorization of Φ(x). et ∗ (d) ht /et ← vΦ (ϕ) with gcd(ht , et ) = 1; Et+ ← gcd(e ; Et ← Et+ · Et−1 . t ,E) t−1 ∗ ∗ (ψt ) = Et+ vΦ (ϕt ), sπ ∈ N, (e) Construct ψt (x) = π sπ i=1 ϕi (x)si with vΦ + 0 ≤ si < Ei (1 ≤ i ≤ t − 1), deg ψt < Ei Fi . (f ) Compute the associate polynomial At (z). (g) Find a factorization of At (z) ∈ Kt (z). (h) If At (z) has two co-prime factors: return a proper factorization of Φ(x).

Factoring Polynomials over Local Fields II

311

(i) Ft+ ← deg ρ where ρt (z)r = At (z), ρt (z) ∈ Kt−1 [z] irreducible; Ft ← Ft+ · Ft−1 , Kt ← K[x]/(ρt (x)). (j) If Et Ft = deg Φ: return a two-element certiﬁcate for Φ(x). + (k) Find ϕt+1 (x) ∼ ρt ϕt (x)Et ψt (x)deg(ρ) of degree nt+1 = Et Ft in OK [x]. (l) t ← t + 1.

Φ

Certificates for Irreducibility If Φ(x) is irreducible we will have Et Ft = N for some t. We obtain the two element certiﬁcate (Deﬁnition 6) for the irreducibility of Φ(x) as follows. A poly∗ nomial Π(x) ∈ K[x] with vΦ (Π) = 1/Et can be found using Algorithm 14. If Ft = 1 we can choose Γ (x) = x. If Ft = 1, let i be maximal with Fi+ = 0. We + ﬁnd Γ (x) ∈ K[x] with Γ (x) ∼ ϕi (x)Ei /ψi (x). Φ

9

Complexity

We restrict our analysis of the complexity of the algorithm to the main loop. The ﬁrst complexity estimate for the Montes algorithm, restricted to irreducibility testing, was given by Veres [17] and improved by Ford and Veres [5]. The complexity estimate for determining the irreducibility of a polynomial Φ(x) ∈ Zp [x] of degree N using this algorithms is O(N 3+ε ν(disc Φ) + N 2+ε ν(disc Φ)2+ε ). The running time of the Round Four algorithm is analyzed in [14], but without taking into account the precision loss in the computation of greatest common divisors. Both estimates rely on Theorem 5 to bound the number of iterations and the required precision and only diﬀer slightly in the exponent of the discriminant of Φ(x). Lemma 17. Let Φ(x) ∈ OK [x] be of degree N and let ϕ(x) ∈ OK [x] be monic of degree n. Then the ϕ-expansion of Φ(x) can be computed in O(N 2 ) operations in OK . N/n Proof. In order to determine the ϕ-expansion Φ(x) = i=1 ai (x)ϕ(x)i we ﬁrst compute q0 (x), a0 (x) ∈ OK [x] with Φ(x) = ϕ(x)q0 (x) + a0 (x), which can be done in O((N − n)n) operations in OK [x]. Next we determine q1 (x), a1 (x) ∈ OK [x] with q0 (x) = ϕ(x)q1 (x) + a1 (x) (O((N − 2n)n) operations in OK [x]), and so on. Therefore the ϕ-expansion of Φ(x) can be computed in ⎛ ⎛ ⎞⎞ N/n 2 N −n O((N −n)n)+O((N −2n)n)+· · ·+O((2n)n) = O⎝n ⎝ i⎠⎠ = O(N 2 ) n i=0 operations in OK . The computation of the (ϕ1 , . . . , ϕt−1 )-expansion of a polynomial a(x) ∈ OK [x] of degree m ≤ deg ϕt −1 consists of the recursive computation of ϕt−1 , ϕt−2 , . . . ,

312

S. Pauli

ϕ2 , and ϕ1 -expansions. Let ni = deg ϕi (1 ≤ i ≤ t). The ϕt−1 -expansion of a(x) yields up to m/nt−1 polynomials of degree less than nt . The ϕt−2 -expansions of these polynomials yield up to m/nt−1 · nt−1 /nt−2 = m/nt−2 of degree less than nt−2 . Thus the (ϕ1 , . . . , ϕt−1 )-expansion of a(x) can be computed in m 2 2 + · · · + O + O(m) O m2 + O nmt nt 2 + O nm n n t−1 1 n1 t−1 operations in OK . Because ni+1 /ni ≥ 2 this is less than 2 2 log m + O(m) = O m2 i=0 2 2−i = O(m2 ). O m2 + O m2 + · · · + O 2m t−1 Lemma 18. The (ϕ0 , . . . , ϕt−1 )-expansion of a(x) ∈ OK [x] with m = deg a ≤ deg ϕt − 1 can be computed in O(m2 ) operations in OK . ∗ (ϕt ) > 2ν(disc Φ) for By Theorem 5 the polynomial Φ(x) is irreducible, if N vΦ ∗ ∗ some t ∈ N. In every iteration the increase from vΦ (ϕt ) to vΦ (ϕt+1 ) is at least 2/N , unless E = N , but that would imply irreducibility. Thus the algorithm terminates after at most ν(disc Φ) iterations. In our analysis of the cost of the steps in the main loop we exclude the cost of ﬁnding a proper factorization to a desired precision using the methods of section 2 in steps (c) and (h). We assume that two polynomials of degree up to n can be multiplied in O(n log n log log n) = O(n1+ε ) operations in their coeﬃcient ring [15].

(a,b,c,d) By Lemma 18 the ϕt -expansion Φ(x) = ϕt (x)N/nt +

N/nt −1

ai (x)ϕt (x)i

i=0

of Φ(x) and the (ϕ1 , . . . , ϕt )-expansion of the ai (x) can be computed in O(N 2 ) operations in OK . (e) The exponents sπ , s1 , . . . , st−1 in ψt (x) = π sπ ϕ1 (x)s0 · · · ϕt−1 (x)st−1 with ∗ vΦ (ψ) = ht /et can be computed with Algorithm 14. The most expensive computation is the extended Euclidean construction, which for integers less than N runs in time O((log N )2 ), at most log2 N times. (f ) We have a representation of ai (x)ψt (x)i−(N/nt ) (1 ≤ i ≤ N/nt ) as nt sums of power products of π, ϕ1 (x), . . . , ϕt−1 (x). In this representation only the exponents of ϕi (x) where Ei+ Fi+ = 1 are non-zero. There are at most log2 N such indices i. Let mt be the number of i < t with Ei+ Fi+ = 1. Reducing the coeﬃcients of the associated polynomial in this representation using the mt + relations ϕi (x)Ei /ψi (x) ∼ γi (1 ≤ i ≤ mt ) takes at most N i=1 i = Φ

O(N (log N )2 ) integer additions and N (t − 1) = O(N log N ) multiplications in the ﬁnite ﬁeld Kt with q F elements. (g,h) The factorization of a polynomial of degree at most N/F over a ﬁnite ﬁeld with at most q F elements can be done in O((N/F )2 log q F ) bit operations [6].

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(j) The cost of ﬁnding the exponents for the representation of Π(x) ∈ K[x] with ∗ vΦ (Π) = 1/E as a power product of π, ϕ1 (x), . . . , ϕt (x) is the same as the cost of ﬁnding ψ(x) in step (f ). The polynomial Γ (x) can be computed in the same way as the coeﬃcients ϑi (x) in step (l). + (k) The polynomial ϕt+1 (x) is constructed as a polynomial in ϕt (x)Et of degree Ft+ with coeﬃcients ϑi (x), 0 ≤ i ≤ Ft+ , (see (2)), obtained from the representations of the elements γu as ϕu (x)Eu /ψu (x) and ∗ vΦ ρu (ϕu (x)Eu /ψu (x)) > 0 for 1 ≤ u ≤ t−1. This is done by manipulating the exponents in the representation of the polynomials as sums of power products of π, ϕ1 (x), . . . , ϕt (x). + The computation of ϕt (x)Et takes log2 Et multiplications of polynomials of + j degree up to Et+ Et−1 Ft− t < N . For 2 ≤ j ≤ Ft+ the polynomial ϕt (x)Et can be computed in Ft+ multiplications of polynomials of degree up to Et Ft < N . For 1 ≤ t − 2 the exponent of ϕi (x) in the representation of ϑi (x) as a power product of ϕ1 (x), . . . , ϕt−1 (x) is less than Ei+ Fi+ . This gives less than log N multiplications of polynomials of degree less than N . As in (e) the exponents of at most log N of the ϕi (x) are nonzero. Therefore in total this step can be conducted in O(N 2+ε ) operations in OK [x]. ∗ By Theorem 5 the maximum of the valuations ν(vΦ (ξ)), where ξ is a root of Φ(x), is less than 2 ν(disc Φ) /N . This is also the maximal (absolute) slope of the Newton polygon of the polynomials under consideration. Therefore a precision of 2ν(disc Φ) is suﬃcient for all operations in the main loop.

Theorem 1. Let p be a ﬁxed prime. We can ﬁnd a breaking element or a two element certiﬁcate for the irreducibility of a polynomial Φ(x) ∈ Zp [x] in at most O(N 2+ε ν(disc Φ)2+ε ) operations of integers less than p.

10

Example

We show that Φ(x) = x32 + 16 ∈ Z2 [x] is irreducible using Algorithm 16. Initially we set ϕ1 (x) = x, E0 = 1, F0 = 1, K0 = Q2 . 32 (a) The ϕ1 -expansion of Φ(x) is Φ(x) = i=0 ai (x)ϕ0 (x)i = x32 + 16. ∗ ∗ (b) The valuations of the coeﬃcients are vΦ (a0 ) = 4, vΦ (ai ) = ∞ for 1 ≤ i ≤ 31, ∗ and vΦ (a32 ) = 0. 4 ∗ (c,d) ϕ1 (x) passes the Newton test; we get vΦ (ϕ1 ) = he11 = 32 = 18 , so E1+ = 8 and E1 = 8. E+ ∗ ∗ (e) We set ψ1 (x) = 2 as vΦ (ϕ1 1 ) = vΦ (x8 ) = 1. 4 (f,g) A1 (z) = z + 1 with A1 (z) = (z − 1)4 in F2 [z]. 8 + (h,i) ϕψ11(x) (x) passes the Hensel test; we get F1 = 1, K1 = Q2 , F1 = 1. (k) We obtain the next approximation of an irreducible factor of Φ(x):

8 x − 1 = x8 − 2. ϕ2 (x) = 2 2

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Second iteration: (a) The ϕ2 -expansion of Φ(x) is Φ(x) = ϕ2 (x)4 + 8ϕ2 (x)3 + 24ϕ2 (x)2 + 32ϕ2 (x) + 32. ∗ ∗ ∗ (32) = 5, vΦ (24) = 3, vΦ (8) = 3, (b) The valuations of the coeﬃcients are vΦ ∗ and vΦ (1) = 0. (c,d) ϕ2 (x) passes the Newton test; we get he22 = 54 , so E2+ = 1, E2 = 8. 2

∗ (ψ2 ) = 54 . (e) We set ψ2 (x) = x2 , so that vΦ (f,g) The associated polynomial with respect to ϕ2 (x) is A2 (z) = z 4 + 1 = (z − 1)4 ∈ F2 [z]. + 2 (x) (h,i) ϕ ψ2 (x) passes the Hensel test, we get F2 = 1, K2 = Q2 , F2 = 1. (l) We set

ϕ2 (x) − 1 = x8 − 2x2 − 2. ϕ3 (x) = ψ2 (x) ψ2 (x)

Third iteration: (a) The ϕ3 -expansion of Φ(x) is Φ(x) = ϕ3 (x)4 + a3 (x)ϕ3 (x)3 + a2 (x)ϕ3 (x)2 + a1 (x)ϕ3 (x) + a0 (x) where a3 (x) = 8x2 + 8, a2 (x) = 24x4 + 48x2 + 24, a1 (x) = 32x6 + 96x4 + 96x2 + 48, a0 (x) = 64x6 + 96x4 + 96x2 + 64. ∗ ∗ ∗ (b) The valuations of the coeﬃcients are vΦ (a0 ) = 21 4 , vΦ (a1 ) = 4, vΦ (a2 ) = 3, ∗ ∗ vΦ (a3 ) = 3, and vΦ (1) = 0. + ∗ (c,d) ϕ3 (x) passes the Newton test; we get vΦ (ϕ3 ) = he33 = 21 16 , E3 = 2, E3 = 16. E+

∗ ∗ (e) We ﬁnd ψ3 (x) = 22 x5 ; so that vΦ (ψ3 ) = vΦ (ϕ3 3 ) = 21 8 . (f,g) The associated polynomial with respect to ϕ3 (x) is A2 (z) = z 2 + 3 = (z − 1)3 ∈ F2 [z]. + 3 (x) (h,i) ϕ ψ3 (x) passes the Hensel test; we get F3 = 1, K3 = Q2 , F3 = 1. (l) We set ϕ4 (x) = x16 − 4x10 − 4x8 − 4x5 + 4x4 + 8x2 + 4.

Fourth iteration: (a) Let Φ(x) = ϕ4 (x)2 + a1 (x)ϕ4 (x) + a0 (x) be the ϕ4 -expansion of Φ(x). ∗ ∗ (b) We have vΦ (a0 ) = 85/16 and vΦ (a1 ) = 3. + (c,d) ϕ4 (x) passes the Newton test; we get he44 = 85 32 , E4 = 2, E4 = 32. (g) Now E4 F4 = 32 = deg Φ which implies the irreducibility of Φ(x) = x32 + 16.

Acknowledgments The author would like to thank the anonymous referees and David Ford for their numerous comments. He apologizes to them for the large number of small mistakes.

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References 1. Cannon, J.J., et al.: The computer algebra system Magma. University of Sydney (2010), http://magma.maths.usyd.edu.au/magma/ 2. Cantor, D.G., Gordon, D.: Factoring polynomials over p-adic fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 185–208. Springer, Heidelberg (2000) 3. Ford, D., Letard, P.: Implementing the Round Four maximal order algorithm. Journal de Th´eorie des Nombres de Bordeaux 6, 39–80 (1994) 4. Ford, D., Pauli, S., Roblot, X.-F.: A Fast Algorithm for Polynomial Factorization over Qp . Journal de Th´eorie des Nombres de Bordeaux 14, 151–169 (2002) 5. Ford, D., Veres, O.: On the Complexity of the Montes Ideal Factorization Algorithm. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX, July 19-23. LNCS, vol. 6197, pp. 174–185. Springer, Heidelberg (2010) 6. Kaltofen, E., Shoup, V.: Subquadratic-time factoring of polynomials over finite fields. Math. Comp. 67 (1998) 7. Guardia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory (2008), arXiv:0807.2620 8. Guardia, J., Montes, J., Nart, E.: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields (2008), arXiv:0807.4065 9. MacLane, S.: A Construction for absolute values in polynomial rings. Trans. Amer. Math. Soc. 40, 363–395 (1936) 10. Montes, J., Nart, E.: On a Theorem of Ore. Journal of Algebra 146, 318–334 (1992) 11. Montes, J.: Pol´ıgonos de Newton de orden superior y aplicaciones aritm´eticas, PhD Thesis, Universitat de Barcelona (1999) ¨ Newtonsche Polygone in der Theorie der algebraischen K¨ 12. Ore, O.: orper. Math. Ann. 99, 84–117 (1928) 13. PARI/GP, version 2.3.4, Bordeaux (2008), http://pari.math.u-bordeaux.fr/ 14. Pauli, S.: Factoring polynomials over local fields. J. Symb. Comp. 32, 533–547 (2001) 15. Sch¨ onhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971) 16. Stein, W., et al.: SAGE: Software for Algebra and Geometry Experimentation (2007), http://www.sagemath.org 17. Veres, O.: On the Complexity of Polynomial Factorization over p-adic Fields, PhD Dissertation, Concordia University, Montreal (2009)

On a Problem of Hajdu and Tengely Samir Siksek1 and Michael Stoll2 1 2

Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK [email protected] Mathematisches Institut, Universit¨ at Bayreuth, 95440 Bayreuth, Germany [email protected]

Abstract. We prove a result that ﬁnishes the study of primitive arithmetic progressions consisting of squares and ﬁfth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a2 , b2 , c2 , d5 ) is (1, 1, 1, 1). For the proof, we ﬁrst reduce the problem to that of determining the sets of rational points on three speciﬁc hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We ﬁnd generators for a subgroup of ﬁnite index of the Mordell-Weil group of the last curve. Applying Chabauty’s method, we prove that the only rational points on this curve are the obvious ones.

1

Introduction

Euler ([9, pages 440 and 635]) proved Fermat’s claim that four distinct squares cannot form an arithmetic progression. Powers in arithmetic progressions are still a subject of current interest. For example, Darmon and Merel [8] proved that the only solutions in coprime integers to the Diophantine equation xn + y n = 2z n with n ≥ 3 satisfy xyz = 0 or ±1. This shows that there are no non-trivial three term arithmetic progressions consisting of n-th powers with n ≥ 3. The result of Darmon and Merel is far from elementary; it needs all the tools used in Wiles’ proof of Fermat’s Last Theorem and more. An arithmetic progression (x1 , x2 , . . . , xk ) of integers is said to be primitive if the terms are coprime, i.e., if gcd(x1 , x2 ) = 1. Let S be a ﬁnite subset of integers ≥ 2. Hajdu [11] showed that if (a11 , . . . , akk )

(1)

is a non-constant primitive arithmetic progression with i ∈ S, then k is bounded by some (inexplicit) constant C(S). Bruin, Gy˝ ory, Hajdu and Tengely [2] showed that for any k ≥ 4 and any S, there are only ﬁnitely many primitive arithmetic progressions of the form (1), with i ∈ S. Moreover, for S = {2, 3} and k ≥ 4, they showed that ai = ±1 for i = 1, . . . , k. A recent paper of Hajdu and Tengely [12] studies primitive arithmetic progressions (1) with exponents belonging to S = {2, n} and {3, n}. In particular, they G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 316–330, 2010. c Springer-Verlag Berlin Heidelberg 2010

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show that any primitive non-constant arithmetic progression (1) with exponents i ∈ {2, 5} has k ≤ 4. Moreover, for k = 4 they show that (1 , 2 , 3 , 4 ) = (2, 2, 2, 5) or (5, 2, 2, 2).

(2)

Note that if (ai i : i = 1, . . . , k) is an arithmetic progression, then so is the reverse progression (ai i : i = k, k − 1, . . . , 1). Thus there is really only one case left open by Hajdu and Tengely, with exponents (1 , 2 , 3 , 4 ) = (2, 2, 2, 5). This is also mentioned as Problem 11 in a list of 22 open problems recently compiled by Evertse and Tijdeman [10]. In this paper we deal with this case. Theorem 1. The only arithmetic progression in coprime integers of the form (a2 , b2 , c2 , d5 ) is (1, 1, 1, 1). This together with the above-mentioned results of Hajdu and Tengely completes the proof of the following theorem. Theorem 2. There are no non-constant primitive arithmetic progressions of the form (1) with i ∈ {2, 5} and k ≥ 4. The primitivity condition is crucial, since otherwise solutions abound. Let for example (a2 , b2 , c2 , d) be any arithmetic progression whose ﬁrst three terms are squares — there are inﬁnitely many of one can take a = r2 − 2rs − s2 , these; 2 2 2 2 2 2 b = r + s , c = r + 2rs − s — then (ad ) , (bd2 )2 , (cd2 )2 , d5 ) is an arithmetic progression whose ﬁrst three terms are squares and whose last term is a ﬁfth power. For the proof of Thm. 1, we ﬁrst reduce the problem to that of determining the sets of rational points on three speciﬁc hyperelliptic curves of genus 4. A 2-cover descent computation (following Bruin and Stoll [3]) shows that there are no rational points on two of these curves. We ﬁnd generators for a subgroup of ﬁnite index of the Mordell-Weil group of the last curve. Applying Chabauty’s method, we prove that the only rational points on this curve are the obvious ones. All our computations are performed using the computer package MAGMA [1]. The result we prove here may perhaps not be of compelling interest in itself. Rather, the purpose of this paper is to demonstrate how we can solve problems of this kind with the available machinery. We review the relevant part of this machinery in Sect. 3, after we have constructed the curves pertaining to our problem in Sect. 2. Then, in Sect. 4, we apply the machinery to these curves. The proofs are mostly computational. We have tried to make it clear what steps need to be done, and to give enough information to make it possible to reproduce the computations (which have been performed independently by both authors as a consistency check).

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Construction of the Curves

Let (a2 , b2 , c2 , d5 ) be an arithmetic progression in coprime integers. Since a square is ≡ 0 or 1 mod 4, it follows that all terms are ≡ 1 mod 4, in particular, a, b, c and d are all odd. Considering the last three terms, we have the relation √ √ (−d)5 = b2 − 2c2 = (b + c 2)(b − c 2) . Since b√and c are odd and coprime, the two factors √ on the right are coprime in R = Z[ 2]. Since R× /(R× )5 is generated by 1 + 2, it follows that √ √ √ √ (3) b + c 2 = (1 + 2)j (u + v 2)5 = gj (u, v) + hj (u, v) 2 with −2 ≤ j ≤ 2 and u, v ∈ Z coprime (with u odd and v ≡ j + 1 mod 2). The polynomials gj and hj are homogeneous of degree 5 and have coeﬃcients in Z. Now the ﬁrst three terms of the progression give the relation a2 = 2b2 − c2 = 2gj (u, v)2 − hj (u, v)2 . Writing y = a/v 5 and x = u/v, this gives the equation of a hyperelliptic curve of genus 4, Cj : y 2 = fj (x) where fj (x) = 2gj (x, 1)2 −hj (x, 1)2 . Every arithmetic progression of the required form therefore induces a rational point on one of the curves Cj . We observe that taking conjugates in (3) leads to √ √ √ (−1)j b + (−1)j+1 c 2 = (1 + 2)−j (u + (−v) 2)5 , which implies that f−j (x) = fj (−x) and therefore that C−j and Cj are isomorphic and their rational points correspond to the same arithmetic progressions. We can therefore restrict attention to C0 , C1 and C2 . Their equations are as follows. C0 : y 2 = f0 (x) = 2x10 + 55x8 + 680x6 + 1160x4 + 640x2 − 16 C1 : y 2 = f1 (x) = x10 + 30x9 + 215x8 + 720x7 + 1840x6 + 3024x5 + 3880x4 + 2880x3 + 1520x2 + 480x + 112 C2 : y 2 = f2 (x) = 14x10 + 180x9 + 1135x8 + 4320x7 + 10760x6 + 18144x5 + 21320x4 + 17280x3 + 9280x2 + 2880x + 368 The trivial solution a = b = c = d = 1 corresponds to j = 1, (u, v) = (1, 0) in the above and therefore gives rise to the point ∞+ on C1 (this is the point at inﬁnity where y/x5 takes the value +1). Changing the signs of a, b or c leads to ∞− ∈ C1 (Q) (the point where y/x5 = −1) or to the two points at inﬁnity on the isomorphic curve C−1 .

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319

Background on Rational Points on Hyperelliptic Curves

Our task will be to determine the set of rational points on each of the curves C0 , C1 and C2 constructed in the previous section. In this section, we will give an overview of the methods we will use, and in the next section, we will apply these methods to the given curves. We will restrict attention to hyperelliptic curves, i.e., curves given by an aﬃne equation of the form C : y 2 = f (x) where f is a squarefree polynomial with integral coeﬃcients. The smooth projective curve birational to this aﬃne curve has either one or two additional points ‘at inﬁnity’. If the degree of f is odd, there is one point at inﬁnity, which is always a rational point. Otherwise there are two points at inﬁnity corresponding to the two square roots of the leading coeﬃcient of f . In particular, these two points are rational if and only if the leading coeﬃcient is a square. For example, C1 above has two rational points at inﬁnity, whereas the points at inﬁnity on C0 and C2 are not rational. We will use C in the following to denote the smooth projective model; C(Q) denotes as usual the set of rational points including those at inﬁnity. 3.1

Two-Cover Descent

It will turn out that C0 and C2 do not have rational points. One way of showing that C(Q) is empty is to verify that C(R) is empty or that C(Qp ) is empty for some prime p. This does not work for C0 or C2 ; both curves have real points and p-adic points for all p. (This can be checked by a ﬁnite computation.) So we need a more sophisticated way of showing that there are no rational points. One such method is known as 2-cover descent. We sketch the method here; for a detailed description, see [3]. An important ingredient of this and other methods is the algebra L := Q[T ] =

Q[x] , Q[x] · f (x)

where T denotes the image of x. If f is irreducible (as in our examples), then L is the number ﬁeld generated by a root of f . In general, L will be a product of number ﬁelds corresponding to the irreducible factors of f . We now assume that f has even degree 2g + 2, where g is the genus of the curve. This is the generic case; the odd degree case is somewhat simpler. We can then set up a map, called the descent map or x − T map: x − T : C(Q) −→ H :=

L× . Q× (L× )2

Here L× denotes the multiplicative group of L, and (L× )2 denotes the subgroup of squares. On points P ∈ C(Q) that are neither at inﬁnity nor Weierstrass points (i.e., points with vanishing y coordinate), the map is deﬁned as

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(x − T )(P ) = x(P ) − T mod Q× (L× )2 . Rational points at inﬁnity map to the trivial element, and if there are rational Weierstrass points, their images can be determined using the fact that the norm of x(P ) − T is y(P )2 divided by the leading coeﬃcient of f . If we can show that x − T has empty image on C(Q), then it follows that C(Q) is empty. We obtain information of the image by considering again C(R) and C(Qp ). We can carry out the same construction over R and over Qp , leading to an algebra Lv (v = p, or v = ∞ when working over R), a group Hv and a map (x − T )v : C(Qv ) −→ Hv

(where Q∞ = R).

We have inclusions C(Q) → C(Qv ) and canonical homomorphisms H → Hv . Everything ﬁts together in a commutative diagram x−T

C(Q) v C(Qv )

v (x−T )v

/

/H v

Hv

where v runs through the primes and ∞. If we can show that the images of the lower horizontal map and of the right vertical map do not meet, then the image of x − T and therefore also C(Q) must be empty. We can verify this by considering a ﬁnite subset of ‘places’ v. In general, we obtain a ﬁnite subset of H that contains the image of x−T ; this ﬁnite subset is known as the fake 2-Selmer set of C/Q. It classiﬁes either pairs of (isomorphism classes of) 2-covering curves of C that have points everywhere locally, i.e., over R and over all Qp , or else it classiﬁes such 2-covering curves, in which case it is the (true) 2-Selmer set. Whether it classiﬁes pairs or individual 2-coverings depends on a certain condition on the polynomial f . This condition is satisﬁed if either f has an irreducible factor√ of odd degree, or if deg f ≡ 2 mod 4 and f factors over a quadratic extension Q( d) as a constant times the product of two conjugate polynomials. A 2-covering of C is a morphism π : D → C that is unramiﬁed and becomes Galois over a suitable ﬁeld extension of ﬁnite degree, with Galois group (Z/2Z)2g . It is known that every rational point on C lifts to a rational point on some 2-covering of C. The actual computation splits into a global and a local part. The global computation uses the ideal class group and the unit group of L (or the constituent number ﬁelds of L) to construct a ﬁnite subgroup of H containing the image of x − T . The local computation determines the image of (x − T )v for ﬁnitely many places v. 3.2

The Jacobian

Most other methods make use of another object associated to the curve C: its Jacobian variety (or just Jacobian). This is an abelian variety J (a higherdimensional analogue of an elliptic curve) of dimension g, the genus of C. It

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reﬂects a large part of the geometry and arithmetic of C; its main advantage is that its points form an abelian group, whereas the set of points on C does not carry a natural algebraic structure. For our purposes, we can more or less forget the structure of J as a projective variety. Instead we use the description of the points on J as the elements of the degree zero part of the Picard group of C. The Picard group is constructed as a quotient of the group of divisors on C. A divisor on C is an element of ¯ of all algebraic points on C. The the free abelian group DivC on the set C(Q) absolute Galois group of Q acts on DivC ; a divisor that is ﬁxed by this action is rational. This does not mean that the points occurring in the divisor must be rational; points with the same multiplicity can be permuted. A nonzero rational ¯ has an associated divisor div(h) that function h on C with coeﬃcients in Q records its zeros and poles (with multiplicities). If h has coeﬃcients in Q, then div(h) is rational. The homomorphism deg : DivC → Z induced by sending each ¯ to 1 gives the degree of a divisor. Divisors of functions have degree point in C(Q) zero. Two divisors D, D ∈ DivC are linearly equivalent if their diﬀerence is the divisor of a function. The equivalence classes are the elements of the Picard group PicC deﬁned by the following exact sequence. × div ¯ × −→ Q(C) ¯ 0 −→ Q −→ DivC −→ PicC −→ 0

Since divisors of functions have degree zero, the degree homomorphism descends ¯ is isomorphic as a to PicC . We denote its kernel by Pic0C . It is a fact that J(Q) group to Pic0C . The rational points J(Q) correspond to the elements of Pic0C left invariant by the Galois group. In general it is not true that a point in J(Q) can be represented by a rational divisor, but this is the case when C has a rational point, or at least points everywhere locally. The most important fact about the group J(Q) is the statement of the Mordell-Weil Theorem: J(Q) is a finitely generated abelian group. For this reason, J(Q) is often called the Mordell-Weil group of J or of C. If P0 ∈ C(Q), then the map C P → [P − P0 ] ∈ J is a Q-deﬁned embedding of C into J. We use [D] to denote the linear equivalence class of the divisor D. The basic idea of the methods described below is to try to recognise the points of C embedded in this way among the rational points on J. We need a way of representing elements of J(Q). Let P → P − denote the hyperelliptic involution on C; this is the morphism C → C that changes the sign of the y coordinate. Then it is easy to see that the divisors P + P − all belong to the same class W ∈ PicC . An eﬀective divisor D (a divisor such that no point occurs with negative multiplicity) is in general position if there is no point P such that D − P − P − is still eﬀective. Divisors in general position not containing points at inﬁnity can be represented in a convenient way by pairs of polynomials (a(x), b(x)). This pair represents the divisor D such that its image on the projective line (under the x-coordinate map) is given by the roots of a; the corresponding points on C are determined by the relation y = b(x). The polynomials have to satisfy the relation f (x) ≡ b(x)2 mod a(x). This is

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the Mumford representation of D. The polynomials a and b can be chosen to have rational coeﬃcients if and only if D is rational. (The representation can be adapted to allow for points at inﬁnity occurring in the divisor.) If the genus g is even, then it is a fact that every point in J(Q) has a unique representation of the form [D] − nW where D is a rational divisor in general position of degree 2n and n ≥ 0 is minimal. The Mumford representation of D is then also called the Mumford representation of the corresponding point on J. It is fairly easy to add points on J using the Mumford representation, see [5]. This addition procedure is implemented in MAGMA, for example. There is a relation between 2-coverings of C and the Jacobian J. Assume C is embedded in J as above. Then if D is any 2-covering of C that has a rational point P , D can be realised as the preimage of C under a map of the form Q → 2Q + Q0 on J, where Q0 is the image of P on C ⊂ J. A consequence of this is that two rational points P1 , P2 ∈ C(Q) lift to the same 2-covering if and only if [P1 − P2 ] ∈ 2J(Q). 3.3

The Mordell-Weil Group

We will need to know generators of a ﬁnite-index subgroup of the Mordell-Weil group J(Q). Since J(Q) is a ﬁnitely generated abelian group, it will be a direct sum of a ﬁnite torsion part and a free abelian group of rank r; r is called the rank of J(Q). So what we need is a set of r independent points in J(Q). The torsion subgroup of J(Q) is usually easy to determine. The main tool used here is the fact that the torsion subgroup injects into J(Fp ) when p is an odd prime not dividing the discriminant of f . If the orders of the ﬁnite groups J(Fp ) are coprime for suitable primes p, then this shows that J(Q) is torsion-free. We can ﬁnd points in J(Q) by search. This can be done by searching for rational points on the variety parameterising Mumford representations of divisors of degree 2, 4, . . . . We can then check if the points found are independent by again mapping into J(Fp ) for one or several primes p. The hard part is to know when we have found enough points. For this we need an upper bound on the rank r. This can be provided by a 2-descent on the Jacobian J. This is described in detail in [16]. The idea is similar to the 2cover descent on C described above in Sect. 3.1. Essentially we extend the x − T map from points to divisors. It can be shown that the value of (x − T )(D) only depends on the linear equivalence class of D. This gives us a homomorphism from J(Q) into H, or more precisely, into the kernel of the norm map NL/Q : H → Q× /(Q× )2 . It can be shown that the kernel of this x − T map on J(Q) is either 2J(Q), or it contains 2J(Q) as a subgroup of index 2. The former is the case when f satisﬁes the same condition as that mentioned in Sect. 3.1. We can then bound (x − T )(J(Q)) in much the same way as we did when doing a 2-cover descent on C. The global part of the computation is identical. The local part is helped by the fact that we now have a group homomorphism (or a homomorphism of F2 -vector spaces), so we can use linear algebra. We obtain a bound for the order of J(Q)/2J(Q), from which we can deduce a bound for

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the rank r. If we are lucky and found that same number of independent points in J(Q), then we know that these points generate a subgroup of ﬁnite index. The group containing (x−T )(J(Q)) we compute is known as the fake 2-Selmer group of J [13]. If the polynomial f satisﬁes the relevant condition, then this fake Selmer group is isomorphic to the true 2-Selmer group of J (that classiﬁes 2coverings of J that have points everywhere locally). 3.4

The Chabauty-Coleman Method

If the rank r is less than the genus g, there is a method available that allows us to get tight bounds on the number of rational points on C. This goes back to Chabauty [6], who used it to prove Mordell’s Conjecture in this case. Coleman [7] reﬁned the method. We give a sketch here; more details can be found for example in [15]. Let p be a prime of good reduction for C (this is the case when p is odd 1 and does not divide the discriminant of f ). We use ΩC (Qp ) and ΩJ1 (Qp ) to denote the spaces of regular 1-forms on C and J that are deﬁned over Qp . If P0 ∈ C(Q) and ι : C → J, P → [P − P0 ] denotes the corresponding embedding 1 of C into J, then the induced map ι∗ : ΩJ1 (Qp ) → ΩC (Qp ) is an isomorphism that is independent of the choice of basepoint P0 . Both spaces have dimension g. There is an integration pairing Q 1 ΩC (Qp ) × J(Qp ) −→ Qp , (ι∗ ω, Q) −→ ω = ω, log Q . 0

In the last expression, log Q denotes the p-adic logarithm on J(Qp ) with values in the tangent space of J(Qp ) at the origin, and ΩJ1 (Qp ) is identiﬁed with the dual of this tangent space. If r < g, then there are (at least) g −r linearly independent 1 diﬀerentials ω ∈ ΩC (Qp ) that annihilate the Mordell-Weil group J(Q). Such a diﬀerential can be scaled so that it reduces to a non-zero diﬀerential ω ¯ mod p. Now the important fact is that if ω ¯ does not vanish at a point P¯ ∈ C(Fp ), then there is at most one rational point on C(Q) whose reduction is P¯ . (There are more general bounds valid when ω ¯ does vanish at P¯ , but we do not need them here.)

4

Determining the Rational Points

In this section, we determine the set of rational points on the three curves C0 , C1 and C2 . To do this, we apply the methods described in Sect. 3. We ﬁrst consider C0 and C2 . We apply the 2-cover-descent procedure described in Sect. 3.1 to the two curves and ﬁnd that in each case, there are no 2-coverings that have points everywhere locally. For C0 , only 2-adic information is needed in addition to the global computation, for C2 , we need 2-adic and 7adic information. Note that the number ﬁelds generated by roots of f0 or f2 are suﬃciently small in terms of degree and discriminant that the necessary class and unit group computations can be done unconditionally. This leads to the following.

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Proposition 3. There are no rational points on the curves C0 and C2 . Proof. The 2-cover descent procedure is available in recent releases of MAGMA. The computations leading to the stated result can be performed by issuing the following MAGMA commands. > SetVerbose("Selmer",2); > TwoCoverDescent(HyperellipticCurve(Polynomial( [-16,0,640,0,1160,0,680,0,55,0,2]))); > TwoCoverDescent(HyperellipticCurve(Polynomial( [368,2880,9280,17280,21320,18144,10760,4320,1135,180,14]))); We explain how the results can be checked independently. We give details for C0 ﬁrst. The procedure for C2 is similar, so we only explain the diﬀerences. The polynomial f0 is irreducible, and it can be checked √ that the number ﬁeld generated by one of its roots is isomorphic to L = Q( 10 288). Using MAGMA or ˜ pari/gp, one checks that this ﬁeld has trivial class group. The ﬁnite subgroup H × × 2 of H containing the Selmer set is then given as OL,S /(Z× (O ) ), where S L,S {2,3,5} is the set of primes in OL above the ‘bad primes’ 2, 3 and 5. The set S contains two primes above 2, of degrees 1 and 4, respectively, and one prime above 3 and 5 each, of degree 2 in both cases. Since L has two real embeddings and four pairs ˜ is of complex embeddings, the unit rank is 5. The rank (or F2 -dimension) of H then 7. (Note that 2 is a square in L.) The descent map takes its values in the ˜ consisting of elements whose norm is twice a square. This subset is subset of H of size 32; elements of OL representing it can easily be obtained. Let δ be such a representative. We let T be a root of f0 in L and check that the system of equations y 2 = f0 (x), x − T = δcz 2 has no solutions with x, y, c ∈ Q2 , z ∈ L ⊗Q Q2 . The second equation leads, after expanding δz 2 as a Q-linear combination of 1, T, T 2, . . . , T 9 , to eight homogeneous quadratic equations in the ten unknown coeﬃcients of z. Any solution to these equations gives a unique x, for which f0 (x) is a square. The latter follows by taking norms on both sides of x − T = δcz 2 . So we only have to check the intersection of eight quadrics in P9 for existence of Q2 -points. Alternatively, we × × 2 evaluate the descent map on C0 (Q2 ), to get its image in H2 = L× 2 /(Q2 (L2 ) ), where L2 = L ⊗Q Q2 . Then we check that none of the representatives δ map into this image. When dealing with C2 , the ﬁeld L is generated by a root of x10 − 6x5 − 9. Since the leading coeﬃcient of f2 is 14, we have to add (the primes above) 7 to the bad primes. As before, the class group is trivial, and we have the same splitting behaviour of 2, 3 and 5. The prime 7 splits into two primes of degree 1 and two primes of degree 4. The group of S-units of L modulo squares has now ˜ has rank 10, and the subset of H consisting of elements rank 14, the group H whose norm is 14 times a square has 128 elements. These elements now have to be tested for compatibility with the 2-adic and the 7-adic information, which can be done using either of the two approaches described above. The 7-adic check is

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only necessary for one of the elements; the 127 others are already ruled out by the 2-adic check. We cannot hope to deal with C1 in the same easy manner, since C1 has two rational points at inﬁnity coming from the trivial solutions. We can still perform a 2-cover-descent computation, though, and ﬁnd that there is only one 2-covering of C1 with points everywhere locally, which is the covering that lifts the points at inﬁnity. Only 2-adic information is necessary to show that the fake 2-Selmer set has at most one element, so we can get this result using the following MAGMA command. > TwoCoverDescent(HyperellipticCurve(Polynomial( [112,480,1520,2880,3880,3024,1840,720,215,30,1])) : PrimeCutoff := 2); (In some versions of MAGMA this returns a two-element set. However, as can be checked by pulling back under the map returned as a second value, these two elements correspond to the images of 1 and −1 in L× /(L× )2 Q× and therefore both represent the trivial element. The error is caused by MAGMA using 1 instead of −1 as a ‘generator’ of Q× /(Q× )2 . This bug is corrected in recent releases.) The computation can be performed in the same way as for C0 and C2 . The relevant ﬁeld L is generated by a root of x10 − 18x5 + 9; it has class number 1, and the primes 2, 3 and 5 split in the same way as before. The subset H (in fact ˜ consisting of elements with square norm has size 32. Of these, a subgroup) of H only the element represented by 1 is compatible with the 2-adic constraints.√ We remark that by the way it is given, the polynomial f1 factors over Q( 2) into two conjugate factors of degree 5. This implies that the ‘fake 2-Selmer set’ computed by the 2-cover descent is the true 2-Selmer set, so that there is really only one 2-covering that corresponds to the only element of the set computed by the procedure. We state the result as a lemma. We ﬁx P0 = ∞− ∈ C1 as our basepoint and write J1 for the Jacobian variety of C1 . Then, as described in Sect. 3.2, ι : C1 −→ J1 , P −→ [P − P0 ] is an embedding deﬁned over Q. Lemma 4. Let P ∈ C1 (Q). Then the divisor class [P − P0 ] is in 2J1 (Q). Proof. Let D be the unique 2-covering of C1 (up to isomorphism) that has points everywhere locally. The fact that D is unique follows from the computation of the 2-Selmer set. Any rational point P ∈ C1 (Q) lifts to a rational point on some 2-covering of C1 . In particular, this 2-covering then has a rational point, so it also satisﬁes the weaker condition that it has points everywhere locally. Since D is the only 2-covering of C1 satisfying this condition, P0 and P must both lift to a rational point on D. This implies by the remark at the end of Sect. 3.2 that [P − P0 ] ∈ 2J1 (Q). To make use of this information, we need to know J1 (Q), or at least a subgroup of ﬁnite index. A computer search reveals two points in J1 (Q), which are given in Mumford representation (see Sect. 3.2) as follows.

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Q1 = x4 + 4x2 + 45 , −16x3 − 96 5 x 3 3 36 2 48 36 Q2 = x4 + 24 − 1712 5 x + 5 x + 5 x+ 5 , 75 x −

976 2 25 x

−

1728 25 x

−

2336 25

We note that 2Q1 = [∞+ − ∞− ]; this makes Lemma 4 explicit for the known two points on C1 . Lemma 5. The Mordell-Weil group J1 (Q) is torsion-free, and Q1 , Q2 are linearly independent. In particular, the rank of J1 (Q) is at least 2. Proof. The only primes of bad reduction for C1 are 2, 3 and 5. It is known that the torsion subgroup of J1 (Q) injects into J1 (Fp ) when p is an odd prime of good reduction. Since #J1 (F7 ) = 2400 and #J1 (F41 ) = 2633441 are coprime, there can be no nontrivial torsion in J1 (Q). We check that the image of Q1 , Q2 in J1 (F7 ) is not cyclic. This shows that Q1 and Q2 must be independent. The next step is to show that the Mordell-Weil rank is indeed 2. For this, we compute the 2-Selmer group of J1 as sketched in Sect. 3.3 and described in detail in [16]. We give some details of the computation, since it is outside the scope of the functionality that is currently provided by MAGMA (or any other software package). √ We ﬁrst remind ourselves that f1 factors over Q( 2). This implies that the kernel of the x − T map on J(Q) is 2J(Q). Therefore the ‘fake 2-Selmer group’ that we compute is in fact the actual 2-Selmer group of J1 . Since J1 (Q) is torsionfree, the order of the 2-Selmer group is an upper bound for 2r , where r is the rank of J1 (Q). The global computation is the same as that we needed to do for the 2-cover descent. In particular, the Selmer group is contained in the group H from above, consisting of the S-units of L with square norm, modulo squares and modulo {2, 3, 5}-units of Q. For the local part of the computation, we have to compute the image of J1 (Qp ) under the local x− T map for the primes p of bad reduction. We check that there is no 2-torsion in J1 (Q3 ) and J1 (Q5 ) (f1 remains irreducible both over Q3 and over Q5 ). This implies that the targets of the local maps (x − T )3 and (x − T )5 are trivial, which means that these two primes need not be considered as bad primes for the descent computation. The real locus C1 (R) is connected, which implies that there is no information coming from the local image at the inﬁnite place. (Recall that C1 denotes the smooth projective model of the curve. The real locus of the aﬃne curve y 2 = f1 (x) has two components, but they are connected to each other through the points at inﬁnity.) Therefore, we only need to use 2-adic information in the computation. We set L2 = L⊗Q Q2 and compute the natural homomorphism μ2 : H −→ H2 =

L× 2 × 2 Q2 (L× 2)

.

Let I2 be the image of J1 (Q2 ) in H2 . Then the 2-Selmer group is μ−1 2 (I2 ).

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It remains to compute I2 , which is the hardest part of the computation. The 2-torsion subgroup J1 (Q2 )[2] has order 2 (f1 splits into factors of degrees 2 and 8 over Q2 ); this implies that J1 (Q2 )/2J1 (Q2 ) has dimension g + 1 = 5 as an F2 -vector space. This quotient is generated by the images of Q1 and Q2 and of three further points of the form [Di ] − deg2Di W , where Di is the sum of points on C1 whose x-coordinates are the roots of D1 : x − 12 x − 14 , D2 : x2 − 2x + 6 , D3 : x4 + 4x3 + 12x2 + 36 , respectively. These points were found by a systematic search, using the fact that the local map (x − T )2 is injective in our situation. We can therefore stop the search procedure as soon as we have found points whose images generate a ﬁvedimensional F2 -vector space. We thus ﬁnd I2 ⊂ H2 and then can compute the 2-Selmer group. In our situation, μ2 is injective, and the intersection of its image with I2 is generated by the images of Q1 and Q2 . Therefore, the F2 -dimension of the 2-Selmer group is 2. Lemma 6. The rank of J1 (Q) is 2, and Q1 , Q2 ⊂ J1 (Q) is a subgroup of finite odd index. Proof. The Selmer group computation shows that the rank is ≤ 2, and Lemma 5 shows that the rank is ≥ 2. Regarding the second statement, it is now clear that we have a subgroup of ﬁnite index. The observation stated just before the lemma shows that the given subgroup surjects onto the 2-Selmer group under the x − T map. Since the kernel of the x − T map is 2J1 (Q), this implies that the index is odd. Now we want to use the Chabauty-Coleman method sketched in Sect. 3.4 to show that ∞+ and ∞− are the only rational points on C1 . To keep the computations reasonably simple, we want to work at p = 7, which is the smallest prime of good reduction. For p a prime of good reduction, we write ρp for the two ‘reduction mod p’ maps J1 (Q) → J1 (Fp ) and C1 (Q) → C1 (Fp ). Lemma 7. Let P ∈ C1 (Q). Then ρ7 (P ) = ρ7 (∞+ ) or ρ7 (P ) = ρ7 (∞− ). Proof. Let G = Q1 , Q2 be the subgroup of J1 (Q) generated by the two points Q1 and Q2 . We ﬁnd that ρ7 (G) has index 2 in J1 (F7 ) ∼ = Z/10Z ⊕ Z/240Z. By Lemma 6, we know that (J1 (Q) : G) is odd, so we can deduce that ρ7 (G) = ρ7 (J1 (Q)). The group J1 (F7 ) surjects onto (Z/5Z)2 . Since ρ7 (J1 (G)) has index 2 in J1 (F7 ), ρ7 (G) = ρ7 (J1 (Q)) also surjects onto (Z/5Z)2 . This implies that the index of G in J1 (Q) is not divisible by 5. We determine the points P ∈ C1 (F7 ) such that ι(P ) ∈ ρ7 (2J1 (Q)) = 2ρ7 (G). We ﬁnd the set X7 = {ρ7 (∞+ ), ρ7 (∞− ), (−2, 2), (−2, −2)} .

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Note that for any P ∈ J1 (Q), we must have ρ7 (P ) ∈ X7 by Lemma 4. Now we look at p = 13. The image of G in J1 (F13 ) ∼ = Z/10Z ⊕ Z/2850Z has index 5. Since we already know that (J1 (Q) : G) is not a multiple of 5, this implies that ρ13 (G) = ρ13 (J1 (Q)). As above for p = 7, we compute the set X13 ⊂ C1 (F13 ) of points mapping into ρ13 (2J1 (Q)). We ﬁnd X13 = {ρ13 (∞+ ), ρ13 (∞− )} . Now suppose that there is P ∈ C1 (Q) with ρ7 (P ) ∈ {(−2, 2), (−2, −2)}. Then ι(P ) is in one of two speciﬁc cosets in J1 (Q)/ ker ρ7 ∼ = G/ ker ρ7 |G . On the other hand, we have ρ13 (P ) = ρ13 (∞± ), so that ι(P ) is in one of two speciﬁc cosets in J1 (Q)/ ker ρ13 ∼ = G/ ker ρ13 |G . If we identify G = Q1 , Q2 with Z2 , then we can ﬁnd the kernels of ρ7 and of ρ13 on G explicitly, and we can also determine the relevant cosets explicitly. It can then be checked that the union of the ﬁrst two cosets does not meet the union of the second two cosets. This implies that such a point P cannot exist. Therefore, the only remaining possibilities are that ρ7 (P ) = ρ7 (∞± ). Remark 8. The use of information at p = 13 to rule out residue classes at p = 7 in the proof above is a very simple instance of a method known as the MordellWeil sieve. For a detailed description of this method, see [4]. Now we need to ﬁnd the space of holomorphic 1-forms on C1 , deﬁned over Q7 , that annihilate the Mordell-Weil group under the integration pairing, compare Sect. 3.4. We follow the procedure described in [14]. We ﬁrst ﬁnd two independent points in the intersection of J1 (Q) and the kernel of reduction mod 7. In our case, we take R1 = 20Q1 and R2 = 5Q1 + 60Q2 . We represent these points in the form Rj = [Dj − 4∞− ] with eﬀective divisors D1 , D2 of degree 4. The coeﬃcients of the primitive polynomial in Z[x] whose roots are the x-coordinates of the points in the support of D1 have more than 100 digits and those of the corresponding polynomial for D2 ﬁll several pages, so we refrain from printing them here. (This indicates that it is a good idea to work with a small prime!) The points in the support of D1 and D2 all reduce to ∞− modulo the prime above 7 in their ﬁelds of deﬁnition (which are degree 4 number ﬁelds totally ramiﬁed at 7). Expressing 1 a basis of ΩC (Q7 ) as power series in the uniformiser t = 1/x at P0 = ∞− 1 times dt, we compute the integrals numerically. More precisely, the diﬀerentials η0 =

dx , 2y

η1 =

x dx , 2y

η2 =

x2 dx 2y

and η3 =

x3 dx 2y

1 form a basis of ΩC (Q7 ). We get 1

ηj = t3−j

1 2

−

15 145385 4 2764899 5 t + 115t2 − 1980t3 + t − t + . . . dt 2 4 4

as power series in the uniformiser. Using these power series up to a precision of t20 , we compute the following 7-adic approximations to the integrals.

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⎛

Rj

0

ηi

0≤i≤3,1≤j≤2

⎞ −20 · 7 + O(74 ) −155 · 7 + O(74 ) ⎜−150 · 7 + O(74 ) −13 · 7 + O(74 ) ⎟ ⎟ =⎜ ⎝−130 · 7 + O(74 ) −83 · 7 + O(74 ) ⎠ −19 · 7 + O(74 ) 163 · 7 + O(74 )

From this, it follows easily that the reductions mod 7 of the (suitably scaled) 1 diﬀerentials that kill J1 (Q) ﬁll the subspace of ΩC (F7 ) spanned by 1 ω1 = (1 + 3x − 2x2 )

dx 2y

and ω2 = (1 − x2 + x3 )

dx . 2y

Since ω2 does not vanish at the points ρ7 (∞± ), this implies that there can be at most one rational point P on C1 with ρ7 (P ) = ρ7 (∞+ ) and at most one point P with ρ7 (P ) = ρ7 (∞− ) (see for example [15, Prop. 6.3]). Proposition 9. The only rational points on C1 are ∞+ and ∞− . Proof. Let P ∈ C1 (Q). By Lemma 7, ρ7 (P ) = ρ7 (∞± ). By the argument above, for each sign s ∈ {+, −}, we have #{P ∈ C1 (Q) : ρ7 (P ) = ρ7 (∞s )} ≤ 1. These two facts together imply that #C1 (Q) ≤ 2. Since we know the two rational points ∞+ and ∞− on C1 , there cannot be any further rational points. We can now prove Thm. 1. Proof (of Thm. 1). The considerations in Sect. 2 imply that if (a2 , b2 , c2 , d5 ) is an arithmetic progression in coprime integers, then there are coprime u and v, related to a, b, c, d by (3), such that (u/v, a/v 5 ) is a rational point on one of the curves Cj with −2 ≤ j ≤ 2. By Prop. 3, there are no rational points on C0 and C2 and therefore also not on the curve C−2 , which is isomorphic to C2 . By Prop. 9, the only rational points on C1 (and C−1 ) are the points at inﬁnity. This translates into a = ±1, u = ±1, v = 0, and we have j = ±1. We deduce a2 = 1, b2 = g1 (±1, 0)2 = 1, whence also c2 = d5 = 1.

References 1. Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System I: The User Language. J. Symb. Comp. 24, 235–265 (1997), http://magma.maths.usyd.edu.au/magma 2. Bruin, N., Gy˝ ory, K., Hajdu, L., Tengely, S.: Arithmetic progressions consisting of unlike powers. Indag. Math. 17, 539–555 (2006) 3. Bruin, N., Stoll, M.: 2-cover descent on hyperelliptic curves. Math. Comp. 78, 2347–2370 (2009) 4. Bruin, N., Stoll, M.: The Mordell-Weil sieve: Proving non-existence of rational points on curves. LMS J. Comput. Math. (to appear), arXiv:0906.1934v2 [math.NT] 5. Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48, 95–101 (1987) 6. Chabauty, C.: Sur les points rationnels des courbes alg´ebriques de genre sup´erieur a l’unit´e. C. R. Acad. Sci. Paris 212, 882–885 (1941) (French) `

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7. Coleman, R.F.: Eﬀective Chabauty. Duke Math. J. 52, 765–770 (1985) 8. Darmon, H., Merel, L.: Winding quotients and some variants of Fermat’s last theorem. J. Reine Angew. Math. 490, 81–100 (1997) 9. Dickson, L.E.: History of the theory of numbers. Vol. II: Diophantine Analysis. Chelsea Publishing Co., New York (1966) 10. Evertse, J.-H., Tijdeman, R.: Some open problems about Diophantine equations from a workshop in Leiden in (May 2007), http://www.math.leidenuniv.nl/~ evertse/07-workshop-problems.pdf 11. Hajdu, L.: Perfect powers in arithmetic progression. A note on the inhomogeneous case. Acta Arith. 113, 343–349 (2004) 12. Hajdu, L., Tengely, S.: Arithmetic progressions of squares, cubes and n-th powers. Funct. Approx. Comment. Math. 41, 129–138 (2009) 13. Poonen, B., Schaefer, E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997) 14. Stoll, M.: Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math. 11, 367–380 (2008) 15. Stoll, M.: Independence of rational points on twists of a given curve. Compositio Math. 142, 1201–1214 (2006) 16. Stoll, M.: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98, 245–277 (2001)

Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures Jonathan P. Sorenson Butler University, Indianapolis IN 46208, USA [email protected] http://www.butler.edu/~ sorenson

Abstract. We extend the known tables of pseudosquares and pseudocubes, discuss the implications of these new data on the conjectured distribution of pseudosquares and pseudocubes, and present the details of the algorithm used to do this work. Our algorithm is based on the spacesaving wheel data structure combined with doubly-focused enumeration, run in parallel on a cluster supercomputer.

1

Introduction

It is well-known that testing for primality can be done in polynomial time [1,3]. However, the fastest known deterministic algorithms are conjectured to be the pseudosquares prime test of Lukes, Patterson, and Williams [6], and its generalizations, the pseudocube prime test of Berrizbeitia, M¨ uller, and Williams [4], and the Eisenstein pseudocube test [13,15], all of which run in roughly cubic time, if a suﬃciently large pseudosquare or pseudocube is available. In particular, the pseudosquares prime test is very useful in the context of ﬁnding all primes in an interval [10], where sieving can be used in place of trial division. This, then, motivates our search for larger and larger peudosquares and pseudocubes, and our attempts to predict their distribution. See, for example, Wooding and Williams [14] and also [7,12,8,2,11]. In this paper, we present extensions to the known tables of pseudosquares and pseudocubes in §2. We discuss the implications of this new data on the conjectured distribution of pseudosquares and pseudocubes in §3, and give a minor reﬁnement of the current conjectures. Then we describe our parallel algorithm, based on Bernstein’s doubly-focused enumeration [2], which is used in a way similar, but not identical to the work of Wooding and Williams [14], combined with the space-saving wheel data structure presented in [10, §4.1]. We then suggest ideas for future work in §5.

Supported by a grant from the Holcomb Awards Committe, and computing resources provided by the Frank Levinson Supercomputing Center at Butler University.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 331–339, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Computational Results

Let (x/y) denote the Legendre symbol [5]. For an odd prime p, let Lp,2 , the pseudosquare for p, be the smallest positive integer such that 1. Lp,2 ≡ 1 (mod 8), 2. (Lp,2 /q) = 1 for every odd prime q ≤ p, and 3. Lp,2 is not a perfect square. In other words, Lp,2 is a square modulo all primes up to p, but is not a square. We found the following new pseudosquares: p Lp,2 367 36553 34429 47705 74600 46489 373 42350 25223 08059 75035 19329 379 > 1025 The two pseudosquares listed were found in 2008 in a computation that went up to 5 × 1024 , taking roughly 3 months wall time. The ﬁnal computation leading to the lower bound of 1025 ran for about 6 months, in two 3-month pieces, the second of which ﬁnished on January 1st, 2010. Wooding and Williams [14] had found a lower bound of L367,2 > 120120 × 264 ≈ 2.216 × 1024 . (Note: a complete table of pseudosquares, current as of this writing, is available at http://cr.yp.to/focus.html care of Dan Bernstein). Note that 1025 may be used as a lower bound for L379,2 in the pseudosquares prime test. Together with trial division to guarantee there are no divisors below, say, 1010 , this means the pseudosquares prime test is practical on integers of 35 decimal digits, especially in the context of a prime sieve [10]. Similarly, for an odd prime p, let Lp,3 , the pseudocube for p, be the smallest positive integer such that 1. 2. 3. 4.

Lp,3 ≡ ±1 (mod 9), (q−1)/3 Lp,3 ≡ 1 (mod q) for every prime q ≤ p, q ≡ 1 (mod 3), gcd(Lp,3 , q) = 1 for every prime q ≤ p, and Lp,3 is not a perfect cube.

We found the following new pseudocubes (only listed for p ≡ 1 (mod 3)): p 499 523,541 547 571,577 1 601,607 2 613 67 619

601 1166 41391 62485 41913 44415

25695 14853 50561 73199 74719 80981

21674 91487 50994 87995 36148 24912

Lp,3 16551 89317 02789 15947 78852 27899 69143 39717 42758 90677 90374 06633 > 1027

These pseudocubes were found in about 6 months of total wall time in 2009. Wooding and Williams [14] had found a lower bound of L499,3 > 1.45152 × 1022. For a complete list of known pseudocubes, see [14,4,11].

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3

333

The Distribution of Pseudosquares and Pseudocubes

Let pi denote the ith prime, and qi denote the ith prime such that qi ≡ 1 (mod 3). In [6] it was conjectured that, for a constant c2 > 0, we have Lpn ,2 ≈ c2 2n log pn .

(1)

Using similar methods, in [4] it was conjectured that, for a constant c3 > 0, we have Lqn ,3 ≈ c3 3n (log qn )2 . (2) In a desire to test the accuracy of these conjectures, for integers n > 0 let us deﬁne Lpn ,2 , 2n log pn Lq ,3 . c3 (n) := n n 3 (log qn )2 c2 (n) :=

(3) (4)

We calculated c2 (n) and c3 (n) from known pseudosquares and pseudocubes. We present these computations in Table 1, for pseudosquares, and in Table 2, for pseudocubes, below. From Table 1, we readily see that c2 (n) appears to be bounded between roughly 5 and 162, with an average value near 45. There is no clear trend toward zero or inﬁnity. Due to the common occurence of values of n where Lpn ,2 = Lpn+1 ,2 (for example, n = 56), it should also be clear c2 (n) does not have a limit. Similarly for the pseudocubes, in Table 2 we see that 0.05 < c3 (n) < 6.5 for 10 ≤ n ≤ 53, with an average value of roughly 1.22. And again, there is no clear trend toward zero or inﬁnity, nor can there be a limit for c3 (n). This leads us to the following reﬁnements, if you will, of the conjectures (1),(2) above. Conjecture. For the pseudosquares, we conjecture that Lpn ,2 > 0, 2n log pn Lp ,2 < ∞. lim sup n n 2 log pn n→∞ lim inf n→∞

(5) (6)

Similarly, for the pseudocubes, we conjecture that Lqn ,3 n n→∞ 3 (log qn )2

lim inf

lim sup n→∞

> 0,

Lqn ,3 < ∞. 3n (log qn )2

(7) (8)

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Table 1. Values of c2 (n) based on known pseudosquares

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

pn 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109

Lpn ,2 c2 (n) 73 16.61 241 18.72 1009 32.41 2641 34.42 8089 49.28 18001 49.64 53881 71.48 87481 54.49 117049 33.95 515761 73.34 1083289 73.24 3206641 105.41 3818929 61.97 9257329 73.38 22000801 84.55 48473881 90.70 48473881 44.98 175244281 79.49 427733329 95.70 427733329 47.54 898716289 49.04 2805544681 75.69 2805544681 37.25 2805544681 18.28 10310263441 33.29 23616331489 37.96 85157610409 67.89 85157610409 33.81

n 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

pn Lpn ,2 c2 (n) 113 196265095009 38.67 127 196265095009 18.87 131 2871842842801 137.15 137 2871842842801 67.95 139 2871842842801 33.88 149 26250887023729 152.68 151 26250887023729 76.14 157 112434732901969 161.79 163 112434732901969 80.30 167 112434732901969 39.96 173 178936222537081 31.58 179 178936222537081 15.69 181 696161110209049 30.45 191 696161110209049 15.07 193 2854909648103881 30.84 197 6450045516630769 34.70 199 6450045516630769 17.32 211 11641399247947921 15.46 223 11641399247947921 7.65 227 190621428905186449 62.42 229 196640148121928601 32.14 233 712624335095093521 58.06 239 1773855791877850321 71.92 241 2327687064124474441 47.12 251 6384991873059836689 64.15 257 8019204661305419761 40.11 263 10198100582046287689 25.40 269 10198100582046287689 12.65 271 10198100582046287689 6.32 277 69848288320900186969 21.54 281 208936365799044975961 32.14 283 533552663339828203681 40.99 293 936664079266714697089 35.76 307 936664079266714697089 17.73 311 2142202860370269916129 20.23 313 2142202860370269916129 10.10 317 2142202860370269916129 5.04 331 13649154491558298803281 15.94 337 34594858801670127778801 20.14 347 99492945930479213334049 28.81 349 99492945930479213334049 14.39 353 295363187400900310880401 21.32 359 295363187400900310880401 10.63 367 3655334429477057460046489 65.54 373 4235025223080597503519329 37.86

Sieving for Pseudosquares and Pseudocubes in Parallel

Table 2. Values of c3 (n) based on known pseudocubes n 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

qn Lqn ,3 79 7235857 97 8721539 103 8721539 109 91246121 127 91246121 139 98018803 151 1612383137 157 1612383137 163 7991083927 181 7991083927 193 7991083927 199 20365764119 211 2515598768717 223 6440555721601 229 29135874901141 241 29135874901141 271 29135874901141 277 406540676672677 283 406540676672677 307 406540676672677 313 406540676672677 331 75017625272879381 337 75017625272879381 349 75017625272879381 367 996438651365898469 373 2152984914389968651 379 12403284862819956587 397 37605274105479228611 409 37605274105479228611 421 37605274105479228611 433 205830039006337114403 439 1845193818928603436441 457 7854338425385225902393 463 12904554928068268848739 487 13384809548521227517303 499 60125695216741655189317 523 116614853914870278915947 541 116614853914870278915947 547 4139150561509947885227899 571 16248573199879956914339717 577 16248573199879956914339717 601 24191374719361484275890677 607 24191374719361484275890677 613 674441580981249129037406633

c3 (n) 6.42 2.35 0.764 2.6 0.813 0.281 1.49 0.488 0.795 0.254 0.0827 0.0695 2.8 2.34 3.49 1.14 0.365 1.69 0.558 0.181 0.0598 3.61 1.2 0.394 1.71 1.23 2.34 2.33 0.77 0.254 0.459 1.37 1.91 1.04 0.355 0.527 0.336 0.111 1.31 1.69 0.56 0.274 0.0912 0.845

335

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It has been pointed out, both by one of the referees and by Rich Schroeppel [9], that a value for k > 0 such that Lpn ,2 = Lpn+1 ,2 = · · · = Lpn+k ,2 likely is not bounded. This applies to pseudocubes as well. It implies that we, most likely, cannot simultaneously have both (5) and (6), nor both of (7) and (8). This might be avoided if we, say, multiply our upper bounds by n and divide our lower bounds by n in our conjectures. Our data also has implications on the relative eﬃciently of primality testing. In particular, several researchers have pointed out that if conjectures (1),(2) are true, then the running time of the pseudocube prime test, which depends on 2/3 the value of Lqn ,3 , should eventually outperform the pseudosquare prime test, whose running time depends on Lpn ,2 . In particular, one infers from conjectures (1) and (2) that 2/3 n 2/3 Lqn ,3 3 > 1 (9) Lpn ,2 2 for suﬃciently large n (see [14, §9.1]). This inference follows from our reﬁned conjectures as well. We have our ﬁrst speciﬁc value of n to support (9), namely with n = 48, where 2/3 Lqn ,3 ≈ 2.214 · Lpn ,2 . However, given that c2 (n) averages about 45, and c3 (n) averages just over 1.2, we would reasonably expect (9) to largely be true only for n larger than about 75, under the assumption these averages are maintained. To test this, more pseudosquares and, in particular, more pseudocubes are needed.

4

Algorithm Details

We begin with a review of doubly-focused enumeration, explain how we employ parallelism, and how the space-saving wheel datastructure is utilized. We also discuss the details of our implementation, including the hardware platform and software used. 4.1

Doubly-Focused Enumeration

The main idea is that every integer x, with 0 ≤ x ≤ H, can be written in the form x = t p M n − tn M p (10) where gcd(Mp , Mn ) = 1,

0 ≤ tp ≤

H + Mn Mp , Mn

and 0 ≤ tn < Mn .

(11)

(See [2] or [14, Lemma 1].) This is an explicit version of the Chinese Remainder Theorem.

Sieving for Pseudosquares and Pseudocubes in Parallel

337

To ﬁnd pseudosquares, we set Mn and Mp to be products of small odd primes and 8, choose tp to be square modulo Mp , and −tn to be square modulo Mn . To be precise, in our implementation we set Mp = 7 · 11 · 13 · 17 · 19 · 23 · 29 · 31 · 37 · 41 · 43 · 53 · 89 = 2057 04617 33829 17717 and Mn = 8 · 3 · 5 · 47 · 59 · 61 · 67 · 71 · 73 · 79 · 83 · 97 = 4483 25952 77215 26840. Note that both Mp , Mn < 264 , allowing us to work in 64-bit machine arithmetic. To ﬁnd pseudocubes, the same idea applies, only note that if −tn is a cube modulo Mn , so is tn . We used only 2, 9 and primes congruent to 1 (mod 3) for better ﬁlter rates: Mp = 2 · 7 · 13 · 31 · 43 · 73 · 79 · 127 · 139 · 157 · 181 = 701 85635 61110 39402 and Mn = 9 · 19 · 37 · 61 · 67 · 97 · 103 · 109 · 151 · 163 = 693 11050 43291 92503 4.2

Parallelism and Main Loop

Each processor core was assigned an interval of tp values to process by giving it values of H − and H + . For ﬁnding pseudosquares, H + − H − ≈ Mn · 4.76 × 1011 . For ﬁnding pseudocubes, H + − H − ≈ Mn · 4.99 × 1012 . Parallelism was achieved by having diﬀerent processors working on diﬀerent intervals simultaneously. Once all processors had ﬁnished their current intervals, the work was saved to disk (allowing restarts as needed) and new intervals were assigned. To process an interval, each processor core did the following: 1. Using the wheel datastructure, generate all square or cube values of tp with H − ≤ tp Mn ≤ H + , and store these in an array A[]. 2. The wheel datastructure does not generate the tp values in order, so sort A[] in memory using quicksort. Note that H − and H + are chosen close enough together so that this array held no more than 40 million integers, using at most 320 megabytes of RAM per processor core. 3. Using the ﬁrst and last entries in A[], compute a range of valid tn values to process, and then use a wheel datastructure to generate all tn values in that range such that −tn is square modulo Mn for pseudosquares, or tn is a cube modulo Mn for pseudocubes. We use an outer loop over tn values in the order enumerated by the wheel data structure for Mn , and an inner loop over consecutive tp values drawn from A[].

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4. For each tn generated, we normalize sieve tables for the next 4 primes (101, 103, 107, 109 for pseudosquares, and 193, 199, 211, 223 for pseudocubes) to allow for constant-time table lookup to see if an x-value (see below) is a square/cube modulo these primes, indexed by tp value. The number of primes to use for this depends on how many tp values will be processed for each tn – in our case, it was several hundred on average, so this step improves performance. If it were fewer, say 50, then normalizing the sieve tables would require more work than is saved by having constant-time lookup. 5. For each tn generated, using binary search on A[] to ﬁnd all the tp values it can match with, generate an x = tp Mn − tn Mp within our global search range. (For example, in our last run for pseudosquares, we searched for x values between 7.5 × 1024 and 1025 .) Note: at this point we do not actually compute the value of x. 6. Lookup each tp value in the normalized tables mentioned above. If it fails any of the 4 sieve tests, move on to the next tp value. For pseudosquares, a tp values passes these tests with probability roughly (1/2)4 = 1/16, and for pseudocubes, roughly (1/3)4 = 1/81. Note that this step is the running time bottleneck of the algorithm. 7. The next batch of primes q have precomputed sieve tables that are not normalized, but we precompute Mp and Mn modulo each q so the we can compute x mod q without exceeding 64-bit arithmetic. Continue only if our tp value passes all these sieve tests as well. The expected number of primes q used in this step is constant. 8. Finally, compute x using 128-bit hardware arithmetic, and see if it is a perfect square or perfect cube. If it passes this test, append x to the output ﬁle for this processor core. We had two wheel datastructures, one each for Mp and Mn . For details on how this datastructure works, see [10]. We leave the details for how to modify the datastructure to handle cubes in place of squares to the reader. 4.3

Implementation Details

To compute the tables presented in §2, we used Butler University’s cluster supercomputer, BigDawg, which has 24 compute nodes, each of which has four AMD Opteron 8354 quad-core CPUs at 2.2GHz with 512KB cache, for a total of 384 compute cores. As might be expected, we did not have sole access to this machine for over a year, so the code was designed, and ran, using anywhere from 10 to 24 nodes, or from 160 to 384 cores, depending on the needs of other users. This ﬂexibility is one advantage of our parallelization method – by tp intervals. In [14], they parallelized over residue classes, which restricts the CPU count to a ﬁxed number (180 in their case). BigDawg runs a Linux kernel on its head node and compute nodes, and the code was written in C++ using the gnu compiler (version 4.1.2) with MPI. It has both 10GB ethernet and Inﬁniband interconnect, but inter-processor communication was not a bottleneck for our programs.

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We tested our code by ﬁrst ﬁnding known pseudosquares (all but the highest few) and known pseudocubes, in the process verifying previous results.

5

Future Work

We plan to port our code to work with 8 NVidia GPUs recently added to Butler’s supercomputer, giving it roughly 2-3 times the raw computing power. This will require a major restructuring of the code, and the removal of recursion in the wheel datastructure.

References 1. Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. of Math. 160(2), 781– 793 (2004), http://dx.doi.org/10.4007/annals.2004.160.781 2. Bernstein, D.J.: Doubly focused enumeration of locally square polynomial values. In: High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, pp. 69–76. Amer. Math. Soc., Providence (2004) 3. Bernstein, D.J.: Proving primality in essentially quartic random time. Math. Comp. 76(257), 389–403 (2007), http://dx.doi.org/10.1090/S0025-5718-06-01786-8 (electronic) 4. Berrizbeitia, P., M¨ uller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004) 5. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979) 6. Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Math. Comp. 65(213), 361–372, S25–S27 (1996) 7. Pomerance, C., Shparlinski, I.E.: On pseudosquares and pseudopowers. In: Combinatorial Number Theory, pp. 171–184. Walter de Gruyter, Berlin (2009) 8. Schinzel, A.: On pseudosquares. New Trends in Prob. and Stat. 4, 213–220 (1997) 9. Schroeppel, R.: Private communication (February 2010) 10. Sorenson, J.P.: The pseudosquares prime sieve. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 193–207. Springer, Heidelberg (2006) 11. Stephens, A.J., Williams, H.C.: An open architecture number sieve. In: Number theory and cryptography (Sydney, 1989). London Math. Soc. Lecture Note Ser., vol. 154, pp. 38–75. Cambridge Univ. Press, Cambridge (1990) ´ 12. Williams, H.C.: Edouard Lucas and primality testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. John Wiley & Sons Inc, New York (1998), A Wiley-Interscience Publication 13. Wooding, K.: The Sieve Problem in One- and Two-Dimensions. Ph.D. thesis, The University of Calgary, Calgary, AB (April 2010) http://math.ucalgary.ca/~ hwilliam/files/wooding10thesis.pdf 14. Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006) 15. Wooding, K., Williams, H.C.: Improved primality proving with Eisenstein pseudocubes. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 372–384. Springer, Heidelberg (2010)

On the Extremality of an 80-Dimensional Lattice Damien Stehl´e1,2 and Mark Watkins2 2

1 CNRS and Macquarie University Magma Computer Algebra Group, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia [email protected], [email protected]

Abstract. We show that a speciﬁc even unimodular lattice of dimension 80, ﬁrst investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal lattice in this dimension. The known part of its automorphism group is isomorphic to SL2 (F79 ), which is smaller (in cardinality) than the two previous examples. The technique to show extremality involves using the positivity of the Θ-series, along with fast vector enumeration techniques including pruning, while also using the automorphisms of the lattice.

1

Introduction

We show that a speciﬁc 80-dimensional even unimodular lattice is extremal, that is, that it has no (nonzero) vectors of norm less than 8. It follows that the kissing number of this lattice is 1 250 172 000.1 Although two other even unimodular extremal lattices in dimension 80 are known [3], the one we describe has a construction related to coding theory, and has an automorphism group that contains SL2 (F79 ). In Section 2 we recall some facts and results about extremal lattices. In Section 3 we follow the method of Schulze-Pillot [40] to construct our lattice N80 as a 2-neighbour of a lattice derived from a length 80 extended quadratic residue code over F19 . The prime 19 here is not overly signiﬁcant; the construction√produces ﬁve unimodular lattices in correspondence with the class group of Q( −79), and the ideal class that yields N80 (the only extremal one among the ﬁve) has an ideal of norm 19 in it.2 Alternatively, a variation (see [1]) on a method of Gross [18, §11] can be used to construct N80 , and deals more directly with the ideals of this imaginary quadratic ﬁeld. Via either method, it is fairly immediate that N80 has an automorphism group that contains SL2 (F79 ). In Section 4 we note that various choices of bases make the group action nice (doubly transitive as signed permutations on the coordinates), and then make a speciﬁc basis choice that relates directly to the construction in [1]. 1

2

We do not describe herein any features of these minimal vectors. In fact, the 2 555 orbits of these vectors under the known automorphisms were ﬁrst found (without proof of completeness) by the authors of [1], with whom we started this project. We could also have chosen l = 5 (as indicated in [40, Example 3]), but for technical reasons (in lattice generation) wanted l not to be too small.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 340–356, 2010. c Springer-Verlag Berlin Heidelberg 2010

On the Extremality of an 80-Dimensional Lattice

341

In Section 5 we ﬁrst brieﬂy outline our method of proof that the lattice N80 is extremal. We need to show that N80 has no nonzero vectors of norm 6 or smaller. We can almost immediately eliminate vectors of norm 2, while a slightly more involved argument is necessary to show there are no vectors of norm 4. We then use the nonnegativity of the coeﬃcients of the Θ-series of the lattice to reduce the problem of showing that there is no vector of norm 6 to the problem of ﬁnding (almost) all the vectors of norm 10. The latter is feasible due to the fact that we need only ﬁnd one representative in each orbit class under the known automorphisms, whereas the more direct method of an exhaustive search for norm 6 vectors would be signiﬁcantly more time-consuming. After ﬁrst cataloguing the norm 10 orbits that have a nontrivial stabiliser, all the other vectors will have a full orbit under the known automorphisms, and so we can reduce the problem by a factor of approximately #SL2 (F79 ) = 492 960. This leaves us with only 15.3 million orbits of norm 10 to ﬁnd. In Section 6 we describe our method to ﬁnd all the norm 10 orbits. One principal idea is to prune the tree corresponding to the Kannan-Fincke-Pohst enumeration algorithm that ﬁnds all short lattice vectors [21,12]. Our tree pruning strategy, which generalizes that of [38, §7] and improves the one from [39], considers a truncated search domain that is much smaller but still ﬁnds a signiﬁcant proportion of the desired vectors. Note that the pruning strategy we describe and its analysis have been independently discovered by Gama, Nguyen, and Regev [15, §4]. In our case, we need only ﬁnd one vector in each orbit class, so the fact we miss some vectors when searching is unimportant. Another idea to speed the search is to periodically apply a random perturbation to the basis and re-apply lattice reduction (namely LLL with deep insertions [38]), before again searching with tree pruning. As our lattices are of quite high dimension, the new basis is very likely to be diﬀerent than the previous ones. This can help in two ways: ﬁrstly, searching with a given lattice basis for short vectors, even with pruning available, tends to become less cost-eﬀective over time, in terms of the number of vectors found per second; and secondly, and rather surprisingly to us, a “good basis” for searching can sometimes have many orbit classes which will not show up until quite deep in the search. We still do not understand this latter phenomenon, but it is easily overcome via the random perturbations. Section 7 gives our results and veriﬁcation methods, plus related questions. Computations. All timings are given for 2.3Ghz Opteron 8356 processors. If otherwise unspeciﬁed, only one processor is used.

2

Extremal Lattices

The extremality of a lattice is typically deﬁned using Θ-series, as for instance in [7, §7.4].3 In particular, an extremal unimodular even lattice in dimension d with 8|d has a minimum nonzero vector norm of 2(1+d/24), as this is twice the 3

The precise notion of “extremal” seems to vary over time; for instance [6] is more demanding, asking that the minimum be at least 1 + d/8.

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dimension of the associated space of modular forms. For odd lattices, shadow theory is typically used to obtain satisfactory bounds [8]. A relatively recent survey on extremality appears in [14]. In particular, there were already two extremal even unimodular lattices known in dimension 80, both due to Bachoc and Nebe [3] via a coding theory construction. The ﬁrst lattice L80 has an automorphism group 2.A7 ⊗√−7 2.M22 .2 of size 212 34 52 72 11 = 4 470 681 600, and this group is known to be a maximal ﬁnite subgroup of GL80 (Z) (see [3, Theorem 3.2]). The second extremal lattice M80 has known automorphisms [3, Lemma 4.11] of order 212 34 52 = 8 294 400. For comparison, the number of known automorphisms of our lattice is 492 960. Our lattice N80 is isometric neither to L80 nor M80 . The argument for L80 is immediate, as its automorphism group is known to be maximal but 79 does not divide the order. For M80 we can compute the minimal vectors in a few days, and perhaps argue via some property of them versus those for N80 . We can also argue via Aschbacher’s theorem on maximal subgroups of ﬁnite classical groups, and in an appendix, we sketch a proof along these lines, showing that Aut(N80 ) is a maximal ﬁnite subgroup of GL80 (Z) up to a possible index of 4. The idea of extremality can also be extended to include other lattices which are isomorphic to their dual(s). In this case, the full space of modular forms is typically replaced by the subspace that is ﬁxed under the Atkin-Lehner involutions [36]. This then relates the question to a simultaneous maximisation of the minimum of a lattice and that of its shadow; see [13] and [32] for instance. Finally, we note that [28] shows that there are only ﬁnitely many extremal lattices, though the most easily computed bound on maximal dimension still seems to be quite high.4 In the other direction, King [22] classiﬁes all (even) unimodular lattices in dimension 32 with no roots, and ﬁnds there to be at least 107 such; as the lack of roots implies that the lattices have no vectors of norm 2, it follows that each is extremal. Similarly, Peters [33] shows there are at least 1051 extremal lattices in dimension 40.

3

Construction of the Lattice N80

We follow the paper [40] of Schulze-Pillot on quadratic residue codes and cyclotomic lattices, which builds on works from Thompson, Feit [9], and Quebbemann [35, §3] about unimodular lattices with an automorphism of prime order. 4

The proof therein is similar in ﬂavour to the idea we exploit, that is, for suﬃciently large dimension, the ﬁrst form in a triangular basis will have coeﬃcients that are negative, and thus positivity precludes the existence of an extremal lattice. See the recent [42, p. 36] for a brief sketch. Our computations give that the q n+2 term in the expansion is negative for n ≥ 6 775, 6 789, 6 803 for the respective 0, 8, 16 mod 24 classes, which gives an upper bound of 163 264 = (6802· 24) + 16 for the dimension of an even unimodular extremal lattice. Finally, Rains [37] has followed upon the work of Krasikov and Litsyn [27] to obtain that the minimal norm of a unimodular lattice is (asymptotically with dimension d → ∞) smaller than the Siegel bound ∼ d/12 by at least a constant factor (see N = 1 in the Remark after Theorem 4.2 in [37]).

On the Extremality of an 80-Dimensional Lattice

343

The construction gives a unimodular lattice as a sublattice of index p in a (rescaled) direct sum of two lattices of dimensions 2 and (p − 1). In this, the 2-dimensional lattice T2 can be taken as any integral lattice of determinant p. The lattice Up−1 of dimension (p − 1) comes about from an (unpublished) construction of Thompson (see [9, §9]). We let E = Q(ζp ) be cyclotomic, and take ¯ = (d) with d ∈ E + totally positive. This ideal an ideal A ⊆ OE such that AA induces a (positive deﬁnite) lattice of dimension (p − 1) via a basis for the ring of integers Z[ζp ], with the quadratic form given by Q1 (u) = trE ud−1 ). Via a Q (u¯ computation (with the diﬀerent as in [9, Theorem 9.3], or with a Vandermonde determinant) one can show that the lattice Up−1 has determinant pp−2 . To obtain a unimodular lattice of dimension (p + 1), we start with the direct sum T2 ⊕ Up−1 , and take the sublattice of this consisting of all vectors whose norm is a multiple of p. Upon dividing the whole lattice by p, the result will be integral and unimodular, the latter since (p·pp−2 )·p2 /pp+1 = 1. We need to show that this actually yields a sublattice, that is, the resulting subset of the original lattice satisﬁes the group law, and this is most easily done via homomorphic projection maps. We take the lattice N(T2 , Up−1 ) = {(m, u) ∈ T2 ⊕ Up−1 | π(m) = ρ(u)} under the quadratic form Q (m, u) = Q0 (m) + Q1 (u) /p, with the projection maps being π : T2 → R/radQ0 (R) where R = T2 /pT2 , and ρ˜ : A → A/(1 − ζp )A (here ρ˜ is on A, with ρ on Up−1 ). Since (1 − ζp ) has norm p, both images will be vector spaces over Fp of dimension 1, and we can identify them (arbitrarily) by taking m0 ∈ T2 and u0 ∈ A with Q0 (m0 ) ≡ 1 (mod p) and u0 u¯0 d−1 ≡ 1 (mod (1 − ζp )OE ). The lattice N(T2 , Up−1 ) will be even if and only if T2 is even. 3.1

An Odd Lattice

Rather than derive our desired even unimodular lattice directly, we again follow Schulze-Pillot, who ﬁrst constructs an odd lattice for which the automorphism group can be determined via a relation to coding theory, and then passes to an even lattice via Kneser’s neighbouring construction. √ We let K be the imaginary quadratic ﬁeld Q( −79), and d = l = 19 an auxiliary prime that splits. Writing (l)OK = l¯l, the location of l in the class group of K will have a determining factor on the lattice we derive in the end, and so the choice of l is not completely arbitrary. Welet a be the ideal of K generated by l and the twisted Gauss sum 12 1 − 33 a χp (a)ζpa where χp is the quadratic character modulo p. Using the notation of Schulze-Pillot, we have p = −j 2 + 8ml with p = 79, j = 15, m = 2, and l = 19, so that yj ≡ 1 (mod l) together with y ≡ 1 (mod 4) yields y = 33.5 Noting that a¯a = (l) and ¯ = (19) in OE . Letting T2 be taking E = Q(ζ79 ), we write A = aOE so that AA the 2-dimensional lattice (in a basis {w1 , w 2 }) of determinant 79 given by the 5

The import of this numerology only becomes clear when proofs are included, as this choice of y for the scaling factor of the Gauss sum allows one to show that the cyclotomic and coding theory constructions agree.

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19 15 l j , we ﬁx the gluing via π(w 1 ) = ρ( lζp ), = Gram matrix Q0 = 15 16 j 8m where here · gives the map from A to Up−1 . We let No = N(T2 , Up−1 ) with these choices, noting that No is odd.

3.2

Relation to Coding Theory

We can obtain correspondence with coding theory by taking p coordinates the as ei = w1 ⊕ lζpi for 0 ≤ i ≤ p − 1 and an additional one e∞ = jw1 − lw2 , from which a computation shows that these ei form a scaled root system of type 80A1 in No , that is, each ei has the same norm, and they are all mutually orthogonal. Indeed, for all 0 ≤ i ≤ p − 1 we have ei = Q0 (w 1 ) + (p − 1) · (l2 /l) /p = l since Q0 (w 1 ) = l, while e∞ = Q0 (jw 1 − lw2 )/p = l(8ml − j 2 )/p = l. For the inner products, we have

ei , ek = ei + ek − ei − ek i

1 k ¯i ¯k Q0 (2w1 ) + (l2 /l) · trE − 2l = Q (ζ + ζ )(ζ + ζ ) p

1 i−k 4l + l · trE + ζ i+k − 2l = Q 2+ζ p 1 = 4l + l · [2(p − 1) − 1 − 1] − 2l = 0 p when i = k and i, k = ∞, while for i = ∞ we have

ei , e∞ = ei + e∞ − 2l 1 = Q0 (j + 1)w1 − lw2 + (p − 1) · (l2 /l) − 2l p 1 = l(1 + 8ml − j 2 ) + l(p − 1) − 2l = 0. p Using this root system, it follows that the extended quadratic residue code C ⊆ F80 l (or indeed, any self-dual code) gives an integral unimodular lattice via

1 ai ) ∈ C ai ei (¯ (1) NC = l i where the sum is over all 80 coordinates, and a ¯i is reduction mod l of ai . The proof that NC is the same lattice as our lattice No is given in [40, Proposition 1], using the generator matrix and idempotent of the code.6 The appearance of the value y = 33 with the Gauss sum is of relevance therein. 6

We have taken a sublattice of index lp+1 via the scaled root system, and then taken a superlattice of the same index via the construction from coding theory, and so just have to check that these operations are compatible.

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One nicety of this re-visioning is that the code automorphism (of order 4) given by a∞ → a0 , a0 → −a∞ , ai → −χp (i)aj , where ij ≡ −1 (mod p), can be seen to lift to the lattice. Combined with the order p automorphism induced via ζp , which ﬁxes a∞ and cycles a0 → a1 → · · · → ap−1 → a0 , this gives SL2 (Fp ) as a subgroup of the automorphism group Aut(No ) of the lattice. In an appendix, we use the classiﬁcation of ﬁnite simple groups to show that this realisation of SL2 (F79 ) is within a factor of 4 of being a maximal ﬁnite subgroup of GL80 (Z), so that [Aut(No ) : SL2 (F79 )] ≤ 4. 3.3

The Even 2-Neighbours

The above lattice No is odd, while we wish to get an even unimodular lattice. The method of passing to this is given by the neighbouring method of Kneser [26]. Again following Schulze-Pillot, we want to ﬁnd v ∈ No with Q(v) ∈ 4Z, and then take the lattice spanned by v/2 and the sublattice of No whose inner product with v is even. Via linear algebra over F2 , we ﬁnd that there is a 2-dimensional space of such v satisfying the conditions (Schulze-Pillot notes this in general via genus theory). Obviously v = 0 does not help us, while we also need Q(v) ∈ 8Z if the resulting neighbouring lattice is to be even, and this eliminates another of the initial 4 possibilities. This leaves but 2 choices for v, one of which gives a lattice with many vectors of norm 4 (note that v itself must have norm at least 32 if the new lattice is to have minimum 8) and the other of which is our desired lattice N80 . As in [40, Proposition 2], we could construct N80 directly using a diﬀerent choice with T2 in the cyclotomic construction, though the relation to coding theorythen becomes less clear. For instance, [40, Example 3] takes l = 5 and 8 1 to get the same N80 . Finally, the last Remark of [40] notes the Q0 = 1 10 automorphisms of No given by SL2 (Fp ) all transfer to N80 . As noted above, we show in an appendix that [Aut(N80 ) : SL2 (F79 )] ≤ 4 so that in particular N80 and M80 are not isometric, but our proof of extremality does not use this.

4

Nice Bases for N80

We next link N80 to the construction given in [1] that modiﬁes the method of Gross. The authors of [1] construct the lattice from a representation that is irreducible away from 2. In particular, in the basis they obtain, all the coordinates are of the same parity. Furthermore, the automorphisms are given by a doubly transitive signed permutation action on the coordinates. From our construction, we have a lattice N80 with automorphisms generated by two matrices O79 and O4 . We wish to transform this so that the automorphisms are generated by signed permutations σ79 and σ4 (as in the end of Section 3.2), thus giving a doubly transitive coordinate action. One way to achieve this is just to solve the 802 -dimensional linear algebra problem given by

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equating the automorphisms, that is, solve O79 X = Xσ79 and O4 X = Xσ4 for the unknown matrix X (we try solving this with both σ4 and σ43 ). It turns out that the resulting solution space is 2-dimensional, and if we write X1 and X2 for generators of it, then the determinant of the matrix X1 t + X2 u is given by 240 f (t, u)40 where f is a binary quadratic form of discriminant −79 corresponding to the ideal of above. To obtain the representation of [1] we choose the pair (t, u) so that f (t, u) = 8, so that the transform maps vectors of norm 10 in N80 to vectors of norm 16 · 10 in the resulting sublattice of Z80 . The resulting basis has the property that every vector has coordinates all of the same parity. We denote this transform matrix from N80 to Z80 by T16 , and the resulting lattice basis by B80 . 4.1

Identifying Orbits

As noted above, the action of σ79 and σ4 is doubly transitive, and we can exploit this to expedite the ﬁnding of a canonical representative for a given orbit. We ﬁrst ﬁnd the largest coordinate in absolute value, and move it to the front, and then cycle the latter 79 coordinates until the second largest is in the second position. This movement uses 80 · 79 elements of the group, and after modding out by the centre {±1}, we only have 39 possibilities left to check for their 78 latter coordinates (we use a lexicographic ordering). Of course, we could have many ties amongst the two largest coordinates (this is basis-dependent, and we can map to another choice of (t, u) if desired), but this method will still be much faster than looping over all 492 960 possibilities.

5

Method of Proof

We now describe how we shall show that N80 is indeed extremal. Since the lattice N80 is even and unimodular, its Θ-series Θ80 lies in the vector space of modular forms of level 1 and weight 40 (see [30]). This space has dimension 4, and a triangular integral basis is: f0 = 1 + 1 250 172 000 q 4 + 7 541 401 190 400 q 5 + O(q 6 ), f1 = q + 19 291 168 q 4 + 37 956 369 150 q 5 + O(q 6 ), f2 = q 2 + 156 024 q 4 + 57 085 952 q 5 + O(q 6 ), f3 = q 3 + 168 q 4 − 12 636 q 5 + O(q 6 ). We thus know that Θ80 = f0 + a1 f1 + a2 f2 + a3 f3 for some integers ai . We shall derive that a1 = a2 = 0 by showing that there are no vectors of norm 2 or 4 in the lattice. We will then have Θ80 = 1 + a3 q 3 + (· · · )q 4 + (7 541 401 190 400 − 12 636 a3) q 5 + O(q 6 ). By positivity we have a3 ≥ 0, and so by ﬁnding 7 541 401 190 400 vectors of norm 10 in the lattice, we deduce that a3 = 0 so that N80 is extremal as claimed.

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The reader might wonder why we do not simply search for norm 6 vectors, but instead aim to ﬁnd all those of norm 10, as the latter (at ﬁrst glance) seems much harder. However, the search in norm 6 has to be exhaustive, while with norm 10 it need not be: we ﬁnd one vector in each orbit, and apply automorphisms to get the whole set. We estimate an exhaustive search for norm 6 vectors would take more than 1 000 times as much work as our method using norm 10 vectors. 5.1

The Lattice N80 Has No Vectors of Norm 2 or 4

As we noted above in Section 4, we can change the basis by a transform T16 so that each vector has its norm multiplied by 16, with the resulting basis having the property that all the coordinates of any vector will have the same parity. In particular, a vector of norm 2 or 4 will have the square-sum of its coordinates as 32 or 64, with necessarily all coordinates being even. Also, the inner product of any two vectors in this basis will need to be a multiple of 16, a fact we exploit below. Finally, the lattice automorphisms in this new basis are given by signed permutations, with the action doubly transitive. No vectors of norm 2 (roots). One proof (from Elkies) ﬁrst notes that the only root systems with compatible automorphisms are A80 1 and D80 . With the former, any automorphism of order 79 would necessarily ﬁx at least one of the 160 roots, but the 2-dimensional sublattice of N80 ﬁxed by a 79-cycle has no roots. The latter is similarly impossible; a 39-cycle must ﬁx a root since gcd(39, 12 640) = 1, but the 4-dimensional sublattice therein lacks roots. Another way (similar to a comment in [40, Example 3]) would be to use l = 5 and note that we must have i a2i = 2l = 10 in (1), while the minimal distance7 of the extended quadratic residue code of length 80 over F5 is > 10, though care needs to be made here when working with both N80 and the odd lattice L. A direct computation also easily shows that N80 has no roots. After applying suitable reduction, the veriﬁcation typically takes less than 30 minutes. We did not try a similar computation with norm 4, as we estimate that it would likely take a few months. e No vectors of norm 2 or 4. We let B80 be the sublattice of B80 given by e e vectors with even coordinates in the T16 basis, and map B80 → B80 /2 → F80 2 via the additive coordinate map generated by ±2 → ±1 → 1. The image in F80 2 is a binary code C2 , and this inherits the automorphisms from the lattice. e We have 16| v, w for any v, w ∈ B80 , which implies that C2 is doubly-even, that is, each codeword has weight divisible by 4. Similarly, we see that C2 ⊆ C2⊥ , as the inner product between any two codewords is 0 (in F2 ). We then show e equality here by ﬁnding enough vectors in B80 to show that dim(C2 ) ≥ 40. As C2 is self-dual and has automorphism group PSL2 (F79 ), it follows from either [25, Theorem 6.2] or [24, Satz 3.4] that C2 is equivalent to the extended 7

It seems that showing the minimal distance exceeds 20 would take about 58 days, though the computation should parallelise.

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binary quadratic residue code,8 and thus has minimal weight of 16 with 97 565 minimal codewords which lie in 3 orbits under the automorphisms.9 We now check that the preimages of codewords of weight 0 and 16 in C2 do not yield vectors of norm 2 or 4 in N80 .10 This is done using the explicit form −1 −1 of T16 . For weight 0, we need to check that T16 w is non-integral for w = 8, 0, . . . , 0, 4, ±4, 0, . . . , 0, 4, ±4, (. . .) where in this third expression exactly two of the latter 78 coordinates have size 4. By the doubly transitive nature of the automorphism action, this suﬃces. There are thus 3 + 23 78 = 24 027 possibilities to check here. 2 For weight 16, we have 3 orbits of codewords. For each orbit we take a representative, and lift its nonzero coordinates in 216 ways to every choice of sign −1 for ±2. We then apply T16 to each, and note that none are integral. This completes the proof that there are no vectors of norm 2 or 4 in the lattice N80 . e Presumably we could similarly show that B80 has no vectors of norm 96, but extending our observations to odd-coordinate vectors in B80 looks more diﬃcult. 5.2

Vectors with a Nontrivial Stabiliser

We now describe how to use the known automorphisms to reduce our vectorﬁnding quota from 7.5 trillion vectors down to about 15.3 million. We make a separate computation of the norm 10 vectors that have nontrivial stabiliser. If a vector v has a nontrivial stabiliser under the above action of G = SL2 (F79 ), there is some nontrivial element g ∈ G such that the kernel of g − id contains v. So we loop over nontrivial elements (or conjugacy classes) of G, compute this kernel (which is a sublattice), and then search for short vectors in it. The elements of order 3 give a kernel sublattice of dimension 28, for which it takes a few seconds to ﬁnd the vectors of norm ≤ 10. These yield 465 orbit classes under the action. The elements of order 5, 39, and 79 give lattices of dimensions 16, 4, and 2, and yield 15, 2, and 1 orbits respectively. Upon computing the stabilisers, we obtain – – – – 8 9

10

1 orbit with stabiliser size 79 · 39 = 3081 (order 79), 2 orbits with stabiliser size 39 (order 39), 15 orbits with stabiliser size 5 (order 5), 465 orbits with stabiliser size 3 (order 3).

We thank Elkies for recalling this fact, and J. Cannon for the Klemm reference. Here is an alternative method. Assume ﬁrst that there is a codeword w of weight 4 or 8. Take a 79-cycle σ and note that since (8−1)2 < 79 there is some iterate of σ such that w and σw intersect only in the ﬁxed coordinate. This implies that w, σw = 1, which contradicts that C2 is self-dual. Since there are no codewords of weight 4 or 8, we can then apply Gleason’s theorem [16] and get that the weight enumerator is of the form q 0 + (a + 15 200) q 12 + (127 965 + 2a) q 16 + (11 347 488 − 101a) q 20 + . . . for some a ∈ Z, and in an echo of our proof of lattice extermality, show code extremality (no codewords of weight 12) via ﬁnding 12 882 688 codewords of weight 20; for this, we ﬁnd short vectors in the lattice, map to the code, and apply automorphisms. We do not explicitly need the fact that the code is extremal for this step, but only that we have all codewords of length 16 or less.

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None of the other 78 nontrivial conjugacy classes of SL2 (F79 ) yields an orbit with vectors of norm 10. We can also note that there no vectors of norm 6 with a nontrivial stabiliser (though this is not strictly necessary for our proof). An accounting then tells us that there are presumably 7 541 323 277 280 vectors of norm 10 yet unfound, and dividing by #SL2 (F79 ) = 492 960 predicts 15 298 043 orbits with trivial stabiliser. Via a standard coupon-collecting analysis [11, p. 213] we expect that about 250 million suitably random vectors of norm 10 should suﬃce to hit each orbit at least once. In fact, for the purposes of proving the lattice extremal, we need only ﬁnd (15 298 043 − 12 635) orbits (see the q 5 coeﬃcient of f3 , and use the fact that 492 960|a3 as we ﬁnd no vectors of norm 6 with nontrivial stabiliser), and due to the lengthy ﬁnal part of coupon-collecting,11 this reduces the expected running time by about 55%. However, for completeness, we still chose to ﬁnd all orbits.

6

General Search for Vectors of Norm 10

The general method to enumerate short vectors in a lattice is due to Kannan [21] and Fincke and Pohst [12]. This corresponds to the computation of the leaves of a huge tree. As noted by Schnorr and Euchner [38], this tree can be pruned to some extent. This can be thought of as searching ﬁrst in the areas of the search region which are more likely to contain short vectors, or, equivalently, removing the tree nodes that are less likely to produce useful leaves. The initial pruning strategy was later improved in [39]. We describe below a further improvement. 6.1

The Full KFP Tree Search

The basic method iteratively looks at the projections to the span of the ﬁrst i coordinates for decreasing i. We have a basis given by {bi } and wish to solve the inequality i xi bi 2 ≤ 10. Borrowing the common notation for lattice reduction, we take the Gram-Schmidt orthogonalisation, and translate the xi ’s by the μj,i ’s: bi = bi −

μi,j bj so that μi,j =

j

d

bi , bj for i > j, and y = x + μj,i xj . i i bj 2 j=i+1

Here d is the dimension. By substituing yi for xi , we get by positivity leads to the series of inequalities:

i

yi2 bi 2 ≤ 10, which

yd2 bd 2 ≤ 10, 2 bd−1 2 ≤ 10 − yd2 bd 2 , yd−1 ... y12 b1 2

≤ 10 −

d

yi2 bi 2 .

i=2 11

The comparison is between

N

N n=1 n

and

N

N n=12636 n

for N = 15 298 043.

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Note that for all i, the variable xi is an integer, while yi is a shift of xi by a ﬁxed amount (once xi+1 , . . . , xd have been chosen). The KFP method proceeds by look ing at all yd ’s satisfying the ﬁrst inequality, then all pairs yd−1 , yd satisfying the second, etc. In particular, the vectors with yi ≈ 0 for all i up to a given point will be found most easily (and these often correspond to small xi ’s). Also, to ﬁnd more short vectors earlier in the search procedure, it is useful to run over the diﬀerent possible the centre of the interval implied by the inequality yi2 b∗i 2 ≤ xi ’s2 from ∗ 2 10 − j>i yj bj : the variable xi will run across the integers by decreasing proximity to − j>i μj,i xj . This “zig-zag” strategy, introduced by Schnorr and Euchner [38], allows one to split the search of the tree in diﬀerent stages: in the ﬁrst stage, we have xj = 0 for all j > 1; then in the second stage we have xj = 0 for all j > 2 but x2 = 0; etc. We call stage i the period of time during which xj = 0 for all j > i but xi = 0. Stage i means that we have already reached level i in the KFP tree but not yet been in level i + 1 (level 1 corresponding to the leaves). The arithmetic operations corresponding to Gram-Schmidt orthogonalisation computations can be quite slow. The Magma [5] implementation of the KFP tree search replaces them by double precision ﬂoating-point arithmetic operations, in a fully reliable way (using [34]). 6.2

Tree Pruning

Our pruning strategy consists in restricting the above inequalities by a “pruning factor” that depends on the level. So the above inequalities become d

yi2 bi 2 ≤ 10 · Pj , ∀j

i=j

where Pj is the jth pruning factor. A version of this with a speciﬁc choice of Pj appears in [38, §7], and the general description as well as its analysis below have been independently obtained in [15, §4]. In the latter, the authors also introduce the concept of “extreme pruning”, which resembles but diﬀers from our bases switching strategy (see subsection below). The “best” choice for the pruning factors appears to be something like Pj = (d − j + 1)/d. We happened to choose Pj = 1 − (j − 1)/100 in practise. The idea here can be phrased as follows: we have a given quantity of “norm” (here 10) to spend on a vector; if we spend a lot on the coordinates xj to xd , there will then be a lesser chance that we can form an integral vector via some possible choice of the other coordinates, due to positivity and the fact that most coordinates will have at least some nonzero contribution. Eﬃcacy of pruning. To give an idea of the eﬃcacy of pruning, we can use the notion, from [19], of expected enumeration cost for a given lattice basis {bi } and for vectors of norm A (a function EnumerationCost is available in Magma [5]): d d π d−j+1 k=j A/ bk 2 . (2) Γ 1 + (d − j + 1)/2 j=1

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A typical enumeration cost for our bases with N80 was around 1023 . This is the expected number of nodes of the KFP tree. For comparison, the implementation in Magma [5] has a traversal rate of about 7.5 million nodes per second. By comparing this enumeration cost estimate to the expected 7.5·1012 vectors of norm 10, we ﬁnd that more than 1010 nodes are expected to be searched for each vector found. In the case of the pruned enumeration, the jth summand in (2) should be multiplied by the volume of the truncated hypersphere {(zj , . . . , zd ) : ∀i ≥ j, k≥i zk2 ≤ Pi }. By estimating these volumes with a Monte-Carlo rejection method (uniformly sampling points in the full hypersphere and counting how many belong to the truncation), we expect our pruning to gain a factor of around 104 here, at the cost of missing about 60% of the short vectors. These speedup and miss ratios are not constant across all levels of the search: they seem to be closer to 100 and 25% respectively for the levels of our interest (due to the early abort and perturbation strategy described below). 6.3

Switching Bases

The early stages of the tree search can have a signiﬁcantly better chance of providing short vectors, due primarily to the relative paucity of “uninteresting” branches that tend to become more numerous at higher levels. In practice, we would ﬁnd 105 vectors in about 30 minutes, for a ratio of about 150 000 nodes searched for each vector found, more than an order of magnitude lower than the above estimate, even with the pruning included. Every 15-30 minutes we would switch the basis by applying a random permutation to the coordinates of the current basis, and then multiplying by a random upper triangular matrix with ones on the diagonal and oﬀ-diagonal entries in {−1, 0, +1}. We then re-apply LLL (with a δ-value nearly 1) to the perturbed basis, and then LLL with deep insertions [38]. Overall, this takes only a few seconds. This basis switching also makes parallelisation essentially trivial. A second reason for periodically changing the basis is that (a phenomenon we found experimentally) there are some bases which “hide” many of the orbits, in the sense that every vector in such an orbit would not be found until we reach one of the latter stages. We currently have no explanation of this.

7

Conclusion and Related Work

We implemented the above in a combination of Magma [5] and C. As we typically found 105 vectors of norm 10 in about 30 minutes, the estimated time was around 52 days. Using 14 processors in parallel, it took us about 4 days in April 2009. 7.1

Software to Check Our Data

A veriﬁcation of our proof can be done in much less time than the computation itself. We provide software12 that takes less than 10 hours to verify that N80 12

The code is checkit80.c (to be run with arguments “10 ﬁlename ”) and the data is LAT80.n10.sc16.bz2 in the directory http://magma.maths.usyd.edu.au/~watkins

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is indeed extremal. The input consists of 15 298 526 entries that correspond to coordinate vectors in the T16 basis of Section 4. The following checks are run: – Each entry lexicographically follows its predecessor, −1 – Each entry has norm 160 and is integral when multiplied by T16 , – Each entry is lexicographically the ﬁrst in its orbit. The ﬁrst condition ensures that all entries are distinct, while the last ensures that each corresponds to a distinct orbit, with the middle condition implying that the vectors have norm 10 and are in N80 . We can also list the 483 orbits with nontrivial stabiliser, whose provenance can be checked separately. 7.2

Three Lattices of Dimension 72

The work in progress [1] investigates three lattices of dimension 72. Two of these are 2-neighbours of a lattice constructed via the extended quadratic residue code over F3 , and the other involves a code over Z/4Z. None of these turned out to be extremal (minimal norm of 8), and indeed, we know of no extremal lattice of this dimension. In fact, a recent preprint of Griess [17] claims to be the ﬁrst to prove a minimal norm as large as 6 for an even unimodular lattice of dimension 72. 7.3

Other Candidate Lattices for Extremality in Dimension 80

In [3], the authors note three other candidates for extremality amongst even unimodular lattices in dimension 80. One candidate comes from a cyclo-quaternionic construction given in [31, Remark 5.2], and its automorphism group contains SL2 (F41 ) ⊗ S˜3 , which is of comparable size to our SL2 (F79 ). We do not see how to facilitate the calculation of canonical orbit representatives as readily as in our case, but the fact that canonicalising took only about 5% of our running time indicates that our methods could work in this case, with suﬃcient eﬀort. The other two candidates come from a cyclotomic construction explored in [4], and have an automorphism group containing the general aﬃne linear group ∗ F+ 41 F41 . Our initial opinion is that the automorphism group (even if augmented by an order 4 element) is too small for our method to work well here. Acknowledgments. We thank the authors of [1], with whom we started this research, and S. R. Donnelly who shared some of his ideas with us. We also thank the anonymous reviewers for their recommendation to add a proof that the automorphism group of N80 diﬀers from those of L80 and M80 . The present work is part of the Australian Research Council Discovery Project DP0880724 “Integral lattices and their theta series”.

References 1. Abel, Z., Elkies, N.D., Kominers, S.D.: On 72-dimensional lattices (in preparation) 2. Aschbacher, M.: On the maximal subgroups of the ﬁnite classical groups. Invent. Math. 76(3), 469–514 (1984), http://dx.doi.org/10.1007/BF01388470

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35. Quebbemann, H.-G.: Zur Klassiﬁkation unimodularer Gitter mit Isometrie von Primzahlordnung (German) [On the classiﬁcation of unimodular lattices with an isometry of prime order]. J. Reine Angew. Math. 326, 158–170 (1981), http://resolver.sub.uni-goettingen.de/purl?GDZPPN002198681; Quebbemann, H.-G.: Unimodular lattices with isometries of large prime order. II. Math. Nachr. 156, 219–224 (1992), http://dx.doi.org/10.1002/mana.19921560114 36. Quebbemann, H.-G.: Atkin-Lehner eigenforms and strongly modular lattices. Enseign. Math. 43(1-2), 55–65 (1997), http://retro.seals.ch/digbib/view?rid=ensmat-001:1997:43::263 37. Rains, E.M.: New asymptotic bounds for self-dual codes and lattices. IEEE Trans. Inform. Theory 49(5), 1261–1274 (2003), http://dx.doi.org/10.1109/TIT.2003.810623 38. Schnorr, C.P., Euchner, M.: Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. Math. Program. 66, 181–191 (1994), http://dx.doi.org/10.1007/BF01581144 39. Schnorr, C.P., H¨ orner, H.H.: Attacking the Chor-Rivest cryptosystem by improved lattice reduction. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 1–12. Springer, Heidelberg (1995), http://dx.doi.org/10.1007/3-540-49264-X_1 40. Schulze-Pillot, R.: Quadratic residue codes and cyclotomic lattices. Arch. Math. (Basel) 60(1), 40–45 (1993), http://dx.doi.org/10.1007/BF01194237 41. Weisfeiler, B.: On the size of structure of ﬁnite linear groups. Notes from 1984, Parts 1-17, A1-A10, totalling 91 typewritten and 63 handwritten pages, http://weisfeiler.com/boris/papers/papers.html 42. Zagier, D.: Elliptic modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of modular forms. Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, Universitext, June 2004, x+266 pp. Springer, Berlin (2008)

A

Appendix: Proof That M80 and N80 Are Not Isometric

We wish to show that M80 is not isometric to our lattice N80 . Bachoc and Nebe list a subgroup of Aut(M80 ) of order 212 34 52 , while we have S ∼ = SL2 (F79 ) as a subgroup of Aut(N80 ). We wish to show that there is no ﬁnite matrix group in GL80 (Z) that is a supergroup of both of these (possibly after conjugation). We let G be such a putative supergroup, and note that [G : S] ≥ 27 33 5. From a classical theorem of Minkowski [29] on the modular reduction of matrix groups, we have injective maps ιp : G → GL80 (Fp ) for all odd primes p. By taking a gcd over all odd p this gives a bound of #G 2198 358 524 714 118 136 175 194 233 292 312 372 412 ·43·47·53·59·61·67·71·73·79, though here we really a divisibility result at a speciﬁc prime.13 only need such We write H = ι7 G ∩ SL80 (Z) , and since every matrix in S ∼ = SL2 (F79 ) has determinant 1 we have ι7 (S) ⊆ H. As every matrix in G has determinant ±1, 13

We note in passing that the best upper bound on the size of a ﬁnite matrix group is due to Feit [10], relying on unpublished notes of Weisfeiler [41].

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we get [ι7 (G) : H] ≤ 2, and since [G : S] > 4 and ι7 is injective, this implies that [H : ι7 (S)] > 2. The use of a theorem of Aschbacher (see below) now implies that 7780 #H, which contradicts the above bound. Thus G cannot exist, and so M80 and N80 are not isometric. Indeed, this argument almost shows that S is maximal ﬁnite in GL80 (Z), though a low-index extension could still exist. We now use Aschbacher’s theorem [2] on maximal subgroups of ﬁnite classical groups (see also [23]). Let l be an odd prime (to be speciﬁed below) and suppose that ιl (S) ⊂ H ⊆ SL80 (Fl ). We note that S splits into a pair of conjugate √ absolutely irreducible unitary 40-dimensional representations deﬁned over Q( −79). We know that H lies in some maximal (proper) subgroup of SL80 (Fl ), and the theorem of Aschbacher lists the possibilities. For any inert prime l that does not divide #S, we can eliminate class 1 of Aschbacher since ιl (S) acts irreducibly (we could consider split primes also, but choosing an inert prime simpliﬁes the argument slightly). Classes 2 and 4-7 are not possible simply because 79 must divide #H. This leaves subgroups of class 3 (splitting as above) or class 8 (inclusions of classical groups), or class 9 (other simple groups, handled below). The inclusions of classical groups give us G80 (Fl ) for G = Sp, SO± and SU40 (Fl ), while the splitting of class 3 yields SL40 (Fl2 ).2. where the notation indicates that we have a 2-extension – in this case, we continue the analysis after replacing H by H ∩ SL40 (Fl2 ), where this subgroup has index at most 2 in H. We iteratively apply Aschbacher’s theorem to each classical group obtained; either H is isomorphic to this classical group, or is contained in a maximal subgroup of it. We again use 79|#H, and ﬁnd that the only possible maximal subgroup of Sp80 (Fl ) that could contain H is SU40 (Fl ).2, and similarly with the others. Any maximal subgroup chain of classical groups must end here, since H contains ιl (S) and S → SU40 (Fl ) is absolutely irreducible. So we end in one of the following cases: H is isomorphic to one of SU40 (Fl ). or SL40 (Fl2 ). with = 1, 2, or G80 (Fl ) with G = Sp, SO± , SL; or [H : ιl (S)] = 2, in correspondence to a 2-extension as above; or (sometimes called “class 9” for Aschbacher) we have PSL2 (F79 ) ⊂ K ⊂ P, where K is simple and P is the associated simple group of one of the above classical groups. There is sundry general knowledge for this latter situation, but for us a caseby-case analysis (with l = 7 for concreteness) using the known orders of the ﬁnite simple groups is suﬃcient to show that no such K can exist.14 We conclude that either [H : ι7 (S)] = 2, or that H contains a copy of SU40 (F7 ) and so 7780 #H.

14

One can also proceed via degrees of representations, and D. F. Holt indicated to us that the tables of Hiss and Malle [20] should suﬃce for this.

Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number John Voight Department of Mathematics and Statistics University of Vermont 16 Colchester Ave Burlington, VT 05401, USA [email protected]

Abstract. We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve deﬁned over a totally real ﬁeld with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real ﬁeld of odd degree. We conclude with two examples which illustrate the eﬀectiveness of our algorithms.

The development and implementation of algorithms to compute with automorphic forms has emerged as a major topic in explicit arithmetic geometry. The ﬁrst such computations were carried out for elliptic modular forms, and now very large and useful databases of such forms exist [2,13,14]. Recently, eﬀective algorithms to compute with Hilbert modular forms over a totally real ﬁeld F have been advanced. The ﬁrst such method is due to Demb´el´e [4,5], who worked initially under the assumption that F has even degree n = [F : Q] and strict class number 1. Exploiting the Jacquet-Langlands correspondence, systems of Hecke eigenvalues can be identiﬁed inside spaces of automorphic forms on B × , where B is the quaternion algebra over F ramiﬁed precisely at the inﬁnite places of F —whence the assumption that n is even. Demb´el´e then provides a computationally eﬃcient theory of Brandt matrices associated to B. This method was later extended (in a nontrivial way) to ﬁelds F of arbitrary class number by Demb´el´e and Donnelly [6]. When the degree n is odd, a diﬀerent algorithm has been proposed by Greenberg and the author [8], again under the assumption that F has strict class number 1. This method instead locates systems of Hecke eigenvalues in the (degree one) cohomology of a Shimura curve, now associated to the quaternion algebra B ramiﬁed at all but one real place and no ﬁnite place. This method uses in a critical way the computation of a fundamental domain and a reduction theory for the associated quaternionic unit group [16]; see Section 1 for an overview. In this article, we extend this method to the case where F has arbitrary (strict) class number. Our main result is as follows; we refer the reader to Sections 1 and 2 for precise deﬁnitions and notation. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 357–371, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Theorem 1. There exists an (explicit) algorithm which, given a totally real field F of degree n = [F : Q], a quaternion algebra B over F ramified at all but one real place, an ideal N of F coprime to the discriminant D of B, and a weight k ∈ (2Z>0 )n , computes the system of eigenvalues for the Hecke operators Tp with p DN and the Atkin-Lehner involutions Wpe with pe DN acting on the space of quaternionic modular forms SkB (N) of weight k and level N for B. In other words, there exists an explicit ﬁnite procedure which takes as input the ﬁeld F , its ring of integers ZF , a quaternion algebra B over F , an ideal N ⊂ ZF , and the vector k encoded in bits (each in the usual way), and outputs a ﬁnite set of number ﬁelds Ef ⊂ Q and sequences (af (p))p encoding the Hecke eigenvalues for each cusp form constituent f in SkB (N), with af (p) ∈ Ef . From the Jacquet-Langlands correspondence, applying the above theorem to the special case where D = (1) (and hence n = [F : Q] is odd), we have the following corollary. Corollary 2. There exists an algorithm which, given a totally real field F of odd degree n = [F : Q], an ideal N of F , and a weight k ∈ (2Z>0 )n , computes the system of eigenvalues for the Hecke operators Tp and Atkin-Lehner involutions Wpe acting on the space of Hilbert modular cusp forms Sk (N) of weight k and level N. This corollary is not stated in its strongest form: in fact, our methods overlap with the methods of Demb´el´e and his coauthors whenever there is a prime p which exactly divides the level; see Remark 5 for more detail. Combining these methods, Donnelly and the author [7] are systematically enumerating tables of Hilbert modular forms, and the details of these computations (including the dependence on the weight, level, and class number, as well as a comparison of the runtime complexity of the steps involved) will be reported there [7], after further careful optimization. A third technique to compute with automorphic forms, including Hilbert modular forms, has been advanced by Gunnells and Yasaki [9]. They instead use the theory of Vorono˘ı reduction and sharbly complexes; their work is independent of either of the above approaches. This article is organized as follows. In Section 1, we give an overview of the basic algorithm of Greenberg and the author which works over ﬁelds F with strict class number 1. In Section 2, using an adelic language we address the complications which arise over ﬁelds of arbitrary class number, and in Section 3 we make this theory concrete and provide the explicit algorithms announced in Theorem 1. Finally, in Section 4, we consider two examples, one in detail; our computations are performed in the computer system Magma [1]. The author would like to thank Steve Donnelly and Matthew Greenberg for helpful discussions as well as the referees for their comments. The author was supported by NSF Grant No. DMS-0901971.

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1

359

An Overview of the Algorithm for Strict Class Number 1

In this section, we introduce the basic algorithm of Greenberg and the author [8] with a view to extending its scope to base ﬁelds of arbitrary class number; for further reading, see the references contained therein. Let F be a totally real ﬁeld of degree n = [F : Q] with ring of integers ZF . Let × × F+× be the group of totally positive elements of F and let Z× F,+ = ZF ∩ F+ . Let B be a quaternion algebra over F of discriminant D. Suppose that B is split at a unique real place v1 , corresponding to an embedding ι∞ : B → B ⊗ R ∼ = M2 (R), and ramiﬁed at the other real places v2 , . . . , vn . Let O(1) ⊂ B be a maximal order and let × × O(1)× + = {γ ∈ O(1) : v1 (nrd(γ)) > 0} = {γ ∈ O(1) : nrd(γ) ∈ ZF,+ }

denote the group of units of O(1) with totally positive reduced norm. Let × + Γ (1) = ι∞ (O(1)× + /ZF ) ⊂ PGL2 (R) ,

so that Γ (1) acts on the upper half-plane H = {z ∈ C : Im(z) > 0} by linear fractional transformations. Let N ⊂ ZF be an ideal coprime to D, let O = O0 (N) × be an Eichler order of level N, and let Γ = Γ0 (N) = ι∞ (O0 (N)× + /ZF ). n Let k = (k1 , . . . , kn ) ∈ (2Z>0 ) be a weight vector; for example, the case k = (2, . . . , 2) of parallel weight 2 is of signiﬁcant interest. Let SkB (N) denote the ﬁnite-dimensional C-vector space of quaternionic modular forms of weight k and level N for B. Roughly speaking, a form f ∈ SkB (N) is an analytic function f : H → Wk (C) which is invariant under the weight k action by the group γ ∈ Γ , where Wk (C) is an explicit right B × -module [8, (2.4)] and Wk (C) = C when k is parallel weight 2. The space SkB (N) comes equipped with the action of Hecke operators Tp for primes p DN and Atkin-Lehner involutions Wpe for prime powers pe DN. The Jacquet-Langlands correspondence [8, Theorem 2.9] (see Hida [10, Proposition 2.12]) gives an isomorphism of Hecke modules ∼

SkB (N) − → Sk (DN)D-new , where Sk (DN)D-new denotes the space of Hilbert modular cusp forms of weight k and level DN which are new at all primes dividing D. Therefore, as Hecke modules one can compute equivalently with Hilbert cusp forms or with quaternionic modular forms. We compute with the Hecke module SkB (N) by identifying it as a subspace in the degree one cohomology of Γ (1), as follows. Let Vk (C) be the subspace of the algebra C[x1 , y1 , . . . , xn , yn ] consisting of those polynomials q which are homogeneous in (xi , yi ) of degree wi = ki − 2. Then Vk (C) has a right action of the group B × given by n γ −wi /2 q((x1 y1 )γ 1 , . . . , (xn yn )γ n ) (1) (det γi ) q (x1 , y1 , . . . , xn , yn ) = i=1

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for γ ∈ B × , where denotes the standard involution (conjugation) on B and γi = vi (γ) ∈ M2 (C). By the theorem of Eichler and Shimura [8, Theorem 3.8], we have an isomorphism of Hecke modules + ∼ SkB (N) − → H 1 Γ, Vk (C) where the group cohomology H 1 denotes the (ﬁnite-dimensional) C-vector space of crossed homomorphisms f : Γ → Vk (C) modulo coboundaries and + denotes the +1-eigenspace for complex conjugation. By Shapiro’s lemma [8, §6], we then have a further identiﬁcation + ∼ SkB (N) − → H 1 Γ, Vk (C) ∼ (2) = H 1 (Γ (1), V (C))+ , Γ (1)

where V (C) = CoindΓ Vk (C). In the isomorphism (2), the Hecke operators act as follows. Let p be a prime of ZF with p DN and let Fp denote the residue class ﬁeld of p. Since F has strict class number 1, by strong approximation [15, Theor`eme III.4.3] there exists π ∈ O such that nrd π is a totally positive generator for p. It follows that there × are elements γa ∈ O+ , indexed by a ∈ P1 (Fp ), such that × × O+ πO+ =

a∈P1 (F

× O+ αa

(3)

p)

where αa = πγa . Let f : Γ (1) → V (C) be a crossed homomorphism, and let γ ∈ Γ (1). The decomposition (3) extends to O(1) as × O(1)× O(1)× + πO(1)+ = + αa . a∈P1 (Fp ) 1 Thus, there are elements δa ∈ O(1)× + for a ∈ P (Fp ) and a unique permutation ∗ 1 γ of P (Fp ) such that αa γ = δa αγ ∗ a (4)

for all a. We then deﬁne f | Tp : Γ (1) → V (C) by (f | Tp )(γ) = f (δa )αa .

(5)

a∈P1 (Fp )

The space SkB (N) similarly admits an action of Atkin-Lehner operators Wpe for primes pe DN. From this description, we see that the Hecke module H 1 (Γ (1), V (C))+ is amenable to explicit computation. First, we compute a ﬁnite presentation for Γ (1) with a minimal set of generators G and a solution to the word problem for the computed presentation using an algorithm of the author [16]. Given such a set of generators and relations, one can explicitly ﬁnd a basis for the C-vector space H 1 (Γ (1), V (C)) [8, §5].

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We then compute the action of the Hecke operator Tp on H 1 (Γ (1), V (C)). We ﬁrst compute a splitting ιp : O → M2 (ZF,p ). The elements αa in (4) are then generators with totally positive reduced norm of the left ideals

xy + Op (6) Ia = Oι−1 p 00 and are obtained by principalizing the ideals Ia ; here again we use strong approximation and the hypothesis that F has strict class number 1. Then for each a ∈ P1 (Fp ) and each γ ∈ G, we compute the permutation γ ∗ [8, Algorithm 5.8] and the element δa = αa γα−1 γ ∗ a ∈ Γ (1) as in (4). Using the solution to the word problem, we then write δa as a word in the generators G for Γ (1), and then for a basis of crossed homomorphisms f we compute f | Tp by computing (f | Tp )(γ) ∈ V (C) for each γ ∈ G as in (5). In a similar way, we compute the action of complex conjugation and the Atkin-Lehner involutions. We then decompose the space H 1 (Γ, V (C)) under the action of these operators into Hecke irreducible subspaces, and from this we compute the systems of Hecke eigenvalues using linear algebra.

2

The Indefinite Method with Arbitrary Class Number

In this section, we show how to extend the method introduced in the previous section to the case where F has arbitrary class number [8, Remark 3.11]. We refer the reader to Hida [11] for further background. 2.1

Setup

We carry over the notation from Section 1. Recall that O = O0 (N) is an Eichler order of level N in the maximal order O(1) ⊂ B. Let H± = {z ∈ C : Im(z) = 0} = C \ R be the union of the upper and lower half-planes. Then via ι∞ , the group B × acts on H± by linear fractional transformations. In this generality, we ﬁnd it most elucidating to employ adelic notation. Let = lim Z/nZ and let denote tensor with Z over Z. Consider the double coset Z ←−n × /O × ), X(C) = B × \(H± × B × /O × by left multiplication via the diagonal embedding. where B × acts on B Then X(C) has the structure of a complex analytic space [3] which fails to be compact if and only if B ∼ = M2 (Q), corresponding to the classical case of elliptic modular forms—higher class number issues do not arise in this case, so from now we assume that B is a division ring. We again write SkB (N) for the ﬁnite-dimensional C-vector space of quaternionic modular forms of weight k and level N: here, again roughly speaking, a quaternionic modular form of weight k ∈ (2Z>0 )n and level N for B is an analytic function × /O × → Wk (C) f : H± × B which is invariant under the weight k action of B × , with Wk (C) as in Section 1.

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Decomposing the Double Coset Space

× By Eichler’s theorem of norms, we have nrd(B × ) = F(+) where × = {a ∈ F × : vi (a) > 0 for i = 2, . . . , n} F(+)

is the subgroup of elements of F which are positive at all real places which are × ∼ ramiﬁed in B. In particular, B × /B+ = Z/2Z, where × = {γ ∈ B × : v1 (nrd(γ)) > 0} = {γ ∈ B : nrd(γ) ∈ F+× }. B+ × acts on the upper half-plane H, therefore we may identify The group B+ × × /O × ). \(H × B X(C) = B+

Now we have a natural (continuous) projection map × × × \B /O , X(C) → B+

and by strong approximation [15, Theor`eme III.4.3] the reduced norm gives a bijection + × × × ∼ × ∼ nrd : B+ \B /O − → F+× \F × /Z (7) F = Cl ZF , where Cl+ ZF denotes the strict class group of ZF , i.e. the ray class group of ZF with modulus equal to the product of all real (inﬁnite) places of F . The space X(C) is therefore the disjoint union of Riemann surfaces indexed by Cl+ ZF , which we identify explicitly as follows. Let the ideals b ⊂ ZF form F be such that b Z F ∩ ZF = b. a set of representatives for Cl+ ZF , and let b ∈ Z For expositional simplicity, choose b = ZF and β = 1 for the representatives of × such that the trivial class. By strong approximation (7), there exists β ∈ B nrd(β) = b. Therefore × × ). X(C) = B+ (H × βO (8) [b]

We have a map × × ) → O× \H (H × βO B+ β,+

× ) → z (z, βO β−1 ∩ B and O× = O× ∩ B × , so that O = O. where Oβ = βO + 1 × β,+ × β + For each β, let Γ = ι∞ O /Z ⊂ PGL2 (R) . Then the Eichler-Shimura β

β,+

F

isomorphism on each component in (8) gives an identiﬁcation of Hecke modules ∼ SkB (N) − → H 1 (Γβ , Vk (C))+ , (9) β

let where + denotes the +1-eigenspace for complex conjugation. For each β, −1 O(1)β = βO(1)β ∩ B be the maximal order containing the Eichler order Oβ ,

Automorphic Forms on Shimura Curves Γ (1)β

× and let Γ (1)β = ι∞ (O(1)× (C) = CoindΓ /ZF ). Further, let Vβ β,+ β Shapiro’s lemma applied to each summand in (9) gives ∼ → H 1 (Γ (1)β , Vβ (C))+ . SkB (N) −

363

Vk (C). Then

(10)

β

2.3

Hecke Operators

H 1 (Γ (1)β , Vβ (C)) in F be the following way. Let p be a prime ideal of ZF with p DN, and let p ∈ Z F ∩ ZF = p. We consider the β -summand in (10), corresponding such that p Z to the ideal class [b ]. Let f : Γ (1)β → Vβ (C) be a crossed homomorphism: we will then obtain a new crossed homomorphism f | Tp : Γ (1) → V (C), where β In the description (10), the Hecke operators Tp act on

β

β

β

corresponds to the ideal class of [pb ] among the explicit choices made above. , be such that nrd( ) = p. Then there are elements γ a ∈ O Let

∈ O β β 1 indexed by a ∈ P (Fp ), such that × × α ×

(11) O O O = a β

β

a∈P1 (Fp )

β

where α a = γa . × , we conclude that there exist unique Let γ ∈ Γβ . Extending (11) to O(1) β × and a unique permutation γ ∗ of P1 (Fp ) such that elements δa ∈ O(1) β

γ ∗ a α a γ = δa α for a ∈ P1 (Fp ). Thus we have a )γ = (β β−1 )δa α γ ∗ a = δa (β β−1 α γ ∗ a ). (β β−1 α where δa = (β β−1 )δa (β β−1 )−1 . β−1 has right order O . has left order O and similarly O Recall that β O β β -ideal Therefore, we may consider the left O β

β O β−1 O α O β β a

(12)

noting that the left and right orders in each case match up, so the product is

compatible. Next, recall that the elements β , β, have reduced norms corresponding to the ideal classes [b ], [pb ], and [p], respectively. Thus the reduced norm of the left ideal (12) has a trivial ideal class. Therefore, by strong approximation (applied now to left ideals of the order Oβ ), for each a ∈ P1 (Fp ), there × exist elements πa ∈ Oβ ∩ B+ such that β β−1 α O a ∩ B = Oβ πa . β

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Hence there exists a unique permutation γ ∗ of P1 (Fp ) such that πa γ = δa πγ ∗ a with δa ∈ O× → Vβ (C) is then ,+ . The new crossed homomorphism f | Tp : Γβ β deﬁned by the formula (f | Tp )(γ) = f (δa )πa a∈P1 (Fp )

for γ ∈ Γβ . 2.4

Complex Conjugation and Atkin-Lehner Involutions

We now deﬁne an operator W∞ which acts by complex conjugation. Let Cl(+) ZF denote the ray class group of ZF with modulus equal to the real (inﬁnite) places of F which are ramiﬁed in B. Then we have a natural map Cl+ ZF → Cl(+) ZF ; this map is an isomorphism if and only if there exists a unit u ∈ Z× F which satisﬁes v1 (u) < 0 and vi (u) > 0 for the other real places vi (i = 2, . . . , n) of F , otherwise the kernel of this map is isomorphic to Z/2Z. Let [m] ∈ Cl+ ZF generate the kernel of this map. Let f : Γ (1)β → Vβ be a crossed homomorphism, and let β correspond to the ideal class [b m−1 ]; we will deﬁne the complex conjugate crossed homomorphism β O β−1 ∩ B has reduced norm (f | W∞ ) : Γ (1)β → Vβ (C). The left Oβ -ideal O β + corresponding to the ideal class [m] ∈ Cl ZF , so there exists a generator μ ∈ Oβ of this ideal such that v1 (nrd(μ )) < 0 but vi (nrd(μ )) > 0 for i = 2, . . . , n. Then given γ ∈ Γ (1)β , we deﬁne

(f | W∞ )(γ) = f (μ γμ−1 )μ . Finally, we deﬁne the Atkin-Lehner involutions Wpe for pe DN. Let p corre F . Then there exists an element π spond to p ∈ Z ∈ Oβ which generates the unique two-sided ideal of Oβ of reduced norm generated by pe . The element 2 ∈ O× F × . Let β correspond to the ideal class [pb ]. π normalizes O and π β

β

× Then as above, by strong approximation there exists an element μ ∈ Oβ ∩ B+ π ∩ B = O μ . Given f : Γ (1) → V , we then deﬁne such that Oβ β β β β β (f | Wpe ) : Γ (1)β → Vβ (C) by

(f | Wp )(γ) = f (μ γμ−1 )μ for γ ∈ Γ (1)β .

3

Algorithmic Methods

In this section, we take the adelic description of Section 2 and show how to compute with it explicitly, proving Theorem 1.

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Our algorithm takes as input a totally real ﬁeld F of degree [F : Q] = n, a quaternion algebra B over F split at a unique real place, an ideal N ⊂ ZF coprime to the discriminant D of B, a vector k ∈ (2Z>0 )n , and a prime p DN, and outputs the matrix of the Hecke operator Tp acting on the space H = +

1 (in the notation of Section 2) with respect to some ﬁxed H Γ (1)β , Vβ (C) β basis which does not depend on p. From these matrices, one decomposes the space H into Hecke-irreducible subspaces by the techniques of basic linear algebra. Our algorithm follows the form given in the overview in Section 1, so we describe our algorithm in steps, with a description of each step along the way. Step 1 (Compute a splitting field): Let K → C be a Galois number ﬁeld containing F which splits B: for example, we can take the normal closure of any quadratic ﬁeld contained in B. Since all computations then occur inside K ⊂ C, we may work then with coeﬃcient modules over K using exact arithmetic. (This step is only necessary if k is not parallel weight 2, for otherwise the action of B × factors through K = Q.) Step 2 (Compute ideal class representatives): Compute a set of representatives [b] for the strict class group Cl+ ZF with each b coprime to pDN. (See Remark 4 below.) Compute a maximal order O(1) ⊂ B. For each representative ideal b, compute a right O(1)-ideal Jb such that nrd(Jb ) = b and let O(1)b be the left order of Jb . (In the notation of Section 2, the right O(1)-ideals Jb represent the elements and O(1)b = O(1) .) β, β Step 3 (Compute presentations for the unit groups): Compute an embedding ι∞ : B → M2 (R) corresponding to the unique split real place. × For each b, compute a ﬁnite presentation for Γ (1)b = ι∞ (O(1)× b,+ /ZF ) consisting of a (minimal) set of generators Gb and relations Rb together with a solution to the word problem for the computed presentation [16]. (Note that the algorithm stated therein [16, Theorem 3.2] is easily extended from units of reduced norm 1 to totally positive units.) For eﬃciency, we start by computing such a presentation with generators G associated to the order O(1) and then for each order O(1)b we begin with the elements in hand formed by short products of elements in G which happen to lie in O(1)b (to aid in the search for units [16, Algorithm 3.2]; note that O(1)∩O(1)b is an Eichler order of level b in O(1)b ). Step 4 (Compute splitting data): Compute a splitting ιN : O(1) → O(1) ⊗ZF ZF,N ∼ = M2 (ZF,N ). Note that since b is coprime to N, we have O(1) ⊗ ZF,N = O(1)b ⊗ ZF,N for all b, so ιN also gives rise to a splitting for each O(1)b . For each b, compute the Eichler order Ob ⊂ O(1)b of level N with respect to ιN . Next, for each b, compute representatives for the left cosets of the group × Γb = ι∞ (Ob,+ /Z× F ) inside Γ (1)b [8, Algorithm 6.1]. Finally, identify Γ (1)b

V (K)b = CoindΓb

Vk (K)

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as a K-vector space given by copies of Vk (K) indexed by these cosets, and compute the permutation action of the representatives of these cosets on this space. In practice, it is more eﬃcient to identify the above coset representatives with elements of P1 (ZF / N) and thereby work directly with the coeﬃcient module V (K)b ∼ = K[P1 (ZF /N)] ⊗ Vk (K). Step 5 (Compute a basis for cohomology): Identify the space of crossed homo morphisms b Z 1 (Γ (1)b , V (K)b ) with its image under the inclusion Z 1 (Γb , V (K)b ) →

V (K)b

g∈Gb

f → (f (g))g∈Gb

consisting of those f ∈ g∈Gb V (K)b which satisfy the relations f (r) = 0 for r ∈ Rb . Compute the space of principal crossed homomorphisms B 1 (Γ (1)b , V (K)b ) in a similar way, and thereby compute using linear algebra a K-basis for the quotient H 1 (Γ (1)b , V (K)b ) = Z 1 (Γ (1)b , V (K)b )/B 1 (Γ (1)b , V (K)b ) for each b. Let H = b H 1 (Γ (1)b , V (K)b ). Step 6 (Compute representatives for left ideal classes): Compute a splitting ιp : O(1) → M2 (ZF,p ). For each ideal b , perform the following steps. First, compute the ideal b with ideal class [b] = [pb ]. Compute the left ideals Ia =

Oι−1 p

xy + Op 00

indexed by the elements a = (x : y) ∈ P1 (Fp ) and then compute the left Ob ideals Ia = Jb J b Ia . × Compute totally positive generators πa ∈ Ob ∩ B+ for Ob πa = Ia [12]. ∗ Now, for each γ ∈ Gb , compute the permutation γ of P1 (Fp ) [8, Algorithm 1 5.8] and then the elements δa = πa γπγ−1 ∗ a for a ∈ P (Fp ); write each such element δa as a word in Gb and from the formula (f | Tp )(γ) =

f (δa )πa

a∈P1 (Fp )

with f in a basis for the b -component of cohomology as in Step 5 compute the induced crossed homomorphism f | Tp in the b-component. Step 7 (Compute the blocks of the intermediate matrix): Assemble the matrix T with rows and columns indexed as in Step 5 with blocks in the (b, b ) position given by the output of Step 6: this matrix describes the action of Tp on H. Step 8 (Decompose H into ±-eigenspaces for complex conjugation): Determine the representative ideal m (among the ideals b) which generates the kernel of the map Cl+ ZF → Cl(+) ZF .

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For each ideal b , perform the following steps. Compute the ideal b such that [b] = [b m−1 ], and compute a generator μ with Ob μ = Jb J b such that v(nrd(μ )) < 0. For each γ ∈ Gb , from the formula

(f | W∞ )(γ) = f (μ γμ−1 )μ , for f in a basis for the b -component of cohomology as in Step 5 compute the induced crossed homomorphism f | Tp in the b-component. Assemble the matrix with blocks in the (b, b ) position given by this output: this matrix describes the action of complex conjugation W∞ on H. Compute a K-basis for the +1-eigenspace H + of H for W∞ . Finally, compute the matrix T + giving the action of Tp restricted to H + and return T + . This completes the description of the algorithm. In a similar way, one computes the Atkin-Lehner involutions, replacing Step 6 with the description given in Section 2.4, similar to the computation of complex conjugation in Step 8. Remark 3. Note that Steps 1 through 3 do not depend on the prime p nor the level N and Steps 4, 5, and 8 do not depend on the prime p, so these may be precomputed for use in tabulation. Remark 4. To arrange uniformly that the ideals b representing the classes in Cl+ ZF are coprime to the prime p in advance for many primes p, one has several options. One possibility is to choose suitable ideals b of large norm in advance. Another option is to make suitable modiﬁcations “on the ﬂy”: if p is not coprime to b, we simply choose a diﬀerent ideal c coprime to p with [b] = [c], a new ideal Jc with nrd(Jc ) = c, and compute an element ν ∈ Ob such that νOb ν −1 = Oc . Conjugating by ν where necessary, one can then transport the computations from one order to the other so no additional computations need to take place.

4

Examples

In this section, we compute with two examples to demonstrate the algorithm outlined in Section 3. Throughout, we use the computer system Magma [1]. Our ﬁrst and most detailed example is concerned with the smallest totally real cubic ﬁeld F with the property that the dimension of the space of Hilbert cusp forms of parallel weight 2 and level (1) is greater than zero and the strict class number of F is equal to 2. This ﬁeld is given by F = Q(w) where w satisﬁes the equation f (w) = w3 − 11w − 11 = 0. The discriminant of F is equal to 2057 = 112 17, and ZF = Z[w]. The roots of f in R are −2.602 . . . , −1.131 . . . , and 3.73 . . . , and we label the real places v1 , v2 , v3 of F into R according to this ordering. We deﬁne the sign of a ∈ F to be the triple sgn(a) = (sgn(vi (a)))3i=1 ∈ {±1}3 . The unit group of F is generated by the elements −1, w + 1 with sgn(w + 1) = (1, −1, −1), and the totally positive unit −w2 + 2w + 12.

368

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We begin by ﬁnding a quaternion algebra B with D = ZF which is ramiﬁed at

w + 1, −1 all but one real place [8, Algorithm 4.1]. We ﬁnd the algebra B = F ramiﬁed only at v1 and v2 , generated by i, j subject to i2 = w + 1, j 2 = −1, and ji = −ij. For forms of parallel weight 2, Step 1 is trivial: we can take K = Q. Next, in Step 2 we compute ideal class representatives. The nontrivial class in Cl+ (ZF ) is represented by the ideal b = (w2 − 2w − 6)ZF , which is principal but does not possess a totally positive generator, since sgn(−w2 + 2w + 6) = (−1, 1, −1) and there is no unit of ZF with this sign. We note that N(b) = 7. Next, we compute a maximal order O = O(1); it is generated over ZF by i and the element k = (1 + (w2 + 1)i + ij)/2. Next, we ﬁnd that the right O-ideal Jb generated by w2 − 2w − 6 and the element (5 + (w2 + 5)i + ij)/2 = 2 + 2i + k has nrd(Jb ) = b. Next, in Step 3 we compute presentations for the unit groups. We take the splitting B → M2 (R)

0 1 s 0 , i, j → −1 0 0 −s

where s = v3 (w + 1). We then compute a fundamental domain for Γ = Γ (1) [16], given below.

We ﬁnd that Γ = Γ (1) is the free group on the generators α, β, γ1 , . . . , γ7 subject to the relations γ12 = γ22 = γ33 = γ42 = γ53 = γ62 = γ72 = αβα−1 β −1 γ1 · · · γ7 = 1.

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For example, we have 2α = (w2 − 14) + (2w2 − 4w − 13)i + (−2w2 + 5w + 9)j + (−4w2 + 8w + 26)ij. The groups Γ and Γb have isomorphic presentations. In particular, we note that both Γ and Γb have genus 1, so we conclude that dim S2 (1) = 1 + 1 = 2. We illustrate the computation of Hecke operators with the primes p3 = (w + 2)ZF of norm 3 and p5 = (w + 3)ZF of norm 5. Note that p3 is nontrivial in Cl+ (ZF ) whereas p5 is trivial. Step Step 4 requires no work, since we work with forms of level (1). In Step 5 we compute with a basis for cohomology, and here we see directly that H 1 (Γ, Q) ∼ = Hom(Γ, Q) ∼ = Zfα ⊕ Zfβ where fα , fβ are the characteristic functions for α and β. We have a similar description for H 1 (Γb , Q). Next, in Step 6 we compute representatives of the left ideal classes. For p3 , for example, for I[1:0] ⊂ O we ﬁnd that Jb I[1:0] = Ob ((w + 1) + i + ij) and for I[1:1] ⊂ Ob we have Jb I[1:1] = O(w + 1 − i + ij); we thereby ﬁnd elements πa , πa for a ∈ P1 (Fp3 ). For the generators γ = α, β of O and Ob , we compute the permutations γ ∗ of P1 (Fp3 ); we ﬁnd for example that α∗ is the identity and π[1:0] α = δ[1:0] π[1:0] with δ[1:0] ∈ Ob , namely, 14δ[1:0] = (7w2 − 98) + (−23w2 + 40w + 167)i+

(−25w2 + 59w + 103)j + (−2w2 + 5w + 20)ij. We then write δ[1:0] as a word in the generators for Γb of length 23. Repeating these steps (reducing a total of 64 units), we assemble the block matrix in Step 7 as the matrix ⎞ ⎛ 0020 ⎜0 0 0 2 ⎟ ⎟ T p3 | H = ⎜ ⎝2 0 0 0 ⎠ . 0200

In a similar way, we ﬁnd that Tp5 is the identity matrix. Finally, in Step 8 we compute the action of complex conjugation. Here we have simply μ = i (whereas μb is more complicated), and thereby compute that ⎞ ⎛ 1 1 0 0 ⎜0 −1 0 0 ⎟ ⎟ W∞ | H = ⎜ ⎝0 0 1 1 ⎠ . 0 0 0 −1 + We verify that W∞ commutes

with Tp3 (and Tp5 ). We conclude that Tp3 | H = 10 02 . and Tp5 | H + = 01 20

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We then diagonalize the space H + , which breaks up into two one-dimensional eigenforms f and g, and compute several more Hecke operators: we list in Table 1 below a generator for the prime p, its norm N p, and the Hecke eigenvalues ap (f ) and ap (g) for the cusp forms f, g. Table 1. Hecke eigenvalues for the Hilbert cusp forms for F = Q(w) with w3 − 11w − 11 = 0 of level (1) and parallel weight 2 p w+2 w+3 2 2w + 7 w w2 − w − 8 w−3 2w2 − 5w − 10 w2 − 3w − 2 w2 − 6 w+4 2w2 − 3w − 16 w2 − 2w − 9 w2 + w − 3

Np 3 5 8 9 11 17 17 23 25 29 31 37 41 49

ap (f ) 2 1 −5 −2 0 −5 −5 2 −9 9 −2 −3 −5 −10

ap (g) −2 1 −5 2 0 5 −5 −2 −9 −9 −2 3 5 10

We note that the primes generated by w and w − 3 are ramiﬁed in F . By work of Deligne [3], the curves X = X(1) and Xb are deﬁned over the + + strict class √ ﬁeld F of F , and Gal(F /F ) permutes them. We compute that + 2 F = F ( −3w + 8w + 12). Therefore the Jacobian Jf , corresponding to the cusp form f , is a modular elliptic curve over F + with #J(Fp ) = N p + 1 − af (p) with everywhere good reduction. The form g is visibly a quadratic twist of f by the character corresponding to the extension F + /F . Unfortunately, this curve does not have any apparent natural torsion structure which would easily allow for its identiﬁcation as an explicit curve given by a sequence of coeﬃcients [6, §4]. As a second and ﬁnal example, we compute with a quaternion algebra deﬁned over therefore ramiﬁed at a ﬁnite prime. We take F = √ a quadratic ﬁeld and √ Q( 65), with ZF = Z[(1 + 65)/2]. The ﬁeld F has # Cl(F ) = # Cl+ (F ) = 2. We compute the space S = S2 (p5 )p5 -new of Hilbert cuspidal new forms of parallel weight 2 and level p5 , where p5 is the unique prime in ZF of norm 5. We compute that dim S = 10, and that the space S decomposes into Heckeirreducible subspaces of dimensions 2, 2, 3, 3. For example, the characteristic polynomial of Tp2 for p2 either prime above 2 factors as (T 2 − 2T − 1)(T 2 + 2T − 1)(T 6 + 11T 4 + 31T 2 + 9). Remark 5. By the Jacquet-Langlands correspondence, the space S2 (p5 )p5 -new also occurs in the space of quaternionic modular forms for an Eichler order of

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level p5 in the deﬁnite quaternion algebra ramiﬁed at the the two real places of F and no ﬁnite place, and therefore is amenable to calculation by the work of Demb´el´e and Donnelly. We use this overlap to duplicate their computations (as well as ours) and thereby give some compelling evidence that the results are correct since they are computed in entirely diﬀerent ways.

References 1. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997) 2. Cremona, J.: The elliptic curve database for conductors to 130000. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 11–29. Springer, Heidelberg (2006) 3. Deligne, P.: Travaux de Shimura. S´eminaire Bourbaki, Lecture notes in Math. 244(389), 123–165 √ 4. Demb´el´e, L.: Explicit computations of Hilbert modular forms on Q( 5). Experiment. Math. 14(4), 457–466 (2005) 5. Demb´el´e, L.: Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76(258), 1039–1057 (2007) 6. Demb´el´e, L., Donnelly, S.: Computing Hilbert modular forms over ﬁelds with nontrivial class group. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 371–386. Springer, Heidelberg (2008) 7. Donnelly, S., Voight, J.: Tables of Hilbert modular forms and elliptic curves over totally real ﬁelds (in preparation) 8. Greenberg, M., Voight, J.: Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Math. Comp. (accepted) 9. Gunnells, P., Yasaki, D.: Hecke operators and Hilbert modular forms. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 387–401. Springer, Heidelberg (2008) 10. Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. American Journal of Mathematics 103(4), 727–776 (1981) 11. Hida, H.: Hilbert modular forms and Iwasawa theory. Clarendon Press, Oxford (2006) 12. Kirschmer, M., Voight, J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. (SICOMP) 39(5), 1714–1747 (2010) 13. Stein, W.A.: Modular forms database (2004), http://modular.math.washington.edu/Tables 14. Stein, W.A., Watkins, M.: A database of elliptic curves—ﬁrst report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002) 15. Vign´eras, M.-F.: Arithm´etique des alg`ebres de quaternions. LNM, vol. 800. Springer, Berlin (1980) 16. Voight, J.: Computing fundamental domains for coﬁnite Fuchsian groups. J. Th´eorie Nombres Bordeaux 21(2), 467–489 (2009)

Improved Primality Proving with Eisenstein Pseudocubes Kjell Wooding and H.C. Williams Institute for Security, Privacy and Information Assurance, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4, Canada [email protected], [email protected]

Abstract. In August 2002, Agrawal, Kayal, and Saxena described an unconditional, deterministic algorithm for proving the primality of an integer N . Though of immense theoretical interest, their technique, even incorporating the many improvements that have been proposed since its publication, remains somewhat slow for practical application. This paper describes a new, highly eﬃcient method for certifying the primality of an integer N ≡ 1 (mod 3), making use of quantities known as Eisenstein pseudocubes. This improves on previous attempts, including the peudosquare-based approach of Lukes et al., and the pseudosquare improvement proposed by Berrizbeitia, et al.

1

Motivation

In [1], Lukes et al., building on the ideas of Hall [2], Shanks [3, p. 414], and Selfridge and Weinberger [4], described a highly eﬃcient method for proving the primality of an integer N using quantities known as pseudosquares. Their test requires a table of least pseudosquares, denoted M2,x , of suﬃcient size to ensure that N < M2,x . If such a table is available, their method certiﬁes the primality of an integer N using only (log N )3+o(1) operations. In [5], Berrizbeitia et al. introduced a conjecturally more eﬃcient test, relying on quantities they termed pseudocubes, denoted M3,x . Though expected to outperform the pseudosquare-based method asymptotically, this test required a 2/3 table of pseudocubes of suﬃent size to ensure that N < M3,x . In [6], we provided numerical data to support the conjectured asymptotic improvement. In the same paper, however, we pointed out that it is unlikely we will obtain pseudocubes large enough to realize the theoretical gains. Recent results of Sorenson [7] further support both the asymptotic beneﬁt and the practical limitations of this method. In this paper, we propose an alternate deﬁnition of pseudocube — the Eisenstein pseudocube — with a conjectured growth rate better than that of the pseudosquares. Furthermore, we propose an algorithm for proving primality of integers N ≡ 1 (mod 3) that eliminates the troublesome 2/3 exponent of Berrizbeitia’s method. In the process, we supply numerical evidence to support the argument that, both asymptotically and practically, proving primality using G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 372–384, 2010. c Springer-Verlag Berlin Heidelberg 2010

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373

Eisenstein pseudocubes will soon be more eﬃcient than the pseudosquare test for primes N ≡ 1 (mod 3).

2

Eisenstein Pseudocubes √

Let ω be a primitive cube root of unity; i.e. ω = −1+2 3i , and consider the ring of Eisenstein integers, Z[ω]. Recall [8, Chap. 9] that Z[ω] is a unique factorization domain with a norm given by N (α) = αα, and six units: ±1, ±ω, ±ω 2. There are three types of primes in Z[ω]: (1 − ω), which lies over 3; the inert rational primes q ≡ −1 (mod 3) with norm q 2 ; and the primes π of norm ππ = p ≡ 1 (mod 3) where p is prime in Z. We say that an element α ∈ Z[ω] is primary if α ≡ −1 (mod 3).1 It is straightforward to show that every prime in Z[ω] except (1 − ω) has exactly one primary associate. For any α, π ∈ Z[ω] with π prime, N (π) = 3, we can deﬁne the cubic residue α character of α modulo π, denoted π , as follows: 3 1. α = 0 if π | α π 3 α 2. π ≡ α(N (π)−1)/3 (mod π) otherwise, where α ∈ {1, ω, ω 2 }. π 3

3

The properties of this symbol are well-known. See, for example [8]. We can extend the notion of cubic residue character to include non-primes as follows. If α, τ ∈ Z[ω] with 3 | N (τ ), we deﬁne 1 if τ is a unit of Z[ω], α = k α τ 3 otherwise i=1 πi 3

k

where τ = i=1 πi and all πi ∈ Z[ω] are prime. Finally, recall the Cubic Reciprocity Law (CRL), as it applies to to the cubic Jacobi symbol [5, §2.3]: Theorem 1. (Cubic Let α, β be primary in Z[ω] and of coprime Reciprocity) β α norm = 3. Then β = α . 3

3

We are now in a position to deﬁne an Eisenstein pseudocube. Definition 1. Let p be a fixed rational prime. Define μp = a + bω ∈ Z[ω], a, b ∈ Z to be an element of Z[ω] of minimal norm such that: 1. 2. 3. 4.

μp is primary gcd (a, b) = 1 q = 1 for all rational primes q ∈ Z, q ≤ p μp 3 μp not a cube in Z[ω].

We will call μp a minimal Eisenstein pseudocube (or simply an Eisenstein pseudocube) for the prime p. 1

That is to say, if we write α = a + bω, a ≡ −1 (mod 3) and 3 | b.

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Congruence Criteria for Eisenstein Pseudocubes

One technique for eﬃciently computing a table of Eisenstein pseudocubes μp = xp + yp ω, is that of congruential sieving. In order to use this technique, we must ﬁrst establish a set of acceptable residue conditions Sq on μp for each of the primes q ≤ p corresponding to the requirements of Deﬁnition 1. There are 3 cases to consider, one for each type of prime in the Eisenstein integers. 3.1

Case 1: q ≡ −1 (mod 3)

In this case, q is inert and primary. μp is by deﬁnition primary, we can Since μ q = qp , and obtain the desired residue invoke cubic reciprocity: 1 = μp 3

3

conditions by simply computing μp ≡ (m + nω)3 (mod q) for all 0 ≤ m, n < q; i.e. the residue classes given by xp ≡ m3 − 3mn2 + n3 (mod q) yp ≡ 3mn(m − n) (mod q). There are q2 − 1 3

(1)

such solutions modulo q. Example 1. The set of acceptable residues for Eisenstein pseudocubes modulo 5 is given by S5 = {(1 + 0ω), (2 + 0ω), (3 + 0ω), (4 + 0ω), (3 + 1ω), (1 + 2ω), (4 + 3ω), (2 + 4ω)}. 3.2

Case 2: q = 3

Observe that −3ω = (1 − ω)2 . By the bimultiplicity of the cubic Jacobi symbol, 2 3 ω(1 − ω) = . μp 3 μp 3 k Write μp = xp + yp ω = (−1)k−1 i=1 αi where αi = ri + si ω are primary primes; i.e. 3 | si and ri ≡ −1 (mod 3). From the properties of the cubic Jacobi symbol, we know that 2(ri +1) ri +1+si ω 3 , and αωi = ω 3 giving 3

1−ω μp

= 3

k i=1

ω

2(ri +1) 3

= ω2

k

i=1 (ri +1)/3

,

1−ω αi

= 3

Improved Primality Proving with Eisenstein Pseudocubes

and hence Thus

ω μp

ω(1−ω) μp

=

3

ω

ri +1+si 3

=ω

k

i=1 (ri +1)/3+

k

i=1

si /3

,

i=1

3

k

375

=ω

k

i=1

si /3

3 μp

.

2

= ω3

k

i=1

si

.

(2)

3

n n−1 Lemma 1. Let μp = xp + yp ω = (−1) i=1 αi where αi = rni + si ω are prin n−1 mary primes. Then xp ≡ (−1) r (mod 9) and y ≡ p i=1 i i=1 si (mod 9). Proof. If n = 1, the statement is trivially true. Let αj = rj + sj ω, αk = rk + sk ω be primary; i.e. rj ≡ rk ≡ −1 (mod 3) and sj ≡ sk ≡ 0 (mod 3). Writing si = 3Si , ri = −1 + 3Ri for some Si , Ri ∈ Z, observe that −(rk + sk ω)(rj + sj ω) = −(rk + 3Sk ω)(rj + 3Sj ω) ≡ −rk rj − 3(Sj rk + Sk rj )ω ≡ −rk rj − 3(−Sk + 3Rj Sk − Sj + 3Rk Sj )ω ≡ −rk rj + (sk + sj )ω (mod 9) n n−1 which ω) ≡ (−1)n−1 i=1 (ri + si n primary. Thus, by induction, (−1) n is again n n−1 i=1 ri + i=1 si ω (mod 9), so writing μp = xp + yp ω = (−1) i=1 αi where αi = ri + si ω are primary primes xp ≡ (−1)n−1 yp ≡

n

n

ri

(mod 9),

i=1

si

(mod 9)

i=1

as desired.

k (mod 9), soyp /3 si /3 (mod 3). Combin ≡ i=1 k 2 3 i=1 si = ω 2yp /3 . Clearly, 3 ing these facts with Equation 2, we obtain μp = ω 3 y 3 = 1 ⇐⇒ 3 | 3p which, when combined with the requirement that μp be μp 3 primary, gives the requisite congruence conditions: 3 = 1 ⇐⇒ 9 | yp and xp ≡ −1 (mod 3). μp 3

From Lemma 1, yp ≡

k

i=1 si

Example 2. The set of acceptable residues for Eisenstein pseudocubes modulo 9 is given by S9 = {(2 + 0ω), (5 + 0ω), (8 + 0ω)}.

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Case 3: q ≡ 1 (mod 3)

3.3

We can write q = πq πq where πq = a + bω and πq is primary. Of course, πq is also primary. Lemma 2. Let q be a rational prime, μqp = 1 and q = πq πq with πq ∈ Z[ω] 3 μ μ prime and primary, then μqp = 1 if and only if πpq = πpq . 3

3

3

Proof. Recall q = πq πq , and that πq , πq , and μp are all primary. From cubic reciprocity and the properties of the cubic Jacobi symbol [8, §9.3] we have that

−1 μp μp μp μp = πq 3 πq 3 πq 3 πq 3 3 3 3 3 3 μ μ And thus it is clear that μqp = 1 if and only if πpq = πpq . q μp

If

=

q μp

πq μp

πq μp

=

μp πq

μp πq

=

3

3

3

= 1, then from Lemma 2 and the properties of the cubic reciprocity 3

q−1

q−1

symbol, μp 3 ≡ μp 3 (mod πq ). By complex conjugation, we have also that q−1 q−1 μp μp = , and hence μp 3 ≡ μp 3 (mod πq ). Combining these facts, we πq πq 3

obtain

3

q μp

q−1

= 1 ⇐⇒ μp 3 ≡ μp

q−1 3

(mod q).

(3)

3

Writing μp = xp +yp ω, we will now endeavour to reduce (3) to a set of congruence conditions on xp and yp . Note that when q is small, these congruence conditions can be computed by exhaustion. A more elegant algorithm, however, can be obtained from the theory of Lucas sequences. First, observe that if q | yp then (3) reduces to the trivial xp ≡ xp (mod q); i.e. x + 0ω ⊂ Sq for x = 1, . . . , q − 1. For the remaining case, consider the recurring sequences Sn (x, y), Tn (x, y) ∈ Z[x, y] given by:2 S1 (x, y) = x T1 (x, y) = y Sn + Tn ω = (S1 + T1 ω)n with Sn , Tn ∈ Z. Clearly, we have also that Sn + Tn ω 2 = (S1 + T1 ω 2 )n . By subtraction, (ω − ω 2 )Tn = (S1 + T1 ω)n − (S1 + T1 ω 2 )n , and thus writing α = μp = xp + yp ω, β = μp = xp + yp ω 2 , we have Tn =

αn − β n , ω − ω2

(4)

a recurrent sequence whose properties are described in [9]. We may parameterize this recurrence by writing G = α + β, H = αβ, and observing that Tn (G, H) is 2

For simplicity, we will usually write Sn and Tn for Sn (x, y) and Tn (x, y), respectively.

Improved Primality Proving with Eisenstein Pseudocubes

given by the second-order recurrence: Tn+2 = GTn+1 − HTn . From (3), 1 if and only if q | (α

q−1 3

q−1 3

−β ) and hence from (4), q = 1 ⇐⇒ q | T q−1 (G, H). 3 μp 3

377 q μp

= 3

(5)

Since only the case q | yp remains, we can rewrite (5) in terms of a single variable by deﬁning zp ≡ xp yp−1 (mod q). Now (xp + yp ω)(q−1)/3 ≡ (xp + yp ω 2 )(q−1)/3 (mod q) if and only if (zp + ω)(q−1)/3 ≡ (zp + ω 2 )(q−1)/3 (mod q). Setting α = zp + ω, β = zp + ω 2 in (4), we obtain q = 1 ⇐⇒ q | T q−1 (G , H ) (6) 3 μp 3 where G = 2zp − 1, H = zp2 − zp + 1. Since this relationship involves only one variable, we are in eﬀect considering polynomials Tn (x) where T0 (x) = 0,

T1 (x) = 1

Tn+1 (x) = (2x − 1)Tn x − (x2 − x + 1)Tn−1 (x) for a ﬁxed x ∈ Z. By induction, we see that Tn (x) is a polynomial over Z with coeﬃcients of degree n − 1 and leading coeﬃcient n. n −β n In fact, Tn (x) = Un (G , H ) where Un is the Lucas function, Un = αα−β , 2 G = α + β = 2x − 1, H = αβ = x − x + 1, and hence α = (x + ω), β = (x + ω 2 ). By drawing on the rich theory of Lucas functions, we can obtain both an eﬃcient algorithm for computing the acceptable congruence conditions on xp , yp (mod q), and the number of acceptable residues for the prime q. To obtain the candidate solutions zp satisfying (6), compute T q−1 (x) for all 3 0 ≤ x < q by the method described in [3, §4.4], retaining solutions for which T q−1 (x) ≡ 0 (mod q). Each zp obtained in this fashion can then be used to 3 produce (q − 1) acceptable values of μp by evaluating xp = 1, 2, . . . , q − 1 and computing the corresponding yp = xp zp (mod q)—a procedure illustrated in Example 3. To obtain a count of these solutions, observe that in (6), we can write Δ = (α −β)2 = (2x − 1)2 − 4(x2 − x + 1) = −3. If q is a prime ≡ 1 (mod 3) then = 1. Thus if x ∈ Z and q | x2 − x + 1 then q | Tq− (x) [3, Equation = Δ q

4.3.3]. It follows that the polynomial Tq−1 (x) of degree q − 2 has precisely q − 2 distinct zeros modulo q. Now T q−1 (x) ∈ Z[x], and so it divides Tq−1 (x) as, from 3 the theory of Lucas functions [3, Equation 4.2.45], we have T3n (2x − 1, x2 − x + 1) = 3Tn ((x2 − x + 1)n − Tn2 ). It follows that T q−1 (x) has exactly q−1 3 − 1 distinct zeros modulo q. 3 By combining the cases when q | yp and q | yp , we see that there are (q − 1)2 q−1 − 1 (q − 1) + (q − 1) = 3 3 acceptable residues for a prime q ≡ 1 (mod 3).

(7)

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Example 3. Consider the case q = 7. We can derive the acceptable residue conditions on μp as follows. If q | yp , then (x + 0ω) is acceptable for x = 1, . . . , (q − 1). If q | yp then from (6), we have that

7 μp

3

= 1 ⇐⇒ 7 | T 7−1 (G , H ) = 3

T2 (G , H ). Further, T2 (G , H ) = G T1 (G , H )−HT0(G , H ) = G −0 = 2zp −1 and hence, 7 = 1 ⇐⇒ 7 | 2zp − 1. μp 3 Thus, zp ≡ 4 (mod 7). Since we deﬁned zp = xp yp−1 (mod q), xp ≡ 4yp (mod 7), and we can obtain all solutions by running xp through all nonzero residue classes (modulo 7) and computing yp ≡ 4−1 xp ≡ 2xp (mod 7); i.e. xp 1 2 3 4 5 6 . yp ≡ 2xp (mod 7) 2 4 6 1 3 5 Combining these solutions with the trivial case (q | yp ), we obtain a complete set of solutions (modulo 7): S7 = {(1 + 0ω), (2 + 0ω), (3 + 0ω), (4 + 0ω), (5 + 0ω), (6 + 0ω), (4 + 1ω), (1 + 2ω), (5 + 3ω), (2 + 4ω), (6 + 5ω), (3 + 6ω)}.

4

Eisenstein Pseudocubes and Primality Testing

Eisenstein pseudocubes may be employed to prove primality for integers N ≡ 1 (mod 3) via the following theorem [10]. Theorem 2. (Berrizbeitia, 2003, personal correspondence) Let ν = a + bω be a primary element of Z[ω], where gcd (a, b) = 1, ν is not a unit, prime, or perfect power in Z[ω], and N (ν) < N (μp ). Then there must exist a rational prime q ≤ p such that

q ν

3

≡ q (N (ν)−1)/3 (mod ν).

Recall that if N ≡ 1 (mod 3) and N is a prime in Z, then N = νν, where ν is a primary prime in Z[ω]. Furthermore, if q is any rational prime, then N −1 q ≡q 3 (mod ν). ν 3 If we have a table of Eisenstein pseudocubes available to us, Berrizbeitia’s result gives us a means to certify the primality of N ≡ 1 (mod 3); i.e. 1. Test that N is not a perfect power; e.g. via [11]. 2. Find a primary ν ∈ Z[ω] such that N (ν) = N . This can be done eﬃciently using Cornacchia’s algorithm [12, §1.5.2] via the method of Williams [13, §5]. If this step fails, then N is composite.3 3

Cornacchia’s algorithm requires the evaluation of a square root modulo N , and hence, usually requires a factorization of N . For our purposes, however, we simply assume that N is prime in this step. If Cornacchia fails, it is because N was composite, which is exactly what we set out to determine.

Improved Primality Proving with Eisenstein Pseudocubes

379

3. From a precomputed table of Eisenstein pseudocubes, choose μp ∈ Z[ω] of minimal norm such that N < (μp ). N −1 N q ≡ q 3 (mod ν). If the test succeeds for 4. For each prime q ≤ p, test ν 3 all q, then N is prime. Step 1 of this algorithm requires (log N )1+o(1) operations. Cornacchia’s algorithm (Step 2) essentially consists of a GCD computation ((log N )2+o(1) operations), and the computation of a square root modulo a prime ((log N )3+o(1) ). Step 3 is a merely a table lookup. Step 4 appears to be the most computationally intensive component of the algorithm, requiring a series of modular exponentiations (each requiring (log N )2+o(1) operations). The precise number of exponentiations is dependent on the expected growth rate of the Eisenstein pseudocubes, something which we will now attempt to estimate.

5

Eisenstein Pseudocube Growth Rate

Let pi denote the ith prime (p1 = 2), and let Sp denote the set of acceptable residues modulo p for the Eisenstein pseudocubes as developed in Section 3. Writing p = pn , and denoting by (a, b) the Eisenstein integer a + bω, we know that S2 = {(1, 0)} , S9 = {(2, 0), (5, 0), (8, 0)} , and p Sp = (a, b) ∈ Z × Z = 1, a + bω 3

p−1 p−1 ≤ a, b ≤ − 2 2

for p > 3

Recall from Equations (1) and (7) that we expect ⎧ (p − 1)2 ⎪ ⎪ ⎨ if p ≡ 1 (mod 3) 3 |Sp | = 2 ⎪ ⎪ ⎩ (p − 1) if p ≡ 2 (mod 3) 3 acceptable residues modulo p. Writing

S1 = p≡1

(mod 3)

S2 = p≡2

(mod 3)

(p − 1)2 3

H1 =

(p2 − 1) 3

H2 =

p≡1

p≡2

p

(mod 3)

p

(mod 3)

for primes p ≤ pn , and invoking the Chinese Remainder Theorem we see that there are S = 3S1 S2 solutions satisfying the congruence criteria of the Eisenstein pseudocubes in the region −H/2 ≤ a, b < H/2, where H = 9H1 H2 .

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K. Wooding and H.C. Williams

Assume the S solutions μ = a + bω are equidistributed in the region −H/2 ≤ a, b < H/2. By a similar argument to that of Lukes et al. [1], we expect the solution of minimal norm, denoted by μp , to be given by a ≈ b ≈ √HS ; i.e. H2 . S

N (μp ) ≈

(8)

Consider the primes p = pn as n → ∞. Making an assumption that the primes are distributed equally between p ≡ 1 (mod 3) and p ≡ 2 (mod 3), we can approximate H 2 /S as follows. Write H12 = S1 H22 = S2

p≡1

3p2 , and (p − 1)2

(mod 3) p≤x

p≡2

3p2 . − 1)

H12 ≈ 3π(x)/2 S1 ≈ 3π(x)/2

1 p≤x 1−1/p

p≡1

p≤x

(10)

(p2

(mod 3) p≤x

From Mertens’s Theorem [14, p. 351], becomes

(9)

(mod 3) p≤x

e−γ log x

∼

p p−1

as x → ∞, so (9)

2

p p−1

γ π(x)/2

log x. ∼e 3 For (10), recall that p≤x 1 − p12 = ζ(2) = H22 ∼ 3π(x)/2 S2

π2 6

as x → ∞.4 Hence

6 π2

Putting these together, and writing n = π(x), c = N (μpn ) ≈

27eγ π

√

6

, we obtain

(9H1 H2 )2 ∼ c3n log pn 3S1 S2

as n → ∞. Thus, we expect (log N )1+o(1) exponentiations in Step 4 of our primality proving algorithm, for a combined (randomized) complexity of (log N )3+o(1) operations.5 4 5

See, for example, [15, Theorem 1.4.1]. The randomized nature of the algorithm stems solely from the requirement for a quadratic nonresidue in Cornacchia’s algorithm. Finding this quadratic nonresidue requires, on average, two evaluations of a Jacobi symbol.

Improved Primality Proving with Eisenstein Pseudocubes

381

90

80

70

60

50

40

30

20 log(M2,p/ log p) f(x)= 0.676694 x + 4.55216 2/3 ln(M3,q / (ln q)4/3) g(x)=0.707752x + 0.633985 ln(N(mup) / ln p)

10

h(x)=1.05557x + 3.79531 0 0

10

20

30

40 n

50

Fig. 1. Growth Rates

60

70

80

382

6

K. Wooding and H.C. Williams

Experimental Results

Our experiment followed the same basic approach as [6]. To test our hypotheses, a table of Eisenstein pseudocubes was developed using the Calgary Scalable Sieve (CASSIE), a software toolkit for congruential sieving on the University of Calgary’s Advanced Cryptography Laboratory (ACL) Beowulf cluster [6]. First, a series of small, non-normalized runs were performs in order to obtain Eisenstein pseudocubes for values of p ≤ 109. Once these runs were completed, a large parallel job was executed. This larger job evaluated all candidate solutions with N (μp ) ≤ 264 . To parallelize this job, the 11520 acceptable residues formed by Table 1. Eisenstein Pseudocube Results p 18 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61, 67 71 73 79, 83 89 97 101 103, 107 109 113 127 131 137 139 149 151 157

10 15 21 596 2127 5736 9708 14102

1 4 11 82 115 2507 3393 9175 21408 81221 70670 84695 44850 03669 62708 34194 82344 28178

N (μp ) 247 643 5113 13507 39199 1 07803 3 60007 39 04969 61 07191 103 18249 273 33067 991 79467 5329 97833 22785 22747 27417 02809 85007 66499 15475 53813 94233 48797 46210 13649 18103 60731 90827 69801 26375 28481 67688 29893 90619 32079 66151 53761 04348 13749 56547 47279 97583 41459 06441 31739 04110 19739 93471 77659 17235 68077 31706 25921

μp 11 + 18ω 29 + 18ω 71 + 72ω 23 + 126ω 227 + 90ω −181 + 198ω 653 + 126ω 443 + 2160ω −1669 + 1170ω 3617 + 2520ω 6023 + 3366ω 4973 + 11466ω −15451 + 11088ω 54017 + 17514ω 47477 + 56160ω 66887 + 156510ω 235061 + 107172ω −139813 + 253764ω −267733 + 744120ω 1227419 + 761670ω 5052689 + 4961880ω −2127709 + 4462200ω 10322861 + 8601732ω 3056387 + 15918570ω −27791551 + 1366560ω 109364777 + 13014540ω −114717193 + 19952010ω 160585853 + 126202050ω 845355437 + 667764090ω −724036477 + 954969030ω 696254903 + 2666049750ω 2979509543 + 3236384556ω 3671532959 + 3833807040ω

Improved Primality Proving with Eisenstein Pseudocubes

383

combining the solution candidates for moduli 18, 5, 7, and 11 were each used as a normalization modulus.6 Each of these jobs required approximately 8000 CPUseconds. Using 250 processing nodes, the complete job required approximately 4.25 days to complete, obtaining Eisenstein pseudocubes μp for p ≤ 157. These results are summarized in Table 1.

7

Analysis and Conclusions

In Figure 1, Eisenstein pseudocube growth is shown as a function of n, where pn is the nth prime. The straight line represents the least squares line ﬁtted to this data, and is given by: y = 1.05557x + 3.79531 a result that is remarkably consistent with the slope predicted by the argument of Section 5; i.e. log 3 = 1.09861. As a basis for comparison, classical pseudocube and pseudosquare results (including the recent work of Sorenson [7]) are also shown. Two conclusions may be drawn from these results. First, even with the relatively modest amount of computing power used to compute our table of Eisenstein pseudocubes, we have already produced a test that is more eﬃcient than the pseudocube method originally proposed by Berrizbeitia, et al. Second, we would expect that with a reasonable amount of computational investment, the Eisenstein pseudocube primality proving method will eventually be more eﬃcient than existing methods involving the pseudosquares.

8

Summary

In this paper, we have adapted a theorem of Berrizbeitia to produce a highly eﬃcient primality proving algorithm for integers N ≡ 1 (mod 3), making use of quantities known as Eisenstein pseudocubes. In addition to theoretical contributions, we have compiled a table of these quantities using an extensive twodimensional sieve calculation, and oﬀered numerical evidence for a conjectured growth rate: N (μpn ) ∼ c3n log pn as n → ∞.

References 1. Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Mathematics of Computation 65(213), S25–S27, 361–372 (1996) 2. Hall, M.: Quadratic residues in factorization. Bulletin of the American Mathematical Society 39, 758–763 (1933) 6

The normalization optimization, ﬁrst proposed by Lehmer in [16], is described in some detail in [6, §3.2].

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´ 3. Williams, H.C.: Edouard Lucas and Primality Testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. Wiley Interscience, Hoboken (1998) 4. Williams, H.C.: Primality testing on a computer. Ars Combinatoria 5, 127–185 (1978) 5. Berrizbeitia, P., M¨ uller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004) 6. Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006) 7. Sorenson, J.P.: Sieving for pseudosquares and pseudocubes in parallel using doublyfocused enumeration and wheel datastructures. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 331–339. Springer, Heidelberg (2010) 8. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, Heidelberg (1990) 9. Williams, H.C.: Some properties of a special set of recurring sequences. Paciﬁc Journal of Mathematics 77(1), 273–285 (1978) 10. Wooding, K.: The Sieve Problem in One- and Two-Dimensions. PhD thesis, The University of Calgary, Calgary, AB (April 2010), http://math.ucalgary.ca/~ hwilliam/files/wooding10thesis.pdf 11. Bernstein, D.J.: Detecting perfect powers in essentially linear time. Mathematics of Computation 67, 1253–1283 (1998) 12. Cohen, H.: A Course in Computational Algebraic Number Theory, 4th edn. Springer, Heidelberg (1993) 13. Williams, H.C.: An m3 public-key encryption scheme. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 358–368. Springer, Heidelberg (1986) 14. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979) 15. Crandall, R., Pomerance, C.: Prime numbers: A computational Perspective, 2nd edn. Springer, New York (2005) 16. Lehmer, D.H.: The sieve problem for all-purpose computers. Mathematical Tables and Other Aids to Computation 7(41), 6–14 (1953)

Hyperbolic Tessellations Associated to Bianchi Groups Dan Yasaki Department of Mathematics and Statistics University of North Carolina at Greensboro, Greensboro, NC 27412, USA d [email protected]

Abstract. Let F/Q be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.

1

Introduction

Let F/Q be a number ﬁeld. The space of positive deﬁnite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into polyhedral cones corresponding to the facets of the Vorono¨ı polyhedron [1, 11, 13]. This has been computationally explored for real quadratic ﬁelds in [16, 12] and the cyclotomic ﬁeld Q(ζ5 ) in [23]. For F an imaginary quadratic ﬁeld, the polyhedral cones give rise to ideal polytopes in H3 , 3-dimensional hyperbolic space. In work of Cremona and his students [6, 7, 5, 14, 22], analogous polytopes have already been computed for class number one imaginary quadratic ﬁelds as well as a few ﬁelds with class number two and three using diﬀerent methods. The structure of the polytopes was used to compute Hecke operators on modular forms for the Bianchi groups over those ﬁelds. These polytopes were used by Goncharov [10] in his study of Euler complexes on modular curves. The data of the polytope and stabilizer could also be used to give explicit presentations of GL2 (O) using results of Macbeath and Weil [15,21]. Swan [20] has computed presentations of these groups, though √ not with the polytopes constructed here, for imaginary quadratic ﬁelds Q( d) for −d ∈ {1, 2, 3, 5, 6, 7, 11, 15, 19}. Such explicit presentations have been used to compute cohomology of Bianchi groups of small discriminant with non-trivial coeﬃcients in work of Berkove, Sengun, and Finis-Grunewald-Tirao [2, 3, 9, 19]. We remark that there are other ways to obtain the fundamental polytope data. Riley [18] wrote the ﬁrst computer implementation of Poincar´e’s Polyhedron Theorem, which works in the more general setting of geometrically ﬁnite Kleinian G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 385–396, 2010. c Springer-Verlag Berlin Heidelberg 2010

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groups. He computed the fundamental polytopes for many Bianchi groups. From this data, he computed presentations for the Bianchi groups and calculated the rank of their abelianizations. Another method is to use reduction theory. An algorithm of Swan [20] has been very recently implemented by Rahm and Fuchs [17], who used it to compute the integral homology groups of all Bianchi groups which are over imaginary quadratic ﬁelds of class number less than three. In this paper, we investigate the structure of these ideal polytopes for a large range of imaginary quadratic ﬁelds. Our approach and implementation works for general imaginary quadratic ﬁelds, but we restrict the range to ease the computation. We compute the ideal polytope classes for all imaginary quadratic ﬁelds of class number one and two, as well as some ﬁelds of higher class number with small √ discriminant. Speciﬁcally, we compute the ideal polytopes for the ﬁelds Q( d) for square-free d, where −d ∈ {1, · · · , 100, 115, 123, 163, 187, 235, 267, 403, 427}. There is no theoretical obstruction to computing these tessellations for higher class number and higher discriminant. The structure of the paper is as follows. We set the notation for the quadratic ﬁelds and Hermitian forms in Section 2. The implementation is described in Section 3. Finally, in Section 4, we summarize some of the data collected so far. Finally, we describe a general result of Macbeath on computing group presentations for groups of homeomorphisms, illustrating one possible use√of this data. We use this technique to give an explicit presentation for GL2 (Q( −14)) in Section 5.

2

Notation and Background

√ Let F = Q( d) ⊂ C be an imaginary quadratic number ﬁeld. We always take d < 0 to be a square-free integer. Let O ⊂ F denote the ring of integers in F . Then O has a Z-basis consisting of 1 and ω, where √ 1+ d if d ≡ 1 mod 4, ω = √2 d if d ≡ 2, 3 mod 4. Let ¯· denote complex conjugation, the nontrivial Galois automorphism of F . Definition 1. A binary Hermitian form over F is a map φ : F 2 → Q of the form φ(x, y) = ax¯ x + bx¯ y + ¯b¯ xy + cy y¯, where a, c ∈ Q and b ∈ F such that φ is positive definite. By choosing a Q-basis for F , φ can be viewed as a quadratic form over Q. In particular, it follows that φ(O2 ) is discrete in Q.

Hyperbolic Tessellations Associated to Bianchi Groups

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Definition 2. The minimum of φ is m(φ) =

inf

v∈O 2 \{0}

φ(v).

A vector v ∈ O2 is minimal vector for φ if φ(v) = m(φ). The set of minimal vectors for φ is denoted M (φ). Definition 3. A Hermitian form over F is perfect if it is uniquely determined by M (φ) and m(φ).

3 3.1

Implementation Cone of Hermitian Forms and Hyperbolic Space

The space of positive deﬁnite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into polyhedral cones corresponding to the facets of the Vorono¨ı polyhedron Π [11, 13, 1]. The top-dimensional cones of this decomposition correspond to perfect forms and descend to ideal polytopes in H3 , 3-dimensional hyperbolic space. Details are given below. Let G be the restriction of scalars G = ResF/Q (GL2 ). Then the group of rational points G(Q) = GL2 (F ), and the group of real points is G = G(R) GL2 (C). Let H3 be hyperbolic 3-space: H3 = {(z, t) : z ∈ C,

t ∈ R>0 }.

Then G acts on H3 by αβ · (z, t) = (z ∗ , t∗ ), γ δ z∗ =

(αz + β)(γz + δ) + (αt)(γt) |γz + δ|2 + |γ|2 t2

where

and t∗ =

|αδ − βγ|t |γz + δ|2 + |γ|2 t2

Note that diagonal matrices act trivially on H3 , and the stabilizer of the point (i, 1) is U (2). Thus one gets an identiﬁcation between H3 and the coset space GL2 (C)/(U (2) · R>0 ). A binary Hermitian form can be identiﬁed with the 4-dimensional real vector space V of Hermitian 2 × 2 matrices. The group GL2 (C) acts on this space via g · A = gAg ∗ and preserves the open cone C ⊂ V of positive deﬁnite Hermitian matrices, and the stabilizer of I is U (2). Thus one has identiﬁcation C GL2 (C)/U (2). Modding out by homotheties, one gets C/R>0 H3 .

(1)

388

3.2

D. Yasaki

Vorono¨ı Decomposition

There is a map q from O2 to the closure C¯ of C ⊂ V given by q(v) = vv ∗ . The Vorono¨ı polyhedron Π is the unbounded polytope gotten by taking the convex hull of {q(v) : v ∈ O2 \ 0}. Taking cones over the facets of Π, one gets a decomposition of C into polyhedral cones known as the Vorono¨ı decomposition of C. By (1), this decomposition descends to a tessellation of H3 by ideal polytopes. Note that the group Γ = G(Z) = GL2 (O) acts on C and preserves this decomposition. 3.3

Perfect Forms

A perfect form φ is uniquely determined by its minimum m(φ) and set of minimal vectors M (φ). By scaling, we can assume m(φ) = 1. Since each minimal vector deﬁnes a linear equation in V , and V is 4-dimensional, generically 4 minimal vectors will uniquely determine φ. Note that this does not imply that #M (φ) = 4. Indeed in many examples, one has M (φ) > 4. There is a bijection between perfect forms over F and the facets of Π. Let P be a facet of Π with vertices {w1 , . . . , wk }. Then there is a unique form φP ∈ C such that m(φP ) = 1 and {q(v) : v ∈ M (φP )} = {w1 , . . . , wk }. There is an algorithm [11] that uses this bijection to compute the GL2 (O)equivalency classes of perfect forms. The algorithm uses linear algebra and convex geometry, but requires an initial input of a perfect form. To this end, we describe the method that √we used to compute an initial perfect form. For each ﬁeld F = Q( d), we need only to ﬁnd a single perfect form to begin the algorithm. Thus we limit our search to a particular family of quadratic forms. Speciﬁcally, let S0 ⊂ C be the subset of quadratic forms φ such that 1 0 1 ⊆ M (φ). , , 1 1 0 For φ ∈ S0 , the Hermitian matrix Aφ associated to φ must have the form 1 1β , where β ∈ F with Re(β) = − and |β| < 1. Aφ = ¯ β1 2 a ∈ O2 , then If φ ∈ S0 and φ has an additional minimal vector b 1 − a1 2 + a2 2 d + a1 b1 − a2 db2 − b1 2 + b2 2 d √ 1 d, β=− + 2 2 da1 b2 − 2 da2 b1 √ √ where a = a1 + a2 d and b = b1 + b2 d. Combined with (2), this implies

2 1 − a1 2 + a2 2 d + a1 b1 − a2 db2 − b1 2 + b2 2 d d 3 − < . 2 4 (2 da1 b2 − 2 da2 b1 )

(2)

(3)

Hyperbolic Tessellations Associated to Bianchi Groups

389

Reduction theory, speciﬁcally the existence of Siegel sets, ensures that the values NF/Q (a), NF/Q (b), and NF/Q (b − a) for a solution are bounded above by a constant depending upon d. Thus we implement a brute force search over a, b ∈ O a is found satisfying (3), we beginning at 0 and moving out. When a vector b check that the corresponding form φ satisﬁes a 1 0 1 ⊆ M (φ). , , , b 1 1 0 This corresponds to a ideal polytope whose vertices contain {∞, 0, 1, ab }. Once the initial form is found, we implement the algorithm of [11] to ﬁnd all the perfect forms over F up to the action of GL2 (O) (and the corresponding structure of the Vorono¨ı polyhedron) in Magma [4]. This descends, via (1), to give a tessellation of H3 by ideal polytopes.

4

Polytope Data

In this section we collect the results of the computations of the GL2 (O)-conjugacy classes of the ideal Vorono¨ı polytopes. Example: d = −14 √ Let F = Q( √ −14). Then F has class number four and ring of integers O = Z[ω], where ω = −14. There are 9 GL2 (O)-classes of polytopes which are of 3 combinatorial types. There are 3 triangular prisms with cuspidal vertices 5 + 2ω 2 + ω 4 + 2ω P1 = ∞, 1, , , ,0 9 4 9 5 + 2ω 4 + 2ω 12 + 4ω 11 + 4ω , 1, , , , 0 , and P2 = 23 9 9 23 8 + 5ω 2 + ω 1 + ω 2 + ω 3 + 2ω 7 + 4ω P3 = , , , , , , 23 5 5 6 10 21 4.1

and 5 tetrahedra with cuspidal vertices 11 + 4ω 2 + ω 4 + 2ω T1 = , , ,0 , 23 5 9 5 + 2ω 3 + ω 12 + 4ω T2 = 1, , , , 9 5 23 11 + 4ω 2 + ω 2 + ω , , ,0 , T3 = 23 5 6 8 + 5ω 2 + ω 4 + 2ω T4 = , , , 0 , and 23 5 9 3 + ω 12 + 4ω 4+ω , 1, , , T5 = 6 5 23

390

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and a square pyramid with cuspidal vertices 8 + 5ω 2 + ω 1 + ω 2 + ω , , , ,0 . S= 23 5 5 6 Given the cuspidal vertices, one can easily compute the stabilizers of each polytope. The stabilizers are all cyclic in this case. For each stabilizer, we compute a generator. The results are given in Table 1. √ Table 1. Stabilizer groups of Vorono¨ı ideal polytopes for Q( −14) Polytope Stabilizer

4.2

P1

C6

P2

C2

P3

C4

T1

C2

T2

C2

T3

C2

T4

C2

T5

C2

S

C2

Generator 1 −1 1 0 −1 0 0 −1 ω + 1 −ω + 6 2 −ω − 1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1

Polytope Summary

We compute √ the Vorono¨ı polytopes for all imaginary quadratic number ﬁelds F = Q( d) with class number one and two as well as higher class number for d > −100. Although there is no reason an arbitrary convex 3-dimensional polytope could not arise, in all of these cases only 8 combinatorial types show up. We give the names and F -vector ([#vertices, #edges, #faces]) for each in Table 2. We also note that the triangular dipyramid shows up in this range much less frequently than the other polytopes. In Table 3, we give the number of GL2 (O)-classes of each polytope type for F with class number one or two. In Table 4, we give the number of GL2 (O)classes of each polytope type for the remaining imaginary quadratic ﬁelds with d > −100.

Hyperbolic Tessellations Associated to Bianchi Groups

391

Table 2. Combinatorial types of ideal polytopes that occur in this range polytope

F -vector picture

tetrahedron

[4, 6, 4]

octahedron

[6, 12, 8]

cuboctahedron

[12, 24, 14]

triangular prism

[6, 9, 5]

hexagonal cap

[9, 15, 8]

square pyramid

[5, 8, 5]

truncated tetrahedron [12, 18, 8] triangular dipyramid

5

[5, 9, 6]

Group Presentation

A general result of Macbeath [15] and analogous result of Weil [21] give a general method of computing group presentations for groups of homeomorphisms. For the convenience of the reader, we recall these results here and describe how the polytope data computed above can be used to compute explicit presentations of GL2 (OF ). Consider a connected space X acted upon by a group of homeomorphisms Γ . Let U ⊂ X be an open set such that Γ · U = X, and let Σ ⊂ Γ denote the set Σ = {g ∈ Γ : g · U ∩ U = ∅}. Let F (Σ) be the free group generated by Σ. For g ∈ Σ, let xg denote the corresponding element of F (Σ). Let W ⊂ Σ × Σ denote the set W = {(g, h) : U ∩ g · U ∩ gh · U = ∅}. Let R ⊂ F (Σ) denote the subgroup generated by xg xh x(gh)−1 for (g, h) ∈ W . Suppose π0 (X) = π1 (X) = π0 (U ) = 1. Then the subgroup R is a normal subgroup of F (Σ) and Γ F (Σ)/R. To apply this result to the polytope data computed above, choose X = H3 . Fix representatives P1 , . . . , Pk of the GL2 (O) classes of polytopes such that D = P1 ∪· · ·∪Pk is a connected set of polytopes meeting along facets. Let U ⊂ H3 be an open neighborhood of D ∩ H3 . We note that since the vertices D are at

392

D. Yasaki

Table 3. GL2 (O)-classes of Vorono¨ı ideal polytopes for class number one and two

hF

d

1 1 1 1 1 1 1 1 1

−1 0 −2 0 −3 1 −7 0 −11 0 −19 0 −43 0 −67 0 −163 11

1 0 0 0 0 0 0 1 0

0 1 0 0 0 1 0 0 1

0 0 0 1 0 1 2 2 8

0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 2 3

0 0 0 0 1 0 1 1 0

0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

−5 −6 −10 −13 −15 −22 −35 −37 −51 −58 −91 −115 −123 −187 −235 −267 −403 −427

0 0 1 0 1 0 4 0 0 0 1 1 1 1 1 1 1 2

0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0

2 0 1 3 0 4 1 8 2 7 5 5 6 4 12 13 16 19

0 1 0 1 0 0 0 1 1 2 0 2 3 1 4 5 2 4

0 0 2 1 0 2 2 8 0 6 3 4 3 9 11 10 20 24

0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0

0 0 0 1 1 5 3 10 1 47 5 3 1 18 13 24 66 65

Hyperbolic Tessellations Associated to Bianchi Groups Table 4. GL2 (O)-classes of Vorono¨ı ideal polytopes with d > −100

hF

d

3 3 3 3

−23 −31 −59 −83

0 0 0 6

1 0 1 0

0 0 1 0

1 3 3 2

0 0 0 2

1 1 2 1

0 0 0 1

0 0 0 0

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

−14 −17 −21 −30 −33 −34 −39 −46 −55 −57 −73 −78 −82 −85 −93 −97

5 5 8 6 9 20 1 32 5 33 57 69 92 56 79 95

0 0 2 0 0 0 0 1 1 1 1 1 0 0 1 0

0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1

3 2 2 6 8 3 3 5 2 10 13 11 8 17 20 19

0 1 1 4 1 1 1 0 0 3 1 4 3 0 7 3

1 3 4 4 6 6 1 9 2 14 14 18 11 28 21 19

0 1 0 0 1 1 0 0 0 2 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0

5 −47 5 5 −79 9

0 0

0 0

1 5

1 0

2 4

0 0

0 0

−26 −29 −38 −53 −61 −87

18 15 33 45 41 6

1 0 1 0 1 0

0 0 0 0 0 0

2 6 2 7 11 6

1 0 1 2 1 2

4 6 6 13 16 3

0 0 1 0 0 0

0 0 0 0 0 0

7 −71 7

1

0

4

0

4

0

0

−41 −62 −65 −66 −69 −77 −94 −95

31 81 69 67 51 81 125 12

0 0 2 1 2 1 1 0

1 0 0 1 0 0 0 0

9 7 9 9 15 9 10 4

0 2 0 4 2 2 2 0

8 7 19 12 21 26 17 9

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0

10 −74 105 10 −86 130

1 0

0 0

9 9

1 1

12 18

0 1

0 0

12 −89 136

0

0

14

1

21

1

0

6 6 6 6 6 6

8 8 8 8 8 8 8 8

393

394

D. Yasaki

inﬁnity, the set U can be chosen so that if g ∈ Σ, then g takes an edge of D to another edge of D. We remark that many redundant generators and relations are created when implementing this result, especially when the stabilizer groups of the polytopes are large. We can compensate for this using Magma’s commands for simplifying ﬁnitely-presented groups. We illustrate the technique in the example below. Example: d = −14 √ Theorem 1. Let F = Q( −14) with ring of integers O = Z[ω], where ω = √ −14. Then the following is a presentation of GL2 (O):

5.1

GL2 (O) = g1 , · · · , g8 : R1 = · · · = R22 = 1,

where

R1 = g72 ,

R2 = g82 ,

R3 = g62 ,

R4 = g32 ,

R5 = g42 ,

R6 = g22 ,

R7 = g54 ,

R8 = (g2 g1−1 )2 ,

R9 = (g4 g1 )2 , R13 = (g6 g5−2 )2 ,

R10 = g5−1 g1−3 g5−1 , R11 = (g7 g5−2 )2 , R12 = (g8 g5−2 )2 , R14 = (g4 g5−2 )2 ,

R17 = (g3 g5−1 g3 g1 g2 )2 ,

R15 = (g3 g5−2 )2 , R16 = (g6 g1−1 g5−1 )2 ,

R18 = (g3 g7 g1 g8 g1−1 )2 ,

R19 = g4 g5 g4 g1−1 g5 g1 ,

R20 = g8 g5−1 g7 g5−1 g3 g1−1 g3 g7 g3 g7 g1 g8 g3 g5 g7 g5−1 , R21 = g1 g5 g7 g5−1 g3 g1−1 g3 g7 g1 g5−1 g7 g5−1 g3 g1−1 g3 g7 , R22 = g6 g5 g7 g5−1 g3 g1−1 g3 g7 g1 g6 g1−1 g7 g3 g1 g3 g5 g7 g5 . Proof. We choose X, U , and D as described above. In fact, one can choose D to be the polytopes given in Section 4.1. Then F (Σ)/R is deﬁned by 235 generators and 3416 relations. √ We can simplify this presentation in Magma to get the presentation of GL2 (Z[ −14]) above, with 1 −1 01 , g2 = , g1 = 1 0 10 ω + 3 −ω + 1 4ω −2ω + 13 , g4 = , g3 = 6 −ω − 3 2ω + 13 −4ω −2ω − 5 2ω − 3 −5ω 3ω − 15 g5 = , g6 = , −10 2ω + 5 −3ω − 15 5ω ω + 9 −2ω − 1 −2ω − 13 4ω + 4 , g8 = . g7 = −2ω + 10 −ω − 9 ω − 14 2ω + 13 The presentation given in the theorem has torsion elements as generators. In particular, GL2 (O) is generated by elements of order 2, 4, and 6. Since any torsion-free quotient must map these generators to the identity, one immediately gets the following corollary. √ Corollary 1. GL2 (Z[ 14]) has no torsion-free quotients. √ One ﬁnds similar results for F = Q( d) for d = −1 and d = −3 in [8].

Hyperbolic Tessellations Associated to Bianchi Groups

395

Acknowledgments. I thank the reviewers for their comments. I would like to thank John Cremona for helpful conversations at the beginning of this project, and Paul Gunnells for introducing me to these techniques. I thank Sebastian Pauli for his advice on the computation, Carlos Nicholas for his help with the polytopes, and Greg Bell for his help with the group presentations. Finally, I thank Steve Donnelly for helpful discussions and the Magma Group at the University of Sydney for their hospitality during a visit, in which part of this research was completed. This work was partially supported by the UNCG New Faculty grant.

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Author Index

Balakrishnan, Jennifer S. 16 Bernard, Aurore 32 Biasse, Jean-Fran¸cois 50 Bos, Joppe W. 66 Bradshaw, Robert W. 16 Brent, Richard P. 83 ´ Brier, Eric 96 Bruin, Nils 110

Lenstra, Arjen K. 66 Levin, Mariana 6 Lubicz, David 251

Clavier, Christophe

96

Pauli, Sebastian Pomerance, Carl

Dahmen, Sander R. Darmon, Henri 1

110

Fieker, Claus 157 Ford, David 174

Hart, William B. Ionica, Sorina

2

Nagao, Koh-ichi 285 Nebe, Gabriele 4 301 6

Regev, Oded 3 Robert, Damien 251

Elsenhans, Andreas-Stephan Enge, Andreas 142

Gama, Nicolas

McKee, James 270 Mestre, Jean-Fran¸cois

32

Siksek, Samir 316 Sorenson, Jonathan P. 331 Soukharev, Vladimir 219 Soundararajan, K. 6 Stehl´e, Damien 157, 340 Stoll, Michael 316 Sutherland, Andrew V. 142 Tibouchi, Mehdi 234 Tornar´ıa, Gonzalo 186

186 201

Jacobson Jr., Michael J. Jahnel, J¨ org 126 Jao, David 219 Joux, Antoine 201 Joye, Marc 234

126

50

Veres, Olga 174 Vergnaud, Damien Voight, John 357

Watkins, Mark 186, 340 Williams, Hugh C. 372 Wooding, Kjell 372 Yasaki, Dan

Kedlaya, Kiran S. 16 Kleinjung, Thorsten 66

234

385

Zimmermann, Paul

83

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

6197

Guillaume Hanrot François Morain Emmanuel Thomé (Eds.)

Algorithmic Number Theory 9th International Symposium, ANTS-IX Nancy, France, July 19-23, 2010 Proceedings

13

Volume Editors Guillaume Hanrot LIP/ENS-Lyon, 46, allée d’Italie 69364 Lyon Cedex 07, France E-mail: [email protected] François Morain LIX/École polytechnique 91128 Palaiseau Cedex, France E-mail: [email protected] Emmanuel Thomé INRIA Nancy, projet CARAMEL 615 rue du jardin botanique 54602 Villers-lès-Nancy Cedex, France E-mail: [email protected]

Library of Congress Control Number: 2010930653 CR Subject Classification (1998): F.2, G.2, E.3, I.1 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-642-14517-5 Springer Berlin Heidelberg New York 978-3-642-14517-9 Springer Berlin Heidelberg New York

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Preface

ANTS-IX was the ninth edition of the biennial International Symposium on Algorithmic Number Theory. The ﬁrst edition of this symposium was held at Cornell University in 1994. ANTS-IX was held July 19-23, 2010 at INRIA in Nancy, France. The ANTS-IX Program Committee consisted of 12 members whose names are listed on the next page. The selection of the accepted papers among the submissions was made from mid-January to end of March 2010. Each paper was thoroughly reviewed by at least two experts, including a Program Committee member. The Program Committee selected 25 high-quality articles, which are excellent representatives of the current state of the art in various areas of algorithmic number theory. The Selfridge Prize in computational number theory was awarded to the authors of the best contributed paper presented at the conference. We gratefully thank the authors of all submitted papers for their hard work which made the selection of a varied program possible. We also thank the authors of the accepted papers for their cooperation in the timely production of the revised versions. Each submitted paper was presented by one of its co-authors at the conference. Besides contributed papers, the conference included ﬁve invited talks by Henri Darmon (McGill University), Jean-Fran¸cois Mestre (Universit´e Paris 7), Gabriele Nebe (RWTH Aachen), Carl Pomerance (Dartmouth College), and Oded Regev (Tel-Aviv University). We thank the invited speakers for having been able to provide abstracts of their talk, which are reproduced in this volume. This list of invited speakers originally included Fritz Grunewald (HHU D¨ usseldorf), who unfortunately passed away on March 21, 2010, four months before the conference. A special lecture was held to honor his memory. The conference organizers wish to thank all the people who made the conference possible. In particular, we gratefully acknowledge the support of the funding institutions. May 2010

Guillaume Hanrot Fran¸cois Morain Emmanuel Thom´e

Organization

Organizing Committee Anne-Lise Charbonnier J´er´emie Detrey Pierrick Gaudry (Chair) Emmanuel Thom´e Paul Zimmermann

INRIA, Nancy, France INRIA, Nancy, France CNRS, Nancy, France INRIA, Nancy, France INRIA, Nancy, France

Program Committee Nigel Boston University of Wisconsin, USA John Cremona Warwick Mathematics Institute, UK Claus Fieker University of Sydney, Australia ´ Guillaume Hanrot (PC Chair) Ecole Normale Sup´erieure, Lyon, France Kevin Hare University of Waterloo, Canada ´ Thorsten Kleinjung Ecole Polytechnique F´ed´erale de Lausanne, Switzerland Kamal Khuri-Makdisi American University of Beirut, Lebanon ´ Fran¸cois Morain (PC Chair) Ecole Polytechnique, France Takakazu Satoh Tokyo Institute of Technology, Japan Igor Shparlinski Macquarie University, Australia Alice Silverberg University of California at Irvine, USA Frederik Vercauteren Katholieke Universiteit Leuven, Belgium

Poster Session Benjamin Smith

´ INRIA Saclay, Ecole Polytechnique, France

Sponsoring Institutions Institut National de Recherche en Informatique et Automatique (INRIA) Laboratoire Lorrain de Recherche en Informatique et Applications (LORIA) ´ Ecole Polytechnique Centre National de la Recherche Scientiﬁque (CNRS) Microsoft Research, USA Nancy Universit´e Groupement de Recherches en Informatique Math´ematique (GDR IM) Communaut´e Urbaine du Grand Nancy Conseil R´egional de Lorraine

VIII

Organization

Conference Website The names of the winners of the Selfridge Prize, material supplementing the contributed papers, and errata for the proceedings (if relevant), as well as the abstracts of the posters and the posters presented at ANTS-IX, can be found at http://ants9.org/.

Table of Contents

Invited papers Putting the Hodge and Tate Conjectures to the Test . . . . . . . . . . . . . . . . . Henri Darmon Curves of Genus 3 With a Group of Automorphisms Isomorphic to S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Fran¸cois Mestre

1

2

Learning with Errors over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oded Regev

3

Lattices and Spherical Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriele Nebe

4

Fixed Points for Discrete Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mariana Levin, Carl Pomerance, and K. Soundararajan

6

Contributed papers Explicit Coleman Integration for Hyperelliptic Curves . . . . . . . . . . . . . . . . Jennifer S. Balakrishnan, Robert W. Bradshaw, and Kiran S. Kedlaya

16

Smallest Reduction Matrix of Binary Quadratic Forms: And Cryptographic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aurore Bernard and Nicolas Gama

32

Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Fran¸cois Biasse and Michael J. Jacobson Jr.

50

On the Use of the Negation Map in the Pollard Rho Method . . . . . . . . . . Joppe W. Bos, Thorsten Kleinjung, and Arjen K. Lenstra

66

An O(M (n) log n) Algorithm for the Jacobi Symbol . . . . . . . . . . . . . . . . . . Richard P. Brent and Paul Zimmermann

83

New Families of ECM Curves for Cunningham Numbers . . . . . . . . . . . . . . ´ Eric Brier and Christophe Clavier

96

Visualizing Elements of Sha[3] in Genus 2 Jacobians . . . . . . . . . . . . . . . . . . Nils Bruin and Sander R. Dahmen

110

X

Table of Contents

On Weil polynomials of K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas-Stephan Elsenhans and J¨ org Jahnel

126

Class Invariants by the CRT Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas Enge and Andrew V. Sutherland

142

Short Bases of Lattices over Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . Claus Fieker and Damien Stehl´e

157

On the Complexity of the Montes Ideal Factorization Algorithm . . . . . . . David Ford and Olga Veres

174

Congruent Number Theta Coeﬃcients to 1012 . . . . . . . . . . . . . . . . . . . . . . . William B. Hart, Gonzalo Tornar´ıa, and Mark Watkins

186

Pairing the Volcano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sorina Ionica and Antoine Joux

201

A Subexponential Algorithm for Evaluating Large Degree Isogenies . . . . David Jao and Vladimir Soukharev

219

Huﬀ’s Model for Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marc Joye, Mehdi Tibouchi, and Damien Vergnaud

234

Eﬃcient Pairing Computation With Theta Functions . . . . . . . . . . . . . . . . . David Lubicz and Damien Robert

251

Small-Span Characteristic Polynomials of Integer Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James McKee

270

Decomposition Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koh-ichi Nagao

285

Factoring Polynomials over Local Fields II . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Pauli

301

On a Problem of Hajdu and Tengely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samir Siksek and Michael Stoll

316

Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures . . . . . . . . . . . . . . Jonathan P. Sorenson On the Extremality of an 80-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . Damien Stehl´e and Mark Watkins

331

340

Table of Contents

Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Voight

XI

357

Improved Primality Proving with Eisenstein Pseudocubes . . . . . . . . . . . . . Kjell Wooding and H.C. Williams

372

Hyperbolic Tessellations Associated to Bianchi Groups . . . . . . . . . . . . . . . . Dan Yasaki

385

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397

Putting the Hodge and Tate Conjectures to the Test Henri Darmon Department of Mathematics, McGill University, Burnside Hall, Montreal, QC, Canada [email protected]

The Hodge conjecture asserts that the presence of algebraic cycles on a (smooth, projective) variety over the complex numbers can be detected in its Betti cohomology equipped with the Hodge structure arising from its relation with complex deRham cohomology. The Tate conjecture makes a similar assertion with -adic cohomology replacing Betti cohomology. One of the diﬃculties with these conjectures is that the predictions that they make are often hard to test numerically, even in speciﬁc concrete instances. Unlike closely related parts of number theory (a case in point being the Birch and Swinnerton-Dyer conjecture) the study of algebraic cycles has therefore not been as strongly aﬀected by the growth of the experimental and computational community as it perhaps could be. In this lecture, I will describe some numerical experiments that are designed to “test” the Hodge and Tate conjectures for certain varieties (of arbitrarily large dimension) which arise from elliptic curves with complex multiplication and theta series of CM Hecke characters.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 1, 2010. c Springer-Verlag Berlin Heidelberg 2010

Curves of Genus 3 with a Group of Automorphisms Isomorphic to S3 Jean-Fran¸cois Mestre Centre de Math´ematiques de Jussieu Projet Th´eorie des Nombres [email protected]

In this talk, we construct curves of genus 3 with automorphism group equal to S3 ; we give some applications of this construction to the problem of optimal curves, i.e. of curves over a finite field Fq having a number of points equal to the Serre-Weil bound Mq ; in particular, we prove that there exists infinitely many fields F3n having optimal curves; we prove also that there exists an integer C such that, for any finite field F7n , there exists a curve of genus 3 defined over having at least Mq − C points.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 2, 2010. c Springer-Verlag Berlin Heidelberg 2010

Learning with Errors over Rings Oded Regev Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel [email protected]

The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather ineﬃcient due to an inherent quadratic overhead in the use of LWE. After a short introduction to the area, we will discuss recent work on making LWE and its applications truly eﬃcient by exploiting extra algebraic structure. Namely, we will deﬁne the ring-LWE problem, and prove that it too enjoys very strong hardness guarantees. Based on joint work with Vadim Lyubashevsky and Chris Peikert.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, p. 3, 2010. c Springer-Verlag Berlin Heidelberg 2010

Lattices and Spherical Designs Gabriele Nebe Lehrstuhl D f¨ ur Mathematik, RWTH Aachen University, Germany [email protected]

A lattice is a ﬁnitely generated discrete subgroup of Euclidean space. Lattices are an important algorithmic tool in number theory, integral representation theory, geometry, information theory, cryptography, crystallography and have various other applications within mathematics and beyond. Any lattice has only ﬁnitely many vectors of a given length, they form the layers of the lattice, which are ﬁnite subsets of spheres in the underlying Euclidean space. A spherical design of strength t is a ﬁnite set X = ∅ in the Euclidean 1 sphere for which the mean value |X| x∈X f (x) equals the integral of f over the sphere for all polynomials f of degree up to t. This condition is equivalent to x∈X f (x) = 0 for all non-constant harmonic polynomials of degree ≤ t. Spherical designs hence consist of well distributed points on a sphere and are relevant for numerical integration, in information theory, geometry, statistics and have applications for instance in medicine. Boris Venkov combined these two concepts in a very fruitful way that allows to use lattices to classify spherical designs and to use designs for ﬁnding good lattices. An introduction to this subject as well as some applications are given in “R´eseaux euclidiens, designs sph´eriques et formes modulaires”, Enseignement Math., Geneva, 2001. There Venkov introduces the notion of a strongly perfect lattice, which is a lattice whose minimal vectors form a spherical 4-design. Using the characterization by Korkine, Voronoi and Zolotarev one shows that strongly perfect lattices realise local maxima of the sphere packing density function on the space of all similarity classes of n-dimensional lattices (in fact in the space of all periodic packings as proved by Sch¨ urmann). All local maxima of this function are known up to dimension 8. In dimension 8 Dutour, Sch¨ urmann, Vallentin and Riener proved that there are 2408 local maxima. The densest lattice sphere packings are known up to dimension 8 and, thanks to recent results by Elkies and Kumar, in dimension 24, where the Leech lattice is the densest lattice. Combining number theory and geometry with combinatorial methods allows classify strongly perfect lattices, where a full classiﬁcation up to dimension 12 is obtained in joined work with Venkov. With one exception all known strongly perfect lattices Λ have the additional property that also the dual lattice Λ∗ is strongly perfect. Such lattices are called dual strongly perfect, the classiﬁcation of dual strongly perfect lattices in small dimension has been completed in dimension 14 and is an ongoing PhD project by Elisabeth Nossek in Aachen. There are two general approaches to study and construct strongly perfect lattices: by modular forms and by invariant theory of ﬁnite groups. Both concepts usually allow to show that all non-empty layers of the lattice form spherical G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 4–5, 2010. c Springer-Verlag Berlin Heidelberg 2010

Lattices and Spherical Designs

5

4-designs. Such lattices are called universally perfect and play a role in Riemannian geometry. If Λ is a universally perfect lattice then the torus Rn /Λ∗ deﬁned by the dual lattice Λ∗ provides a strict local minimum of the height function on the set of all n-dimensional ﬂat tori. R. Coulangeon also shows that universally perfect lattices Λ achieve local minima of Epstein’s zeta function, they are so called ζ-extreme lattices. The question to ﬁnd ζ-extreme lattices has a long history going back to Sobolev’s work on numerical integration and to work of Delon´e. Universally perfect lattices are dual strongly perfect. The relation with modular forms arises, because the condition that the minimal vectors of the lattice form a 4-design means the annihilation of certain coeﬃcients in its theta series with harmonic coeﬃcients. In this way one can prove the strong perfectness of many extremal lattices of small level. For example there are more than 106 even unimodular lattices without roots in dimension 32 (by work of Oliver King) and the theory of modular forms shows that all of them are universally perfect; this is the only known method to prove that all these lattices are locally densest lattices. If a lattice Λ has a big automorphism group G := Aut(Λ) which has no invariant harmonic polynomials of degree 2 and 4, a condition easily expressed in terms of the character of G ≤ O(n), then Λ is universally perfect. There are many interesting lattices such as the Barnes-Wall lattices, the 248-dimensional Thompson-Smith lattice and others which are strongly perfect by this reason. Tiep and others used representation theory to classify certain matrix groups G for which all orbits form spherical 4-designs. On the other hand lattices are an important tool to ﬁnd and classify good spherical designs. Fixing the strength t and the dimension n, one tries to ﬁnd spherical t-designs X ⊂ S n−1 of minimal possible cardinality. If t = 2m is even, then n−1+m n−2+m |X| ≥ + m m−1 and if t = 2m + 1 is odd then n−1+m |X| ≥ 2 . m A t-design X for which equality holds is called a tight t-design. Tight t-designs in Rn with n ≥ 3 are very rare. Bannai has shown that such tight designs only exist if t ≤ 5 and t = 7, 11. The tight t-designs with t = 1, 2, 3 as well as t = 11 are completely classiﬁed whereas their classiﬁcation for t = 4, 5, 7 is still an open problem. It is conjectured that there are just seven tight t-designs of dimension n ≥ 3 and strength 4, 5, 7, namely in dimensions 6,22 (t=4), 3,7,23 (t=5) respectively 8,23 (t=7); each of these is known to be unique. One possible approach to prove that there are no further tight designs X is to investigate the Euclidean lattice Λ generated by X and to obtain properties of Λ (such as its determinant or its minimum) from the design properties of X and then prove the non existence of such a lattice Λ. This strategy has been successfully applied by Bannai, Munemasa and Venkov to show that there are no further tight designs up to dimension 103.

Fixed Points for Discrete Logarithms Mariana Levin1 , Carl Pomerance2, and K. Soundararajan3 1

Graduate Group in Science and Mathematics Education University of California Berkeley, CA 94720, USA [email protected] 2 Department of Mathematics Dartmouth College Hanover, NH 03755, USA [email protected] 3 Department of Mathematics Stanford University Stanford, CA 94305, USA [email protected]

Abstract. We establish a conjecture of Brizolis that for every prime p > 3 there is a primitive root g and an integer x in the interval [1, p − 1] with logg x = x. Here, log g is the discrete logarithm function to the base g for the cyclic group (Z/pZ)× . Tools include a numerically explicit “smoothed” version of the P´ olya–Vinogradov inequality for the sum of values of a Dirichlet character on an interval, a simple lower bound sieve, and an exhaustive search over small cases.

1

Introduction

If g is an element in a group G and t ∈ g, there is some integer n with g n = t. Finding a valid choice for n is known as the discrete logarithm problem. Note that if g has ﬁnite order m, then n is actually a residue class modulo m. We write logg t = n (or logg t ≡ n (mod m)) in analogy to usual logarithmic notation. Thus, the problem in the title of this paper does not seem to make good sense, since if logg x = x, then the ﬁrst x is a member of the group g and the second x is either an integer or a residue class modulo m. However, sense is made of the equation through the traditional conﬂation of members of the ring Z/kZ with least nonnegative members of residue classes.

The work for this paper was begun at Bell Laboratories in 2001 while the first author was a summer student working with the second author. A version of this work was presented as the 2003 Master’s Thesis of the first author at U. C. Berkeley, see [3]. The second author was supported in part by NSF grant DMS-0703850. The third author was supported in part by NSF grant DMS-0500711.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 6–15, 2010. c Springer-Verlag Berlin Heidelberg 2010

Fixed Points for Discrete Logarithms

7

In particular, suppose G = (Z/pZ)× , where p is a prime number. This is known to be a cyclic group of order p − 1. Suppose g is a cyclic generator of this group, known as a primitive root for p. A ﬁxed point for the discrete logarithm modulo p to the base g is then an integer x in the interval [1, p − 1] such that logg x = x, that is, g x ≡ x (mod p). (Note that if x is not restricted to the interval [1, p − 1] it is easy to ﬁnd ﬁxed points. Namely, if x is a solution to the Chinese remainder problem x ≡ 1 (mod p − 1), x ≡ g (mod p), then g x ≡ x (mod p).) Brizolis (see Guy [6, Section F9]) made the conjecture that for every prime p > 3 there is a primitive root g and an integer x in [1, p − 1] with logg x = x, that is, g x ≡ x (mod p). In this paper we prove this conjecture in a somewhat stronger form. Brizolis had noticed that if there is a primitive root x for p with x in [1, p − 1] and gcd(x, p − 1) = 1, then with y the multiplicative inverse of x modulo p − 1 and g = xy , we would have that g is a primitive root for p as well, and g x ≡ xxy ≡ x (mod p), that is, there is a solution to the ﬁxed point problem. We shall prove then the stronger result that for each prime p > 3 there is a primitive root x for p in [1, p − 1] that is coprime to p − 1. Several authors have shown that the Brizolis property holds for all suﬃciently large primes p. In particular, Zhang [12] showed the strong conjecture holds for all suﬃciently large primes p, but did not give an estimate of what “suﬃciently large” is. Cobeli and Zaharescu [4] also showed that the strong conjecture holds for suﬃciently large primes p, and gave the details that it holds for all p > 102070 , but they indicated that their method would support a bound around 1050 . Our method is similar to that of Zhang, who used the P´ olya–Vinogradov inequality for character sums on an interval. Here we introduce a numerically explicit “smoothed” version of this inequality, see §2. In addition, we combine the traditional character-sum approach with a simple lower bound sieve. There is still some need for direct calculation for smaller values of p, which are easily handled by a short Mathematica program. In particular, we directly veriﬁed the strong conjecture for each prime p < 1.25 · 109 . We mention the article by Holden and Moree [8], which considers some related problems. The total number of solutions to g x ≡ x (mod p) as p runs up to some high bound N , where either g is restricted to be a primitive root, and where it is not so restricted, is considered in Bourgain, Konyagin, and Shparlinski [2]. The smoothed version of the P´ olya–Vinogradov inequality that we introduce in the next section is quite simple and the proof is routine, so it may be known to others. We have found it to be quite useful numerically; we hope it will ﬁnd applications in “closing the gap” in other problems where character sums arise. Some notation: ω(n) denotes the number of distinct prime divisors of n.

8

2

M. Levin, C. Pomerance, and K. Soundararajan

A “Smoothed” P´ olya–Vinogradov Inequality

Let χ be a non-principal Dirichlet character to the modulus q. The P´olya– Vinogradov inequality (independently discovered by P´ olya and Vinogradov in 1918) asserts that there is a universal constant c such that ≤ c√q log q χ(a) (1) M≤a≤M+N for any choice of numbers M, N . Let N (p) denote the numer of primitive roots g for p with g ∈ [1, p − 1] and gcd(g, p − 1) = 1. Using (1) one can show (see Zhang [12] and Campbell [3]) that N (p) =

ϕ(p − 1)2 + O(p1/2+ ), p−1

for every ﬁxed > 0, and so N (p) > 0 for all suﬃciently large p. The aim of this paper is to close the gap and ﬁnd the complete set of primes p with N (p) > 0. Towards this end it would be useful to have a numerically explicit version of (1). In [3], the theorem of Bachman and Rachakonda [1] was used (plus a small unpublished improvement on a secondary term in their inequality due to the second author of the present paper). Recently, elaborating on the work in an early paper of Landau [10], plus an idea of Bateman as mentioned in Hildebrand [7], the second author in [11] proved a stronger numerically explicit version of (1). Using this simpliﬁes the approach in [3]. However, we have found a way to simplify even further by using a “smoothed” version of (1). In this section we prove the following theorem. Theorem 1. Let χ be a primitive Dirichlet character to the modulus q > 1 and let M, N be real numbers with 0 < N ≤ q. Then √ a − M N − 1 ≤ q − √ . χ(a) 1 − N q M≤a≤M+2N Proof. We use Poisson summation, see [9, §4.3]. Let H(t) = max{0, 1 − |t|}. We wish to estimate |S|, where S :=

χ(a)H

a∈Z

a−M −1 . N

Towards this end we use the identity q−1

1 χ(a) = χ(j)e(aj/q), ¯ τ (χ) ¯ j=0

Fixed Points for Discrete Logarithms

9

where τ (χ) ¯ is the Gauss sum for χ ¯ and e(x) := e2πix . Thus, q−1

S=

1 χ(j) ¯ e(aj/q)H τ (χ) ¯ j=0 a∈Z

a−M −1 . N

The Fourier transform of H is ∞ 1 − cos 2πs ˆ ˆ H(s) = H(t)e(−st) dt = when s = 0, H(0) = 1, 2π 2 s2 −∞ which is nonnegative for s real. By a change of variables in the integral, we see that the Fourier transform of e(jt/q)H((t − M )/N − 1) is ˆ (s − j/q)N . N e − (M + N )(s − j/q) H Hence, by Poisson summation, we have q−1 N ˆ (n − j/q)N . χ(j) ¯ e − (M + N )(n − j/q) H S= τ (χ) ¯ j=0 n∈Z

Estimating trivially (that is, taking the absolute value of each term) and using ˆ nonnegative and χ(0) = 0, we have H q−1 N ˆ kN N ˆ . H (n − j/q)N = √ H |S| ≤ √ q j=1 q q n∈Z

k∈Z\qZ

ˆ Since (N/q)H(sN/q) is the Fourier transform of H(qt/N ), from the last calculation we have

N kN N ˆ √ N ˆ kN √ ˆ |S| ≤ q ≤ q − H(0) + H H q q q q q k∈Z k∈Z\qZ

N √ √ N ql √ N = − √ + qH(0) = q − √ , = q − + H q N q q l∈Z

by another appeal to Poisson summation and the deﬁnition of H. This completes the proof of the theorem. In our application we will need a version of Theorem 1 with the variable a satisfying a coprimality condition. We deduce such a result below. Corollary 2. Let k be a square-free integer and let χ be a primitive character to the modulus q > 1. For 0 < N ≤ q, we have √ a 2ω(k) q always ≤ χ(a) 1 − − 1 ω(k)−1 √ N 2 q if k is even. 0≤a≤2N (a,k)=1

10

M. Levin, C. Pomerance, and K. Soundararajan

Proof. Since d|(k,a) μ(d) gives 1 if (a, k) = 1 and 0 otherwise, the sum in question equals ad − 1 μ(d)χ(d) χ(a) 1 − N d|k

a≤2N/d

√ and using Theorem 1 this is bounded in size by 2ω(k) q as desired. If (k, q) is even, then χ(d) = 0 for even divisors d of k, so that we achieve the bound √ 2ω(k)−1 q, again as desired. Suppose now that k is even and q is odd. For each odd divisor d of k, we group together the contribution from d and 2d, and so we may write the sum in question as ad − 1 . μ(d)χ(d) χ(a) 1 − N d|k/2

a≤2N/d a odd

We replace a in the inner sum by q + a, and since q is now odd, the condition that a is odd may be replaced with the condition that q + a = 2b is even. Thus, the above sum becomes 2d(b − q/2) μ(d)χ(d)χ(2) χ(b) 1 − − 1 , N d|k/2

q/2≤b≤q/2+N/d

and appealing again to Theorem 1 we obtain the Corollary in this case. Though we will not need it for our proof, we record the following corollary of Theorem 1. Corollary 3. Let χ be a primitive Dirichlet character to the modulus q > 1 and let M, N be real numbers with N > 0. Then, with θ the fractional part of N/q, q 3/2 a − M − 1 ≤ θ(1 − θ). χ(a) 1 − N N M≤a≤M+2N

3

A Criterion for the Brizolis Property

Let us write the largest square-free divisor of p − 1 as uv where u and v will be chosen later. We shall assume that u is even, and have in mind the situation that u is composed of the small prime factors of p − 1, and that v is composed of the large prime factors; we also allow for the possibility that v = 1. For the rest of the paper, the letter will denote a prime number. Let S denote the set of primitive roots in [1, p − 1] that are coprime to p − 1. Thus, an integer g ∈ [1, p − 1] is in S if and only if for each prime | p − 1 we have both g and g is not an -th power (mod p). Let S1 denote the set of integers in [1, p − 1] that are coprime to u and which are not equal to an -th power (mod p) for any prime dividing u. Let S2 denote the set of integers in S1 which are divisible by some prime which divides v. Let S3 denote the set

Fixed Points for Discrete Logarithms

11

of integers in S1 which equal an -th power (mod p) for some prime dividing v. Now S ⊂ S1 , and the elements in S1 that are not in S are precisely those that, for some prime | v, are either divisible by or are an -th power (mod p). Thus, S = S1 \(S2 ∪ S3 ). We seek a positive lower bound for N :=

g∈S

2g − 1 , 1− p−1

since if N > 0, then S = ∅. By our observation above we have N ≥ N1 − N2 − N3 , where, for j = 1, 2, 3, Nj =

g∈Sj

2g − 1 . 1− p−1

If d is a square-free divisor of p − 1 and g is an integer in [1, p − 1], let Cd (g) be 1 if g is a d-th power (mod p) and 0 otherwise. Thus, Cd (g) =

C (g) =

|d

⎛ 1 ⎝ = 1+ d |d

1 |d

χ of order

Note that

χ(g)

χ =χ0

⎞

1 χ(g)⎠ = d

χ(g).

m|d χ of order m

μ(d)Cd (g)

d|u

is 1 if, for each | u, g is not an -th power (mod p), and is 0 otherwise. By the above calculation, this expression is μ(d) d|u

d

χ(g) =

m|d χ of order m

χ(g)

m|u χ of order m

μ(nm) . nm

n|u/m

The inner sum here is (ϕ(u)/u)μ(m)/ϕ(m), so that N1 =

ϕ(u) u

1≤g≤p−1 (g,u)=1

μ(m) 2g − 1 1− p−1 ϕ(m) m|u

χ(g).

(2)

χ of order m

Let m | u with m > 1. Using Corollary 2, the terms above contribute an amount bounded in magnitude by ϕ(u) ω(u)−1 √ 2 p, u

12

M. Levin, C. Pomerance, and K. Soundararajan

so the total contribution over all m | u with m > 1 has magnitude at most ϕ(u) ω(u) √ 2 − 1 2ω(u)−1 p. u The sum over g in (2) with m = 1 (and so χ = χ0 ) is ϕ(u) u

1≤g≤p−1 (g,u)=1

ϕ(u) 2g − 1 = 1− μ(d) p−1 u d|u

h≤(p−1)/d

2dh − 1 . 1− p−1

The inner sum over h can be evaluated explicitly: it equals (p−1)/(2d) if (p−1)/d is even, and it equals (p − 1)/(2d) − d/(2(p − 1)) if (p − 1)/d is odd. It follows that the contribution when m = 1 is 2 ϕ(u) p − 1 ϕ(u) 1 − dμ(d) u 2 u 2(p − 1) ≥

ϕ(u) u

We conclude that 2 ϕ(u) N1 ≥ u 2 ϕ(u) > u

2

d|u (p−1)/d odd

p−1 ϕ(u)2 − ≥ 2 u(p − 1)

ϕ(u) u

2

p ϕ(u) − . 2 u

p ϕ(u) ϕ(u) ω(u) √ − − 2 − 1 2ω(u)−1 p 2 u u p ϕ(u) ω(u) √ − 4 p. 2 2u

Next we turn to N2 . Since an element in S2 must be divisible by some prime |v we have that ϕ(u) μ(m) 2h − 1 1− N2 ≤ χ(h ). p−1 u ϕ(m) |v h≤(p−1)/ (h,u)=1

χ of order m

m|u

If v = 1, then N2 = 0, so assume v > 1. The terms with m > 1 contribute, using Corollary 2, an amount bounded in size by ϕ(u) √ ω(v) 2ω(u) − 1 2ω(u)−1 p. u The main term m = 1 above contributes (arguing as in our evaluation of the main term for N1 above) ϕ(u) u

|v h≤(p−1)/ (h,u)=1

2h ϕ(u) 2 p − 1 . 1− − 1 ≤ + p−1 u 2 v |v

Fixed Points for Discrete Logarithms

Since

|v

N2 ≤ ≤

13

≤ v, and using v > 1, we conclude that

ϕ(u) u ϕ(u) u

2 2

p−1 1 + 2

|v

p1 2

|v

+

ϕ(u) u

2 +

ϕ(u) √ ω(v) 2ω(u) − 1 2ω(u)−1 p u

ϕ(u) ω(u) √ 4 ω(v) p. 2u

Lastly we consider N3 . An element g of S3 must be an -th power for some prime |v, and the indicator function for this condition is 1 ψ =χ0 ψ(g), as seen above. Therefore we have that N3 is at most |v g≤p−1 (g,u)=1

ϕ(u) μ(m) 2g − 1 1− p−1 u ϕ(m) m|u

χ of order m

1 χ(g) ψ(g) . ψ =χ0

Appealing to Corollary 2 for the terms above with χψ = χ0 we ﬁnd that the contribution of such terms is bounded in magnitude by ϕ(u) 2ω(u)−1 √ 2 ω(v) p. u The main term χ = ψ = χ0 gives ϕ(u) ϕ(u) p − 1 2g ϕ(u) 1 ϕ(u) 1 − 1 ≤ + 1− u p−1 u u 2 p−1 |v

g≤p−1 (g,u)=1

=

|v

ϕ(u) u

Thus,

2

N3 ≤

p−1 1 + 2 v

ϕ(u) u

2

|v

1 ≤

ϕ(u) u

2

p1 . 2 |v

p 1 ϕ(u) ω(u) √ + 4 ω(v) p. 2 2u |v

Combining these bounds for N1 , N2 and N3 we obtain that N≥

ϕ(u) u

2 1 ϕ(u) p √ 1−2 − 4ω(u) (1 + 2ω(v)) p. 2 2u |v

We may conclude as follows: The Brizolis property holds for the prime p ≥ 5, if we may write the largest square-free divisor of p − 1 as uv with u even, |v 1/ < 1/2, and with √ 1 + 2ω(v) 4ω(u) u · . p> ϕ(u) 1 − 2 |v 1/

(3)

14

4

M. Levin, C. Pomerance, and K. Soundararajan

Completing the Proof

Our criterion (3) can be used in a straightforward way with v = 1 to get an upper bound for possible counterexamples to the Brizolis conjecture. Indeed, after a small calculation (using 4ω(n) < 1404n1/3 and n/ϕ(n) < 2 log log n for n larger than the product of the ﬁrst eleven primes), it is seen that the Brizolis property holds for all p > 1025 . It is not pleasant to contemplate checking each prime to this point, so instead we use (3) with v > 1. Suppose ω(p − 1) = k ≥ 10, and take v to be the product of the six largest primes dividing p − 1, and u to be the product of the other smaller primes. Since ω(p − 1) ≥ 10, the primes dividing v are all at least 11, and we have that 1−2

1 |v

≥1−2

1 1 1 1 1 1 + + + + + > 0.28. 11 13 17 19 23 29

If pj denotes the j-th prime, then 4ω(u) u/ϕ(u) ≤ k−6 j=1 (4pj /(pj − 1)), and p > k p − 1 ≥ j=1 pj . So from our criterion (3), if we have k k−6 13 4pj √ , pj ≥ 0.28 j=1 pj − 1 j=1

then the Brizolis property holds for all p with ω(p − 1) = k. We veriﬁed that the inequality above holds for k = 10. If k is increased by 1 then the LHS of our √ inequality is increased by a factor of at least 31 > 5, but the RHS is increased only by a factor of at most 4 × (11/10) = 4.4. Thus, the inequality holds for all k ≥ 10. Suppose now that k = ω(p − 1) ≤ 9. If k ≥ 4, we take u to be the product of the four smallest primes dividing p − 1, and otherwise, we take u to be the product ofall the primes dividing p − 1. Then v has at most 5 prime factors, and 1 − 2 |v 1/ ≥ 1 − 2(1/11 + 1/13 + 1/17 + 1/19 + 1/23) ≥ 0.35. Further 4 p|u 4p/(p − 1) ≤ j=1 4pj /(pj − 1) = 1120. Our criterion (3) shows that if 11 2 p ≥ 1120 × = 1,239,040,000, 0.35 then p satisﬁes the Brizolis property. Using the functions Prime[ ] and PrimitiveRoot[ ] in Mathematica, we were able to directly exhibit a primitive root g for each prime 3 < p < 1.25 · 109 with g in [1, p − 1] and coprime to p − 1. Our program runs as follows. The function Prime[ ] allows us to sequentially step through the primes up to our bound. For each prime p returned by Prime[ ], we invoke PrimitiveRoot[p] to ﬁnd the least positive primitive root r for p. We then sequentially check r2k−1 mod p for k = 1, 2, . . . until we ﬁnd a value coprime to p − 1 with 2k − 1 also coprime to p − 1. The exponent being coprime to p − 1 guarantees that the power is a primitive root, and the residue being coprime to p − 1 then guarantees that we

Fixed Points for Discrete Logarithms

15

have found a member of S. If no such primitive root exists, this algorithm would not terminate, but it did, thus verifying the Brizolis property for the given range. There are various small speed-ups that one can use to augment the program. For example, if r = 2 is a primitive root and p ≡ 1 (mod 4), then note that p − 2 is a primitive root coprime to p − 1, and so work with this prime p is complete. The augmented program ran in about 90 minutes on a Dell workstation. This completes our proof of the Brizolis conjecture. Acknowledgment. We thank Richard Crandall for some technical assistance with the Mathematica program and the referees for some helpful comments.

References 1. Bachman, G., Rachakonda, L.: On a problem of Dobrowolski and Williams and the P´ olya–Vinogradov inequality. Ramanujan J. 5, 65–71 (2001) 2. Bourgain, J., Konyagin, S.V., Shparlinski, I.E.: Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm. Int. Math. Res. Notices, art. ID rnn090, 29 (2008) (Corrigendum: ibid. 2009, No. 16, 3146–3147) 3. Campbell, M.E.: On fixed points for discrete logarithms, Master’s Thesis, U. C. Berkeley Department of Mathematics (2003) 4. Cobeli, C., Zaharescu, A.: An exponential congruence with solutions in primitive roots. Rev. Romaine Math. Pures Appl. 44, 15–22 (1999) 5. Crandall, R., Pomerance, C.: Prime numbers: a computational perspective, 2nd edn. Springer, New York (2005) 6. Guy, R.K.: Unsolved problems in number theory. Springer, Berlin (1984) 7. Hildebrand, A.: On the constant in the P´ olya–Vinogradov inequality. Canad. Math. Bull. 31, 347–352 (1988) 8. Holden, J., Moree, P.: Some heuristics and results for small cycles of the discrete logarithm. Math. Comp. 75, 419–449 (2006) 9. Iwaniec, H., Kowalski, E.: Analytic number theory. American Math. Soc., Providence (2004) 10. Landau, E.: Absch¨ atzungen von Charaktersummen, Einheiten und Klassenzahlen. Nachrichten K¨ onigl. Ges. Wiss. G¨ ottingen, 79–97 (1918) 11. Pomerance, C.: Remarks on the P´ olya–Vinogradov inequality (submitted for publication 2010) 12. Zhang, W.-P.: On a problem of Brizolis. Pure Appl. Math. 11(Suppl.), 1–3 (1995) (Chinese. English, Chinese summary)

Explicit Coleman Integration for Hyperelliptic Curves Jennifer S. Balakrishnan1, Robert W. Bradshaw2, and Kiran S. Kedlaya1 1

Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected], [email protected] 2 University of Washington, Seattle, WA 98195, USA [email protected]

Abstract. Coleman’s theory of p-adic integration ﬁgures prominently in several number-theoretic applications, such as ﬁnding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.

1

Introduction

One of the fundamental diﬃculties of p-adic analysis is that the totally disconnected topology of p-adic spaces makes it hard to introduce a meaningful form of antidiﬀerentiation. It was originally discovered by Coleman that this problem can be circumvented using the principle of Frobenius equivariance. Using this idea, Coleman introduced a p-adic integration theory ﬁrst on the projective line [9], then (partly jointly with de Shalit) on curves and abelian varieties [10], [8]. Alternative treatments have been given by Besser [3] using methods of p-adic cohomology, and by Berkovich [2] using the nonarchimedean Gel’fand transform. Although Coleman’s construction is in principle quite suitable for machine computation, this had only been implemented previously in the genus 0 case [5]. The purpose of this paper is to present an algorithm for computing single Coleman integrals on hyperelliptic curves of good reduction over Cp for p > 2, based on the third author’s algorithm for computing the Frobenius action on the de Rham cohomology of such curves [17]. We also describe an implementation of this algorithm in the Sage computer algebra system. For context, we indicate some of the many potential applications of explicit Coleman integration. Some of these will be treated, with additional numerical examples, in the ﬁrst author’s upcoming PhD thesis. (Some of these applications will require additional reﬁnements of our implementation; see Section 5.) – Torsion points on curves. Coleman’s original application of p-adic integration was to ﬁnd torsion points on curves of genus greater than 1. This could potentially be made eﬀective and automatic. – p-adic heights on curves. Investigations into p-adic analogues of the conjecture of Birch and Swinnerton-Dyer for Jacobians of hyperelliptic curves G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 16–31, 2010. c Springer-Verlag Berlin Heidelberg 2010

Explicit Coleman Integration for Hyperelliptic Curves

17

require computation of the Coleman-Gross height pairing [11]. This global p-adic height pairing can, in turn, be decomposed into a sum of local height pairings at each prime. In particular, for C a hyperelliptic curve over Qp with p a prime of good reduction and for D1 , D2 ∈ Div0 (C) with disjoint support, the Coleman-Gross p-adic height pairing at p is given in terms of the Coleman integral [10] hp (D1 , D2 ) = ωD1 , D2

–

–

–

–

for an appropriately constructed diﬀerential ωD1 associated to the divisor D1 . This pairing is eﬀectively computable by work of the ﬁrst author [1]. Using this work, it should be possible (using ideas of Besser [4]) to add in local heights away from p, and thus compute the Coleman-Gross height pairing on Jacobians of hyperelliptic curves. (In genus 1, one can then compare to an alternate computation based on work of Mazur-Stein-Tate [22] and Harvey [16].) p-adic regulators. A related topic to the previous one is the computation of p-adic regulators in higher K-theory of arithmetic schemes, which are expected to relate to special values of L-functions. Some computations in genus 0 have been made by Besser and de Jeu [5]. Rational points on curves: Chabauty’s method. For C a smooth proper curve over Z[ N1 ], the Chabauty condition on C is that rank J(C) Z N1 < dim J(C), where J(C) denotes the Jacobian of the curve. When the Chabauty condiP tion holds, there exists a 1-form ω on J(C)an with 0 ω = 0 for all points 1 P ∈ J(C) Z N . We might be able to compute C(Z[ N1 ]) if we can ﬁnd all P points P ∈ C an such that 0 ω = 0. This method has already been used in many cases, by Coleman and many others; see [23] for a survey (circa 2007). To apply Chabauty’s method in a typical case, one needs the integral of ω at some point in a residue disc, with which one can ﬁnd all zeroes of the integral in the residue disc. Several methods are suggested in [23, Remark 8.3] for doing this, including Coleman integration. However, no serious attempt has been made to use numerical Coleman integration in Chabauty’s method; it seems likely that it can handle cases where the other methods suggested in [23, Remark 8.3] for ﬁnding constants of integration prove to be impractical. Rational points on curves: nonabelian Chabauty. It may be possible to use (iterated) Coleman integration to ﬁnd rational points on curves failing the Chabauty condition, using Kim’s nonabelian Chabauty method [18]. As a demonstration of the method, Kim [19] gives an explicit double integral which vanishes on the integral points of the minimal regular model of a genus 1 curve over Q of Mordell-Weil rank 1. The erratum to [19] includes a corrected formula, together with some numerical examples computed using the methods of this paper. p-adic polylogarithms and multiple zeta values. These have been introduced recently by Furusho [13], but little numerical data exists so far.

18

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

2

Coleman’s Theory of p-adic Integration

In this section, we recall Coleman’s p-adic integration theory (for single integrals only) in the case of curves with good reduction. This theory involves some concepts from rigid analytic geometry which it would be hopeless to introduce in such limited space; some standard references are [6] and [12]. (See also [10, §1].) Let Cp be a completed algebraic closure of Qp , and let O be the valuation subring of Cp . Choose once and for all a branch of the p-adic logarithm, i.e., a homomorphism Log : C× to the disc {x ∈ Cp : |x − 1| < p → Cp whose restriction ∞ 1} is given by the logarithm series log(x) = i=1 (1−x)i /i. (The choice of branch has no eﬀect on the integrals on diﬀerentials of the second kind, i.e., everywhere meromorphic diﬀerentials with all residues zero.) We ﬁrst introduce integrals on discs and annuli within P1 . Definition 1. Let I be an open subinterval A(I) denote the of [0, +∞). Let 1 annulus (or disc) {t ∈ A1Cp : |t| ∈ I}. For i∈Z ci ti dt ∈ ΩA(I)/C and P, Q ∈ p A(I), deﬁne Q ci (Qi+1 − P i+1 ). ci ti dt = c−1 Log(Q/P ) + i+1 P i∈Z

i=−1

This is easily shown not to depend on the choice of the coordinate t. Remark 2. Note that because of the division by i + 1 in the formula for the integral, we are unable to integrate on closed discs or annuli. We next turn to curves of good reduction. Definition 3. By a curve over O, we will mean a smooth proper connected scheme X over O of relative dimension 1. Equip the function ﬁeld K(X) with the p-adic absolute value, so that the elements of K(X) of norm at most 1 constitute the local ring in X of the generic point of the special ﬁbre X of X. Let XQ denote the generic ﬁbre of X as a rigid analytic space. There is a natural specialization map from XQ to X; the inverse image of any point of X is a subspace of XQ isomorphic to an open unit disc. We call such a disc a residue disc of X. Definition 4. Let X be a curve over O. By a wide open subspace of XQ , we will mean a rigid analytic subspace of XQ of the form {x ∈ XQ : |f (x)| < λ} for some f ∈ K(X) of absolute value 1 and some λ > 1. Coleman made the surprising discovery that there is a well-behaved integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. (Note that one needs to consider wide open subspaces even to integrate diﬀerentials which are holomorphic or meromorphic on the entire curve.) In the case of hyperelliptic curves, Coleman’s construction of these integrals using Frobenius lifts will be reﬂected in our technique for computing the integrals. For the general case, see [10, §2], [3, §4], or [2, Theorem 1.6.1].

Explicit Coleman Integration for Hyperelliptic Curves

19

Theorem 5 (Coleman). We may assign to each curve X over O and each 1 wide open subspace W of XQ a map μW : Div0 (W ) × ΩW/C → Cp , subject to p the following conditions. (Here Div(W ) denotes the free group on the elements of W , and Div0 (W ) denotes the kernel of the degree map deg : Div(W ) → Z taking each element of W to 1.) 1 (a) (Linearity) The map μW is linear on Div0 (W ) and Cp -linear on ΩW/C . p (b) (Compatibility) For any residue disc D of X and any isomorphism ψ : W ∩ 1 D → A(I) for some interval I, the restriction of μW to Div0 (W ∩D)×ΩW/C p is compatible with Deﬁnition 1 via ψ. (c) (Change of variables) Let X be another curve over O, let W be a wide open subspace of X , and let ψ : W → W be any morphism of rigid spaces relative to an automorphism of Cp . Then

(1) μW (ψ(·), ·) = μW (·, ψ ∗ (·)). (d) (Fundamental theorem of calculus) For any Q = i ci (Pi ) ∈ Div0 (W ) and any f ∈ O(W ), μW (Q, df ) = i ci f (Pi ). Remark 6. One cannot expect path independence in the case of bad reduction. For instance, an elliptic curve over Cp with bad reduction admits a Tate uniformization, so its logarithm map has nonzero periods in general. In Berkovich’s theory of integration, this occurs because the nonarchimedean analytic space associated to this curve X has nontrivial ﬁrst homology.

3

Explicit Integrals for Hyperelliptic Curves

We now specialize to the situation where p > 2 and X is a genus g hyperelliptic curve over an unramiﬁed extension K of Qp having good reduction. We will assume in addition that we have been given a model of X of the form y 2 = f (x) such that deg f (x) = 2g + 1 and f has no repeated roots modulo p. (This restriction is inherited from [17], where it is used to simplify the reduction procedure. One could reduce to this case after possibly replacing K by a larger unramiﬁed extension of Qp , by performing a linear fractional transformation in x to put one root at inﬁnity, thus reducing the degree from 2g + 2 to 2g + 1.) We will distinguish between Weierstrass and non-Weierstrass residue discs of X, which respectively correspond to Weierstrass and non-Weierstrass points of X. To discuss the diﬀerentials we will be integrating, we review a core deﬁnition from [17]. Let X be the aﬃne curve obtained by deleting the Weierstrass points from X, and let A = K[x, y, z]/(y 2 − f (x), yz − 1) be the coordinate ring of X . Definition 7. The Monsky-Washnitzer (MW) weak completion of A is the ring A† consisting of inﬁnite sums of the form

∞ Bi (x) , Bi (x) ∈ K[x], deg Bi ≤ 2g , yi i=−∞

20

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

further subject to the condition that vp (Bi (x)) grows faster than a linear function of i as i → ±∞. We make a ring out of these using the relation y 2 = f (x). These functions are holomorphic on wide opens, so we will integrate 1-forms ω = g(x, y)

dx , 2y

g(x, y) ∈ A† .

(2)

Note that we only consider 1-forms which are odd, i.e., which are negated by the hyperelliptic involution. Even 1-forms can be written in terms of x alone, and so can be integrated directly as in Deﬁnition 1. (This last statement would fail if we had taken A† to be the full p-adic completion of A, rather than the weak completion. This observation is the basis for Monsky-Washnitzer’s formal cohomology, which is used in [17].) Note that the class of allowed forms includes those meromorphic diﬀerentials on X whose poles all belong to Weierstrass residue discs. For some applications (e.g., p-adic canonical heights), it is necessary to integrate meromorphic diﬀerentials with poles in non-Weierstrass residue discs. These will be discussed in [1]. Note also that for ease of exposition, we describe all of our algorithms as if it were possible to compute exactly in A† . This is not possible for two reasons: the elements of A† correspond to inﬁnite series, and the coeﬃcients of these series are polynomials with p-adic coeﬃcients. In practice, each computation will be made with suitable p-adic approximations of the truly desired quantities, so one must keep track of how much p-adic precision is needed in these estimates in order for the answers to bear a certain level of p-adic accuracy. We postpone this discussion to § 4.1. 3.1

A Basis for de Rham Cohomology

We ﬁrst note that any odd diﬀerential ω as in (2) can be written uniquely as ω = df + c0 ω0 + · · · + c2g−1 ω2g−1

(3)

with f ∈ A† , ci ∈ K, and ωi =

xi dx 2y

(i = 0, . . . , 2g − 1).

(4)

That is, the ωi form a basis of the odd part of the de Rham cohomology of A† . The process of putting ω in the form (3), using the relations y 2 = f (x), dx , d(xi y j ) = 2ixi−1 y j+1 + jxi f (x)y j−1 2y can be made algorithmic; see [17, §3]. (Brieﬂy, one uses the ﬁrst relation to reduce high powers of x, and the second to reduce large positive and negative powers of y.) Using properties from Theorem 5 (linearity and the fundamental

Explicit Coleman Integration for Hyperelliptic Curves

21

theorem of calculus), the integration of ω reduces eﬀectively to the integration of the ωi . It may be convenient for some purposes to use a diﬀerent basis of de Rham cohomology. For instance, the basis xi dx/2y 3 (i = 0, . . . , 2g − 1) is crystalline (see the erratum to [17]), so Frobenius will act via a matrix with p-adically integral entries. 3.2

Tiny Integrals

Q We refer to any Coleman integral of the form P ω in which P, Q lie in the same residue disc (Weierstrass or not) as a tiny integral. As an easy ﬁrst case, we give an algorithm to compute tiny integrals of basis diﬀerentials. Algorithm 8 (Tiny Coleman integrals). Input: Points P, Q ∈ X(Cp ) in the same residue disc (neither equal to the point at inﬁnity) and a basis diﬀerential ωi . Q Output: The integral P ωi . 1. Construct a linear interpolation from P to Q. For instance, in a nonWeierstrass residue disc, we may take x(t) = (1 − t)x(P ) + tx(Q) y(t) = f (x(t)), where y(t) is expanded as a formal power series in t. 2. Formally integrate the power series in t:

Q

P

ωi =

Q

P

xi

dx = 2y

0

1

x(t)i dx(t) dt. 2y(t) dt

Remark 9. One can similarly integrate any ω holomorphic in the residue disc containing P and Q. If ω is only meromorphic in the disc, but has no pole at P or Q, we can ﬁrst make a polar decomposition, i.e., write ω as a holomorphic diﬀerential on the disc plus some terms of the form c/(t − r)i , and integrate the latter terms directly. (If ω is everywhere meromorphic, this is achieved by a partial fractions decomposition.) 3.3

Non-Weierstrass Discs

Q We next compute integrals of the form P ωi in which P, Q ∈ X(Cp ) lie in distinct non-Weierstrass residue discs. The method of tiny integrals is not available; we instead employ Dwork’s principle of analytic continuation along Frobenius, in the form of Kedlaya’s algorithm [17] for calculating the action of Frobenius Q on de Rham cohomology. Note that we calculate the integrals P ωi for all i simultaneously. (We modify the presentation in [17] by keeping track of exact diﬀerentials, which are irrelevant for computing zeta functions.)

22

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Algorithm 10 (Kedlaya’s algorithm). Input: The basis diﬀerentials {ωi }2g−1 i=0 . Output: Functions fi ∈ A† and a 2g × 2g matrix M over K such that φ∗ (ωi ) = 2g−1 dfi + j=0 Mij ωj for a p-power lift of Frobenius φ. 1. Since K is an unramiﬁed extension of Qp , it carries a unique automorphism φK lifting the Frobenius automorphism x → xp on its residue ﬁeld. Extend φK to a Frobenius lift on A† by setting φ(x) = xp ,

1/2 φK (f )(xp ) − f (x)p φ(y) = y p 1 + f (x)p

∞ 1/2 (φK (f )(xp ) − f (x)p )i = yp , y 2pi i i=0 noting the series converges in A† because φK (f )(xp ) − f (x)p has positive valuation. (This choice of φ(y) ensures that φ(y)2 = φ(f (x)), so that the action on A† is well-deﬁned. 2. Use a Newton iteration to compute y/φ(y). Then for i = 0, . . . , 2g−1, proceed as in § 3.1 to write φ∗ (ωi ) = pxpi+p−1

2g−1 y dx = dfi + Mij ωj φ(y) 2y j=0

(5)

for some fi ∈ A† and some 2g × 2g matrix M over K. We may use Algorithm 10 to compute Coleman integrals between endpoints in non-Weierstrass residue discs, as follows. (Note that our recipe is essentially Coleman’s construction of the integrals in this case.) Algorithm 11 (Coleman integration in non-Weierstrass discs). Input: The basis diﬀerentials {ωi }2g−1 i=0 , points P, Q ∈ X(Cp ) in non-Weierstrass residue discs, and a positive integer m such that the residue ﬁelds of P, Q are contained in Fpm . Q Output: The integrals { P ωi }2g−1 i=0 . 1. Calculate the action of the m-th power of Frobenius on each basis element (see Remark 12): 2g−1 (φm )∗ ωi = dfi + Mij ωj . (6) j=0

2. By change of variables (see Remark 13), we obtain 2g−1 j=0

(M − I)ij

Q

P

ωj = fi (P ) − fi (Q) −

φm (P )

P

ωi −

Q

φm (Q)

ωi

(7)

Explicit Coleman Integration for Hyperelliptic Curves

23

(the fundamental linear system). As the eigenvalues of the matrix M are algebraic integers of Cp -norm pm/2 = 1 (see [17, §2]), the matrix M − I is Q invertible, and we may solve (7) to obtain the integrals P ωi . Remark 12. To compute the action of φm , ﬁrst perform Algorithm 10 to write φ∗ ωi = dgi +

2g−1

Bij ωj .

j=0

If we view f, g as column vectors and M, B as matrices, we then have f = φm−1 (g) + Bφm−2 (g) + · · · + BφK (B) · · · φm−2 (B)g K M = BφK (B) · · · φm−1 (B). K Remark 13. We obtain (7) as follows. By change of variables, φm (Q) Q ωi = (φm )∗ ωi φm (P )

P

Q

= P

(dfi +

2g−1

Mij ωj )

j=0

= fi (Q) − fi (P ) + Adding

φm (P ) P

Q P

ωi +

ωi =

j=0

Q

φm (Q)

φm (P )

P

2g−1

Mij

Q

P

ωj .

ωi to both sides of this equation yields

ωi +

Q

φm (Q)

ωi + fi (Q) − fi (P ) +

2g−1 j=0

Mij

Q

P

ωj ,

which is equivalent to (7). Definition 14. A Teichm¨ uller point of XQ is a point ﬁxed by some power of φ. Each non-Weierstrass residue disc contains a unique such point: if (x, y) ∈ X is a non-Weierstrass point, the Teichm¨ uller point in its residue disc has xcoordinate equal to the usual Teichm¨ uller lift of x. This leaves two choices for the y-coordinate, exactly one of which has the correct reduction modulo p. Note that Teichm¨ uller points are always deﬁned over ﬁnite unramiﬁed extensions of Qp . Remark 15. A variant of Algorithm 11 is to ﬁrst ﬁnd the Teichm¨ uller points P , Q in the residue discs of P, Q, then note that from the fundamental linear system (7), we have Q 2g−1 (M − I)ij ωj = fi (P ) − fi (Q ). (8) j=0

P

Q Q Q From (8), we obtain the integrals P ωi . Finally, write P ωi − P ωi as the Q P sum P ωi + Q ωi of tiny integrals.

24

3.4

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Weierstrass Endpoints of Integration

Suppose now that P, Q lie in diﬀerent residue discs, at least one of which is Weierstrass. Since a diﬀerential ω of the form (2) is not meromorphic over Weierstrass Q residue discs, we cannot always even deﬁne P ω, let alone compute it. We will thus assume (to cover most cases arising in applications) that ω is everywhere meromorphic, with no pole at either P or Q. We then make the following observation. Lemma 16. Let ω be an odd, everywhere meromorphic diﬀerential on X. Choose P, Q ∈ X(Cp ) which are not poles of ω, with P Weierstrass. Then for ι the hyQ Q perelliptic involution, P ω = 12 ι(Q) ω. In particular, if Q is also a Weierstrass Q point, then P ω = 0. Proof. Let I := points, we have

Q

ι(Q) P (−ω) = ι(Q) ω. Then by additivity P ω = P Q ι(Q) ω = 2I, from which the result follows.

in the end-

If P belongs to a Weierstrass residue disc while Q does not, we ﬁnd the Weierstrass point P in the disc of P , then apply Lemma 16 to write

Q

P

ω=

P

P

1 ω+ 2

Q

ι(Q)

ω.

(9)

The ﬁrst integral on the right side of (9) is tiny, while the second integral involves two points in non-Weierstrass residue discs, and so may be computed as in the previous section. The situation is even better if P, Q both belong to residue discs Q containing respective Weierstrass points P , Q : in this case, by Lemma 16, P ω Q P equals the sum P ω + Q ω of tiny integrals. Remark 17. Beware that Lemma 16 does not generalize to iterated integrals. For instance, for double integrals, if both integrands are odd, the total integrand is even, so the argument of Lemma 16 tells us nothing. It is thus worth considering alternate approaches for dealing with Weierstrass discs, which may generalize better to the iterated case. We concentrate on the case where P lies in a Weierstrass residue disc but Q does not, as we may reduce to this case by splitting Q R Q ω = P ω + R ω for some auxiliary point R in a non-Weierstrass residue P disc. In Algorithm 11, the form fi belongs to A† and so need not converge at P . However, it does converge at any point R near the boundary of the disc, i.e., in the complement of a certain smaller disc which can be bounded explicitly. We Q R Q may thus write P ωi = P ωi + R ωi for suitable R in the disc of P , to obtain an analogue of the fundamental linear system (7). Similarly, when we write

Explicit Coleman Integration for Hyperelliptic Curves

25

ω as in (3), we can ﬁnd R close enough to the boundary of the disc of P so that f Q R converges at R, use (3) to evaluate R ω, then compute P ω as a tiny integral. One defect of this approach is that forcing R to be close to the boundary of the residue disc of P forces R to be deﬁned over a highly ramiﬁed extension of Qp , over which computations are more expensive. An alternate approach exploits the fact that for P in the inﬁnite residue disc Q but distinct from the point at inﬁnity, we may compute P ω directly using Algorithm 11. This works because both the Frobenius lift and the reduction process respect the subring of A† consisting of functions which are meromorphic at inﬁnity. When P lies in a ﬁnite Weierstrass residue disc, we may reduce to the previous case using a change of variables on the x-line to move P to the inﬁnite disc. However, one still must use the approach of the previous paragraph Q Q to reduce evaluation of P ω to evaluation of the P ωi .

4

Implementation Notes and Precision

We have implemented the above algorithms in Sage [24] for curves deﬁned over Qp . In doing so, we made the following observations. 4.1

Precision Estimates

For a tiny integral, the precision of the result depends on the truncation of the power series computed. Here is the analysis for a non-Weierstrass disc; the analysis for a Weierstrass disc, using a diﬀerent local interpolation, is similar. (For points over ramiﬁed extensions, one must also account for the ramiﬁcation index in the bound, but it should be clear from the proof how this is done.) Q Proposition 18. Let P ω be a tiny integral in a non-Weierstrass residue disc, with P, Q deﬁned over an unramiﬁed extension of K and accurate to n digits of precision. Let (x(t), y(t)) be the local interpolation between P and Q deﬁned by x(t) = x(P )(1 − t) + x(Q)t = x(P ) + t(x(Q) − x(P )) y(t) = f (x(t)). Let ω = g(x, y)dx be a diﬀerential of the second kind such that h(t) = g(x(t), y(t)) belongs to O[[t]]. If we truncate h(t) modulo tm , then the computed value of the Q integral P ω will be correct to min{n, m + 1 − logp (m + 1) } digits of (absolute) precision. Proof. Let t = t(x(Q) − x(P )). As P, Q are in the same residue disc and are deﬁned over an unramiﬁed ∞extension of K, we have vp (x(Q) − x(P )) ≥ 1. If we expand g(x(t ), y(t )) = i=0 ci (t )i , then by hypothesis ci ∈ O. Thus

26

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

Q

P

ω=

Q

P 1

g(x, y)dx

g(x(t), y(t))dx(t)

= 0

=

0

=

0

x(Q)−x(P )

g(x(t ), y(t ))dt

∞ x(Q)−x(P )

ci (t )i dt

i=0

∞ ci (x(Q) − x(P ))i+1 . = i + 1 i=0

The eﬀect of omitting ci (t )i from the expansion of g(x(t ), y(t )) for some i ≥ m is to change the ﬁnal sum by a quantity of valuation at least i+1− logp (i+1) ≥ m+1− logp (m+1) . The eﬀect of the ambiguity in P and Q is that the computed value of (x(Q) − x(P ))i+1 diﬀers from the true value by a quantity of valuation at least i + 1 − logp (i + 1) + n − 1 ≥ n. For Coleman integrals between diﬀerent residue discs, which we may assume are non-Weierstrass thanks to § 3.4, one must ﬁrst account for the precision loss in Algorithm 10. According to [17, Lemmas 2,3] and the erratum to [17] (or [15]), working to precision pN in Algorithm 10 produces the fi , Mij accurately modulo pN −n for n = 1 + logp max{N, 2g + 1} . We must then take into account the objects involved in the linear system (7), as follows. Q Proposition 19. Let P ω be a Coleman integral, with ω a diﬀerential of the second kind and with P, Q in non-Weierstrass residue discs, deﬁned over an unramiﬁed extension of Qp , and accurate to n digits of precision. Let Frob be the matrix of the action of Frobenius on the basis diﬀerentials. Set B = Frobt −I, Q and let m = vp (det(B)). Then the computed value of the integral P ω will be accurate to n − max{m, logp n } digits of precision. Proof. By the linear system (7), the Coleman integral is expressed in terms of tiny integrals, integrals of exact forms evaluated at points, and a matrix inversion. Suppose that the entries of B = Frobt −I are computed to precision n. Then taking B −1 , we have to divide by det(B), which lowers the precision by m = vp (det(B)). By Proposition 18, computing tiny integrals (with the series expansions truncated modulo tn−1 ) gives a result precise up to n− logp n digits. Q Thus the value of the integral P ω will be correct to n−max{m, logp n } digits of precision. 4.2

Complexity Analysis

We assume that asymptotically fast integer and polynomial multiplication algorithms are used; speciﬁcally addition, subtraction, multiplication, and divi sion take O(log N ) bit operations in Z/N Z and O(n) basering operations in

Explicit Coleman Integration for Hyperelliptic Curves

27

R[x]/xnR[x]. In particular, this allows arithmetic operations in Qp to n (rela log p). Using tive) digits of precision, hereafter called ﬁeld operations, in time O(n Newton iteration, both square roots and the Teichm¨ uller character can be com puted to n digits of precision using O(log n) arithmetic operations. (We again consider only points in non-Weierstrass discs deﬁned over unramiﬁed ﬁelds.) Q Proposition 20. Let P ω be a Coleman integral on a curve of genus g over Qp , 2g−i with ω = dfω + i=1 ci ωi a diﬀerential of the second kind and with P, Q in nonWeierstrass residue discs, deﬁned over Qp , and accurate to n digits of precision. Let Frob be the matrix of the action of Frobenius on the basis diﬀerentials, and let m = vp (det(Frobt −I)). Let F (n) be the running time of evaluating fω at P Q and Q to n digits of precision. The value of the integral P ω can be computed 2 2 to n − max{m, logp n } digits of precision in time F (n) + O(pn g + g 3 n log p). (Over a degree N unramiﬁed extension of Qp , the analysis is the same with the runtime multiplied by a factor of N .) Proof. An essential input to the algorithm is the matrix of the action of Frobenius, which can be computed by Kedlaya’s algorithm to n digits of precision 2 2 in running time O(pn g ). Inverting the resulting matrix can be (na¨ıvely) done 3 with O(g ) arithmetic operations in Qp . It remains to be shown that no other step exceeds these running times. For the tiny integral on the ﬁrst basis diﬀerential, the power series x(t)/y(t) = x(t)f (x(t))−1/2 can be computed modulo tn−1 log n) ﬁeld operations. Each other basis using Newton iteration, requiring O(n diﬀerential can be computed from the ﬁrst by multiplication by the linear poly nomial x(t) and the deﬁnite integral evaluated with O(n) ﬁeld operations, for a 2 total of O(gn ) bit operations. Computing φ(P ) and φ(Q) to n digits of preci + log p) ﬁeld sion is cheap; directly using the formula in Algorithm 10 uses O(g operations. The last potentially signiﬁcant step is computing and evaluating the fi at each P and/or Q. The coeﬃcients of the fi can be read oﬀ in the reduction phase of Kedlaya’s algorithm, and have O(png) terms each. Evaluating (or even 2 2 2 recording) all g of these forms takes O(png ) ﬁeld operations, or O(pn g ) bit operations, which is proportional to the cost of doing the reduction. 4.3

Numerical Examples

Here are some sample computations made using our Sage implementation. Additional examples will appear in the ﬁrst author’s upcoming PhD thesis. Example 21. Lepr´evost [21] showed that the divisor (1, −1) − ∞+ on the genus 2 curve y 2 = (2x − 1)(2x5 − x4 − 4x2 + 8x − 4) over Q is torsion of order 29. Consequently, the integrals of holomorphic diﬀerentials against this divisor must vanish. We may observe this vanishing numerically, as follows. Let C : y 2 = x5 +

1 33 4 3 3 3 2 1 x + x + x − x+ 16 4 8 4 16

28

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

be the pullback of Lepr´evost’s curve by the linear fractional transformation x → (1 − 2x)/(2x) taking ∞ to 1/2. The original points (1, −1), ∞+ correspond to the points P = (−1, 1), Q = (0, 14 ) on C. The curve C has good reduction at p = 11, and we compute

Q

P

ω0 =

Q

P

ω1 = O(116 ),

Q

P

ω2 = 7·11+6·112 +3·113 +114 +5·115 +O(116 ),

consistent with the fact that Q − P is torsion and ω0 , ω1 are holomorphic but ω2 is not. Example 22. We give an example arising from the Chabauty method, taken from [23, § 8.1]. Let X be the curve y 2 = x(x − 1)(x − 2)(x − 5)(x − 6), whose Jacobian has Mordell-Weil rank 1. The curve X has good reduction at 7, and X(F7 ) = {(0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, 6), (3, −6), ∞}. By [23, Theorem 5.3(2)], we know |X(Q)| ≤ 10. However, we can ﬁnd 10 rational points on X: the six rational Weierstrass points, and the points (3, ±6), (10, ±120). Hence |X(Q)| = 10. Since the Chabauty condition holds, there must exist a holomorphic diﬀerQ ential ω for which ∞ ω = 0 for all Q ∈ X(Q). We can ﬁnd such a diﬀerential by taking Q to be one of the rational non-Weierstrass points, then computing Q Q a := ∞ ω0 , b := ∞ ω1 and setting ω = bω0 − aω1 . For Q = (3, 6), we obtain a = 6 · 7 + 6 · 72 + 3 · 73 + 3 · 74 + 2 · 75 + O(76 ) b = 4 · 7 + 2 · 72 + 6 · 73 + 4 · 75 + O(76 ). We then verify that

R Q

ω vanishes for each of the other rational points R.

Remark 23. It is worth pointing out some facts not exposed by Example 22. For instance, since ω is already determined by a single rational non-Weierstrass point, we could have used it instead of a brute-force search to ﬁnd other rational points. More seriously, in other examples, the integral ω may vanish at a point deﬁned over a number ﬁeld which has a rational multiple in the Jacobian. Such points may be diﬃcult to ﬁnd by brute-force search; it may be ∗ easier to reconstruct them from p-adic approximations, obtained by writing ∞ ω as a function of a linear parameter of a residue disc, then ﬁnding the zeroes of that function.

5

Future Directions

Here are some potential extensions of our computation of Coleman integrals.

Explicit Coleman Integration for Hyperelliptic Curves

5.1

29

Iterated Integrals

Coleman’s theory of integration is not limited to single integrals; it gives rise to an entire class of locally analytic functions, the Coleman functions, on which antidiﬀerentiation is well-deﬁned. In other words, one can deﬁne integrals Q ωn · · · ω1 P

which behave formally like iterated path integrals 1 t1 tn−1 ··· fn (tn ) · · · f1 (t1 ) dtn · · · dt1 . 0

0

0

These appear in several applications of Coleman integration, e.g., p-adic regulators in K-theory, and the nonabelian Chabauty method. As in the case of a single integral, one can use Frobenius equivariance to compute iterated Coleman integrals on hyperelliptic curves. One obtains a linear system expressing all n-fold integrals of basis diﬀerentials in terms of lower order integrals. Note that the number of such n-fold integrals is (2g)n , so this is only feasible for small n. The cases n ≤ 4 are already useful for applications, but ideas for reducing the combinatorial explosion for larger n would also be of interest. (One must be slightly careful in dealing with Weierstrass residue discs; see Remark 17.) We have made some limited experiments with double Coleman integrals in Sage. The Fubini identity Q

P

ω2 ω1 +

Q

P

ω1 ω2 =

Q

P

ω1

Q

P

ω2

turns out to be a useful consistency check for both single and double integrals. 5.2

Beyond Hyperelliptic Curves

It should be possible to convert other algorithms for computing Frobenius actions on de Rham cohomology, for various classes of curves, into algorithms for computing Coleman integrals on such curves. Candidate algorithms include the adaptation of Kedlaya’s algorithm to superelliptic curves by Gaudry and G¨ urel [14], or the general algorithm for nondegenerate curves due to Castryck, Denef, and Vercauteren [7]. It should also be possible to compute Coleman integrals using Frobenius structures on Picard-Fuchs (Gauss-Manin) connections, extending Lauder’s deformation method for computing Frobenius matrices [20]. 5.3

Heights After Harvey

We noted earlier that our algorithms for Coleman integration over Qp have linear runtime dependence on the prime p, arising from the corresponding dependence

30

J.S. Balakrishnan, R.W. Bradshaw, and K.S. Kedlaya

in Kedlaya’s algorithm. In [15], Harvey gives a variant of Kedlaya’s algorithm with only square-root dependence on p (but somewhat worse dependence on other parameters), by reorganizing the computation so that the dominant step is ﬁnding the p-th term of a linear matrix recurrence whose coeﬃcients are polynomials in the sequence index. Harvey demonstrates the practicality of his algorithm for primes greater than 250 , which may have some relevance in cryptography for ﬁnding curves of low genus with nearly prime Jacobian orders. It should be possible to use similar ideas to obtain square-root dependence on p for Coleman integration, by constructing a recurrence that computes not just the entries of the Frobenius matrix but also the values fi (P ) and fi (Q). However, this is presently a purely theoretical question, as we do not know of any applications of Coleman integration for very large p. Acknowledgments. The authors thank William Stein for access to his computer sage.math.washington.edu (funded by NSF grant DMS-0821725), and Robert Coleman and Bjorn Poonen for helpful conversations. Balakrishnan was supported by a National Defense Science and Engineering Graduate Fellowship and an NSF Graduate Research Fellowship. Bradshaw was supported by NSF grant DMS-0713225. Kedlaya was supported by NSF CAREER grant DMS0545904, the MIT NEC Research Support Fund, and the MIT Cecil and Ida Green Career Development Professorship. Some development work was carried out at the 2006 MSRI Summer Graduate Workshop on computational number theory, and the 2007 Arizona Winter School on p-adic geometry.

References 1. Balakrishnan, J.S.: Local heights on hyperelliptic curves (2010) (in preparation) 2. Berkovich, V.G.: Integration of one-forms on p-adic analytic spaces. Annals of Mathematics Studies, vol. 162. Princeton University Press, Princeton (2007) 3. Besser, A.: Coleman integration using the Tannakian formalism. Math. Ann. 322(1), 19–48 (2002) 4. Besser, A.: On the computation of p-adic height pairings on Jacobians of hyperelliptic curves, Sage Days 5 (2007), http://wiki.sagemath.org/days5/sched 5. Besser, A., de Jeu, R.: Li(p) -service? An algorithm for computing p-adic polylogarithms. Math. Comp. 77(262), 1105–1134 (2008) 6. Bosch, S., G¨ untzer, U., Remmert, R.: Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Springer, Berlin (1984) 7. Castryck, W., Denef, J., Vercauteren, F.: Computing zeta functions of nondegenerate curves. IMRP Int. Math. Res. Pap., Art. ID 72017, 57 (2006) 8. Coleman, R., de Shalit, E.: p-adic regulators on curves and special values of p-adic L-functions. Invent. Math. 93(2), 239–266 (1988) 9. Coleman, R.F.: Dilogarithms, regulators and p-adic L-functions. Invent. Math. 69(2), 171–208 (1982) 10. Coleman, R.F.: Torsion points on curves and p-adic abelian integrals. Ann. of Math. (2) 121(1), 111–168 (1985) 11. Coleman, R.F., Gross, B.H.: p-adic heights on curves. In: Algebraic Number Theory – in honor of K. Iwasawa. Advanced Studies in Pure Mathematics, vol. 17, pp. 73– 81 (1989)

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12. Fresnel, J., van der Put, M.: Rigid analytic geometry and its applications. In: Progress in Mathematics, vol. 218. Birkh¨ auser Boston Inc., Boston (2004) 13. Furusho, H.: p-adic multiple zeta values. II. Tannakian interpretations. Amer. J. Math. 129(4), 1105–1144 (2007) 14. Gaudry, P., G¨ urel, N.: An extension of Kedlaya’s point-counting algorithm to superelliptic curves. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 480–494. Springer, Heidelberg (2001) 15. Harvey, D.: Kedlaya’s algorithm in larger characteristic. Int Math Res Notices, Article ID No. rnm095, 2007, 29 (2007) 16. Harvey, D.: Eﬃcient computation of p-adic heights. LMS J. Comput. Math. 11, 40–59 (2008) 17. Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001); erratum ibid 18, 417– 418 (2003) 18. Kim, M.: The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45(1), 89–133 (2009) 19. Kim, M.: Massey products for elliptic curves of rank 1. J. Amer. Math. Soc. 23, 725–747 (2010); Erratum by Balakrishnan, J.S., Kedlaya, K.S., Kim, M., http://www.ucl.ac.uk/~ ucahmki/ 20. Lauder, A.G.B.: Deformation theory and the computation of zeta functions. Proc. London Math. Soc. 88(3), 565–602 (2004) 21. Lepr´evost, F.: Jacobiennes de certaines courbes de genre 2: torsion et simplicit´e. J. Th´eor. Nombres Bordeaux 7(1), 283–306 (1995) 22. Mazur, B., Stein, W., Tate, J.: Computation of p-adic heights and log convergence. Doc. Math. Extra, 577–614 (2006) (electronic) 23. McCallum, W., Poonen, B.: The method of Chabauty and Coleman (2007) (preprint) 24. Stein, W.A., et al.: Sage Mathematics Software (Version 4.3.5), The Sage Development Team (2010), http://www.sagemath.org

Smallest Reduction Matrix of Binary Quadratic Forms And Cryptographic Applications Aurore Bernard1 and Nicolas Gama2 1

XLIM, Limoges, France [email protected] 2 GREYC Ensicaen, Caen, France [email protected]

Abstract. We present a variant of the Lagrange-Gauss reduction of quadratic forms designed to minimize the norm of the reduction matrix within a quadratic complexity. The matrix computed by our algorithm 1 4 on the input f has norm O f 1 2 Δf , which is the square root of the best previously known bounds using classical algorithms. This new bound allows us to fully prove the heuristic lattice based attack against NICE Cryptosystems, which consists in factoring a particular subclass of integers of the form pq 2 . In the process, we set up a homogeneous variant of Boneh-Durfee-HowgraveGraham’s algorithm which ﬁnds small rational roots of a polynomial modulo unknown divisors. Such algorithm can also be used to speed-up factorization of pq r for large r.

1

Introduction

Binary quadratic forms appeared progressively in the 17-th century, when Descartes and Fermat ﬁrst introduced the concept of coordinates as a tool to algebraically solve geometric problems. Those forms have wide applications in mathematics and physics, especially in geometry, numerical analysis or algebraic topology. A binary quadratic form is a homogeneous polynomial of degree two in two variables, which can be viewed as the Cartesian equation of a surface f x, y ax2 bxy cy 2 on a given basis of R2 . Of course, this equation varies with the basis of expression, and it is natural to deﬁne an equivalence relation to regroup all these possible equations into classes. Over the real ﬁeld, there are six classes corresponding to the Sylvester’s signatures. They can be distinguished by the sign of the discriminant Δf b2 4ac, and the sign of a c. Forms of strictly negative discriminant (imaginary forms) have a unique zero at the origin, which is also their unique local and global extremum. Forms of strictly positive discriminant (real forms) represent a saddle-shape. Meanwhile, quadratic forms were also used over the integer ring by Fermat, Lagrange and Gauss to solve long standing problems from number theory. This time, binary quadratic forms are equations with integer coeﬃcients of discrete G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 32–49, 2010. c Springer-Verlag Berlin Heidelberg 2010

Smallest Reduction Matrix of Binary Quadratic Forms

33

scatter-plots on a given lattice basis of Z2 . One deﬁnes a similar equivalence relation by base change, except that transformation matrices are now unimodular, and that they preserve the value of the discriminant. Problems related to this equivalence are more complicated than on the real ﬁeld: for instance, in both real and imaginary cases, we do not know any polynomial way to compute the number of equivalence classes of a given discriminant. Deciding the equivalence of two forms is easy in the imaginary case, where each class contains a unique reduced representative computable in polynomial time. However, the problem is hard in the real case, where there are, depending on the notion of reduction, either an exponential number of polynomially computable reduced representatives, or a few representatives computable in exponential time. A reduction algorithm takes as input a quadratic form and outputs a reduced form and the reduction matrix, which is a unimodular base-change matrix used to obtain this form. The most famous polynomial time reduction algorithms are Lagrange algorithm [15] (1773) commonly known as ”Gauss reduction” algorithm [11] (1801). In [14] (1980), Lagarias modiﬁed the Gauss reduction algorithm for make it more eﬃcient. This algorithm is the one used in practice, and which we refer as the Gauss reduction algorithm, or Classical Gauss, if we need to diﬀerentiate it from new ﬂavors which we propose. The cryptanalysis of [6] shows experimental evidences that the small size of reduction matrices have important applications to the factorization of some large numbers used in public key cryptosystems, especially those of the NICE cryptosystems (see [12,13]). However the best currently known upper-bounds on the size of reduction matrices [14,1] are by an order too large, and keep all these results on the factorization heuristic. In this paper, we specially design an eﬃcient variant of the Gauss reduction algorithm to minimize the size of transformation matrix, and we prove constructive upper-bounds which are tight both in the worst case and in the average case. These bounds, combined with an improvement of the methods of [6], allows us to prove all the above mentioned heuristics of on the factorization of integers from the NICE cryptosystems.

2

Preliminaries and Notation

In this section we recall some deﬁnitions and properties concerning binary quadratic forms. For a more detailed account of the theory see [5,4,9]. Then, we summarize some results on the norm of a matrix. Quadratic Forms. A binary quadratic form f is a homogeneous polynomial of degree two in two variables f x, y ax2 bxy cy 2 with a, b, c Z3 which we abbreviate as f a, b, c. Throughout this paper the word form will be used in the sense of binary quadratic form. It is said primitive when gcd a, b, c 1. The discriminant of f is Δf b2 4ac. A discriminant Δf is called fundamental if all the forms of discriminant Δf are necessarily primitive: for example, it is the case of all odd and square-free integers. The set of all primitive forms of discriminant Δf is denoted FΔf . We impose that the discriminant is not a perfect square then a and c are always non-zero. The form f can be factored as

34

A. Bernard and N. Gama

f x, y a x yζf x yζf where ζf and ζf are the complex roots of the univariate polynomial f x, 1 which we call the aﬃne representation of f . When Δf 0, each root of f live in RQ and the form is real. In this case, ζf will denote the smallest root and ζf the largest one. When Δf 0, the roots are in CR and the form is imaginary. We note λ f min f x, y : x, y Z2 0, 0 the ﬁrst minimum of f . We note Mt the transpose of amatrix M. The a b 2 polar representation of f is the symmetric matrix b of determinant

2 c αβ Δf 4. Let M M2 Z be a 2 2 matrix with integer enγ δ tries which we often abbreviate as α, β; γ, δ . We note Id the identity matrix of M2 Z. The composition action of M on f is deﬁned as the form g x, y f αx βy, γx δy and it is noted g f.M. The coeﬃcients of g are g f α, γ , b αδ γβ 2 aαβ cγδ, f β, δ. We remark that for each Composition Action.

root ζg of g, t

αζg β γζg δ

is a root of f . Finally, the polar representation of g is

M f M which implies that Δg

det M2 Δf .

Group action. Let GL2 Z be the general linear group of matrices in M2 Z which are invertible and its subgroup SL2 Z the special linear group of matrices which have a determinant equal to one. The action deﬁned with either GL2 Z or SL2 Z on the set of primitive forms FΔf of a given discriminant is a (right) group action. Two forms f and g are equivalent if they belong to the same SL2 Z-orbit. In this case we note f g. We deﬁne Aut f the group of automorphisms of the form f FΔf as M SL2 Z, trace M 0 and f.M f . The set of all automorphisms of f is Aut f . The group Aut f is known to be cyclic, and we call its generator the fundamental automorphism of f . The largest eigenvalue of the fundamental automorphism of f is the fundamental unit. It only depends on the discriminant Δf , and will be denoted Δf .

1 0 Three specials transformations. We deﬁne the symmetry S , the 0 1 01 1h exchange E and the translation by an integer T h . They are 10 01 three (linear) transformations of GL2 Z. All matrices in GL2 Z can be written as a product of powers of these three transformations and SL2 Z is generated by the product ES and T 1. The action of these transformations on f are f.S a, b, c, f.E c, b, a f.T h a, b 2ah, f h. Note the important fact: the roots of f.S are the opposite of the roots of f and the roots of f.E are the inverse of the roots of f , and that T h subtracts h to each roots of f . Norms of matrices and forms. Let M α, β; γ, δ be a matrix in M2 Z. The Euclidean norm is M2 α2 β 2 γ 2 δ 2 , and the maximum norm is M max α, β , γ , δ . The norm M supv2 1 M.v2 is the induced Euclidean norm, which is also the square root of the largest eigenvalue of Mt M. All the norms are equivalent: M M M2 2M.

Smallest Reduction Matrix of Binary Quadratic Forms

35

Additionally, the induced norm is sub-multiplicative: if N M2 Z then MN M N and Id 1, and it is lower-bounded by the spectral radius ρ M, which is the supremum among the absolute values of the eigenvalues of M. By extension, we deﬁne the norms f ,f 2 and f of a form as the corresponding norm of its polar representation.

3

A New Reduction Algorithm for Real Quadratic Forms

A form f a, b, c is reduced if it satisﬁes two conditions simultaneously: a normalization condition, which deﬁnes the choice of the representative of b mod 2a, and a reduction condition, which often upper-bounds the size of a (or c). In the imaginary case, these conditions are very natural: a form is normal if and only if b a, a is minimal, and is reduced if additionally, a is the minimum λ f . A single translation is needed to normalize any form. However, the reduction condition takes more steps to be achieved. The classical Gauss reduction reduces a form by successive swaps SE and normalizations T b 2a (see [1]) until f is reduced. The Gauss reduction algorithm operates in quadratic time (see [1,21,18]). For each form f of discriminant Δf 4, there exists a unique reduced form g in each equivalence class, and a unique reduction matrix M SL2 Z such that f.M g. In this case Aut f Id . In the real case (Δf 0), the previous reduction conditions applied on f a, b, c are too restrictive, since the smallest integers α, β 0, 0 such that f α, β λ f are in general exponential in the size of f . No polynomial time algorithm can output an exponential reduction matrix. Thus, according to classical notions, f is classically normalized if and only if b a, a when a Δf and b Δ 2a, Δf when a Δf , and f is classically f reduced if additionally, Δf 2a b Δf . It is known that only a ﬁnite subset of forms of discriminant Δf are classically-reduced, and that they form a reduced cycle in each class. The Real-Gauss reduction algorithm, which uses the classical normalization, ﬁnds a reduced form equivalent to its input in quadratic time (see [1]). In this paper, given a normalized form f , we will bound the coeﬃcients of the smallest reduction matrix M α, β; γ, δ such that g f.M ag , bg , cg is reduced. The case of imaginary forms is eased by the uniqueness of the reduction a,c . We improve this matrix. Lemma 5.6.1 in [1] give us that M 2 max upper-bound with the following theorem:

Δf

Theorem 1 (Imaginary Bound). Let f a, b, c be a normalized imaginary form of discriminant Δf 0, and M α, β; γ, δ the reduction matrix such that g f.M ag , bg , cg , M satisﬁes these two upper-bounds: 1) M

2) αβγδ

23

14

c

ag 1 γδ 2

2 31 4

ac Δf

14

.

36

A. Bernard and N. Gama

bounded by fore γδ

α γ b 2 Δf2 , which can be lower2a 4a 4 aag 4 cc 2 γ Δf , and similarly δ 2 Δfg . There-

Proof. One has ag f α, γ aγ 2

Δf γ 2 . It follows that 4a 4 ac . The ﬁrst inequality comes from 3ag cg 3Δf

Δf ,

because

g is reduced. Unless the transformation is trivial (Id or SE), the normalization condition induces the inequalities α γ and β δ , which proves 1 1 αβγδ 4 γδ 2 . Thus, the norm of the reduction matrix is in fact basically in O

f

Δf

.

In the real case however, this proof would not apply directly, because the Δf b 2 α term

γ 2a 4a2 can be exponentially close to 0. The problem is that in the real case, each reduced cycle contains a large (often exponential) number of equivalent reduced forms, and some of them are exponentially far from f . A constructive approach is needed to build a polynomial reduction matrix. The analysis of the Gauss reduction algorithm in [1,14] basically proves that the norm of the computed reduction matrix is bounded by O f . In this paper, we study a variant of this algorithm which ﬁnds a reduction matrix of norm O

f

Δf and we verify that it is tight even in the average case. We deﬁne new relaxed notions of reduction and normalization, and express them according to the roots of the forms, which is more intuitive than the classical conditions on the coeﬃcients: Definition 1. A real binary quadratic form f is: – primary normalized if 0 ζf 1 and primary reduced if also ζf 1 – secondary normalized if 1 ζf 0 and secondary reduced if also 1 ζf .

Finally f is largely reduced if it is either primary or secondary reduced. Both primary and secondary notions are exchanged by the action of S, which negates the roots. Furthermore, primary and secondary reductions are exchanged by E, which inverts the roots. As usual, primary and secondary normalization can always be achieved by the action of some T h. Note that a classically normalized form, which has by deﬁnition at least one root in the interval 1, 1, is either primary or secondary normalized. Similarly, a classically reduced form a, b, c is a largely-reduced form satisfying b 0, which can again be ensured by the action of S. Our main contribution is to solve the following problems, which are equivalent. Lemma 1. The two problems are equivalent: 1. Smallest SL2 Z matrix Given a classically-normalized real form f , ﬁnd M SL2 Z such that f.M is classically-reduced and M is minimal. 2. Smallest GL2 Z matrix Given a primary-normalized real form f , ﬁnd M GL2 Z such that f.M is largely-reduced and M is minimal.

Smallest Reduction Matrix of Binary Quadratic Forms

37

Proof. From a solution M GL2 Z of Problem 2, one deduces a solution of Problem 1 by left-multiplication by Id or S to make the normalization of the input correspond, followed by a right-multiplication by Id or S to force the coeﬃcient b of the reduced form to be positive, followed by a right multiplication by Id or E so that the determinant is 1. The reduction of Problem 2 to Problem 1 is similar. Since Id, S and E are permutation matrices, they do not modify these norms or . Remark that, reducing a problem to the other also preserves the absolute value of the product of the coeﬃcients in each row of the reduction matrices. Lemma 1 motivates the search of a reduction algorithm solving the less restrictive Problem 2, since we can use the above permutation matrices to return to classical notions in SL2 Z. 3.1

Algorithm and Analysis

Let f be a real form. We deﬁne the two integers h f and hf as hf hf

ζf

and

ζf . It is easy to show that h f and hf are respectively the unique integers

such that f.T h f is primary-normalized, and f.T hf is secondary-normalized. Among the two integers hf , h f the one of smallest absolute value is noted h f : that is h f hf if hf hf , and h f hf otherwise. In other words, h f is the shortest normalization of f . As a comparison, there is only a single integer νf in the classical case such that f.T νf is classically-normalized, νf being one of the integers hf , h f but not necessarily the one with the smallest absolute value. Our reduction algorithm, is a variant of the Gauss reduction which operates in GL2 Z. It alternates exchange E and the shortest normalization T h f at each loop, and terminates on a largely-reduced form. As we will see later, any kind 1 of normalization by hf or h f would make a reduction algorithm terminate , but the choice of the shortest normalization h f instead of the classical νf (especially during the last steps) is the key element to minimize the reduction matrix. The main result of the section is the following theorem on the quality of the output of our algorithm, which is the real-case analogue of Theorem 1. Algorithm 1. RedGL2 Input: f a, b, c a primary-normalized form Output: f.M a largely-reduced form and M GL2 Z 1: M Id 2: while f not largely-reduced do 3: f f.E and M ME 4: f f.T hf and M MT hf 5: end while 6: return f and M

1

Exchange step Normalization step

The original Gauss algorithm of 1801 used actually the largest normalization at each step. The number of reduction steps is exponential on some entries. Lagarias introduced the classical normalization to obtain a quadratic complexity

38

A. Bernard and N. Gama

Theorem 2 (Real bound). Let f a, b, c be a primary-normalized form of logdiscriminant Δ 0. Given f as input, RedGL2 terminates after at most

a Δ 4 iterations where ω 15 is the gold number. Its output M 2 log ω 2 α, β; γ, δ and fr

1) M 4 2) αβγδ

14

a

f.M ar , br , cr satisﬁes:

ar

γδ 2 1

21

a

Δ.

Before proving this theorem, we remark that the best known upper-bounds achieved by the classical Gauss algorithm under the conditions same (see theo1 rem 4.4 of [1]) are M a 1 1 Δ and γδ 2 a Δ 1 1 Δ. They are basically the square of the upper-bounds of RedGL2. Figure 1 and 2 illustrate respectively the families of forms Fn n, b, 1 and Gn n, n, 1 with n N and b 2n 3 2 3 , which are families of forms where the Gauss reduction algorithm outputs reduction matrices Δ times larger than our variant RedGL2. Finally, note that a multiplicative triangular inequality on the norms of the polar representations of f fr .M 1 yields f fr 2M, which conﬁrms the optimality of Theorem 2 in average. The analysis of Gauss reduction algo log a Δ rithm in [1] upper-bounds the number of iterations by 2 log 2 2 reduction steps. Our upper-bound on the number of iterations of RedGL2 is tight in the worst case, and is only by a multiplicative factor around 1.4 larger than the maximum number of iterations of the Gauss reduction algorithm. However the primary goal of RedGL2 is the minimization of the reduction matrix. 3.2

Proof of Theorem 2

To prove Theorem 2, we ﬁrst study the termination cases, characterized by the presence of integers between the roots of f.E, and where the choice of the shortest normalization is of greatest importance. Eventually, we shall treat the general case and the complexity. Termination cases. We ﬁrst study the two cases where the algorithm terminates in a single step of reduction. The ﬁrst one deals with normal form f containing exactly one integer between its roots. This is the only case where hf h f , so all notions of normalizations (classical, primary, secondary, shortest) coincide. Lemma 2. Let f a, b, c be a real form satisfying 1 ζf 0 ζf 1, and h h f.E . The form fr f.ET h ar , br , cr is largely-reduced, and its coeﬃcients satisfy ar c, cr a, and h2 ar a. Proof. The reduction matrix from f to fr is ET h 0, 1; 1, h. Consider the parabola p x cx2 bx a which is the aﬃne representation of g f.E. Then we have h h g , and ζg hg 1 1 h g ζg , cr p h g b and p 0 a. By deﬁnition of h we have two cases: if 2c 0 then we have h hg 0 b 2c, else we have b 2c 0 h h g . In both cases we

Smallest Reduction Matrix of Binary Quadratic Forms px=cx2 +bx+a

h

px=cx2+bx+a

a

a

h

ζg ζg

0 νg

b

2c

Gauss

ζg

h=hg νg

cr

39

h-1

ζg h =hg

cr

h-1

Red GL2

Fig. 1. Illustration of Lemma 2 This figure illustrates the convexity inequalities of Lemma 2. In this case, the shortest normalization chosen by RedGL2 is hg ζg , which can be O Δ smaller than the classical normalization ν g ζg in Gauss Algorithm. It is clear that cr is in the interval 0, a . Comparison of heights of the two rectangles on the same convex and decreasing branch of the parabola, gives ch2 a .

Fig. 2. Illustration of Lemma 3 This figure is the analogue for Lemma 3. In this case, the shortest normalization chosen by RedGL2 is hg ζg , which can be O Δ smaller than the classical normalization in Gauss algorithm is ν g ζg . Inequality on the slopes of p before and after ζg gives cr a . Comparison of heights of the two rectangles on the same convex and decreasing branch of the parabola, gives ch 12 a .

graphically verify that cr p h p 0 a (see Figure 1). A convexity inequality on p between 0, h and b 2c, b 2c h shows a cr ch2. Since a and cr have the same sign and a is larger, then a ar h2 . Theorem 2 holds in this termination case: the reduction matrix is M 0, 1; 1, h. By Lemma 2, its norm satisﬁes M h a ar . Since f fr .M 1 , its ﬁrst coeﬃcient is a ar h2 br h cr , thus br h a cr ar h2 and br h2 4ar cr h2 Δ h2 a2 c2r a2r h4 2acr 2aar h2 2ar cr h2 a cr ar h2 2 9a2 , which proves the second point of Theorem 2. The second case of single-step termination concerns normalized form f such that at least two integers lie between the roots of f.E (namely hf.E h f.E ). We just write a proof for primary-normalized forms, but it can be easily extended to secondary-normalized forms. Lemma 3. Let f a, b, c be a real form satisfying 0 ζf ζf 1, and such that hf.E h f.E . If h h f.E , then fr f.ET h ar , br , cr is secondaryreduced, and its coeﬃcients satisfy ar c, cr a, and h2 ar 4a. Proof. The proof of this lemma is also based on convexity inequalities. Let g f.E, of aﬃne representation p x cx2 bx a. Note that h ζg 2. Again, one has p 0 a, p h cr . It follows from the deﬁnition that fr is

40

A. Bernard and N. Gama

secondary-reduced. The reduction matrix is M 0, 1; 1, h, which proves ar c. Application of a convexity inequality (see Figure 2) on p in the two intervals b b 0; h 1 and 2c h 1; 2c of same length yields ar h 12 a p h 1 a, therefore ar h2 4ar h 12 4a. Finally, another convexity inequality centered on ζg gives p

h cr .

p ζg

p 0

0

ζg

p ζg , so a p 0 h ζg

ζg

p h

h ζg

Once again, Theorem 2 holds in this termination case, but this time, M 2 h 2 a ar and Δ h2 a cr ar h2 2 6a .

General case. We now prove the general case of Theorem 2. We call fi ai , bi , ci the successive values of f at the beginning of the while loop of Algorithm 1, and hi h fi .E . We suppose that the primary-normalized form f0 does not have any integer between its roots (otherwise it would either already be reduced or as in Lemma 2). Thus 0 ζf0 ζf0 1. For each iteration i in the loop, if there is at least one integer between the roots of fi .E, then we set m i 1 and the algorithm reaches one of the two termination cases above. Otherwise the shortest normalization hi is the primary one hi h fi .E hfi .E . Thus fi is also primary-normalized and 0 ζfi ζfi 1. Note that the

distance between the roots strictly increases ζfi

ζf

i1

1

ζf

i1

i1

ζf

ζf E ζf E

ζf ζf ζf . Such process can not hold for- ever, otherwise the integer sequence of the ﬁrst coeﬃcients ai Δ ζf ζf i1

ζf

i1

i

i1

i1

i1

i

i

would be strictly decreasing. This proves the termination of the algorithm. The integer m is the smallest index, such that fm 1 .E contains at least one integer between its roots. The shortest normalization hm 1 hfm1 .E h fm1 E is in this case secondary, and satisﬁes hm 1 2. We eventually use the following lemma to conclude the proof of Theorem 2. Lemma 4. Let f a, b, c and g ag , bg , cg be two real forms and M α, β; γ, δ GL2 Z such that f.M g. If all the roots of g are positive and γ 0 and δ 1 then ag δ 2 a. Proof. If γ 0, then M is triangular, so α δ 1 and ag a. We now αζ β suppose γ 0. Let ζg be a root of g, then ζf γζgg δ is a root of f . We have 2 α γ ζf 1 γ ζg γδ 1 γδ thanks to the positivity conditions. Since this

bound holds for both roots of f , ag γ 2 a α γ ζf

α γ ζ a δ2 . f

We continue the proof of Theorem 2 by applying this lemma to the main loop of RedGL2. Note that for each i 1; m, the reduction matrix from f0 to fi is Mi

0 1 1 h0

0 1 0 1 ... 1 h1 1 hi 1

αi βi . γi δi

(1)

Smallest Reduction Matrix of Binary Quadratic Forms

41

Their coeﬃcients are all positive, and satisfy these recurrence equalities for i 2: γi1 δi hi 1 δi 1 δi 2 and δ0 , δ1 1, h1 αi1 βi hi 1 βi 1 βi 2 and β0 , β1 0, 1 Since all the hj j 0..i are greater than 1, it follows that αi min βi , γi max βi , γi δi and Mi δi ω i 2 by induction and comparison to the Fibonacci sequence 2 . Applying Lemma 4 on f0 and fm 1 implies that Mm 1 2 a0 am 1 . At iteration m, Lemma 4 can be applied to fm .T 1, which has positive roots and shares its ﬁrst coeﬃcient am with fm . The transformation matrix Mm T 1 Mm 1 0, 1; 1, hm 1 1 still satisﬁes the conditions of Lemma 4 because hm 1 2. We obtain Mm T 12 a0 am, and ﬁnally Mm 2 4a0 am after a backwards translation by T 1. We already know that fm is secondary-normalized and that the largest root of fm is positive. There are two cases: 1. If the largest root of fm is strictly greater than 1, then r m, fr is secondaryreduced, and the reduction matrix is Mm αm , βm ; γm , δm . One already 2 has Mm 2 4a0 ar . From f0 fr .M 1 , we draw a0 ar δm br δm γm 2 2 2 2 2 2 2 2 2 cr γm , so Δδm γm br δm γm 4ar cr δm γm a0 ar δm cr γm . Since 2 2 2 by construction γm δm 1 Mm 1 and by Lemma 3 applied on fm 1 2 2 and fr , cr am 1 , one ﬁnds Δδm γm 6 a0 2 . 2. If the second root of fm is strictly lower than 1, then by Lemma 2, fm1 is αr βr 0 1 Mm , and reduced. The matrix of reduction is M 1 hm γr δr r m 1. Thus M2 Mm 2 1 hm2 4a0 am 4h2m 16a0 ar . One still has Δδr2 γr2 a0 ar δr2 cr γr2 2 21a0 2 , because cr am by Lemma 2. This concludes the proof of items 1 and 2 of Theorem 2. It remains the complexity issue, proved in the following paragraph. Complexity. We now prove the number of iterations performed by RedGL2. Two steps before the end, at iteration r 2 of RedGL2, we know that the form fr 2 ar 2 , br 2 , cr 2 satisﬁes Δ ar 2 , because the distance between the roots of fm1 is smaller than 1. By Lemma 4 we have ω r 4

a0 ar 2

M r 2

a0

Δ. It follows that r 4 is upper-bounded by

log a Δ steps where ω 12 5 . 2 log ω The worst case complexity of algorithm RedGL2 is reached when all the normalizations occurring in the algorithm until the index r 2 are by h 1. For instance, we experimentally verify that it is the case on this family of inputs g. T 1E n where g is reduced and n grows. 2

The ith number of the sequence of Fibonacci numbers is bigger than ω i2 .

42

4

A. Bernard and N. Gama

Proof of Heuristic Cryptanalysis of the NICE Cryptosystems

We propose an application of the results of the previous section to the cryptanalyses of the NICE cryptosystems. There are two variants, which are by chronological order NICE Imaginary [12] (with imaginary forms), and NICE Real [13] (with real forms). Their security relies on the intractability of factorization of the public discriminant N . They were designed for a similar level of security as RSA, but with faster decryption, since the decryption process has quadratic complexity. Both are now considered as broken. The ﬁrst one succumbed by a proved arithmetic attack in [7]. However, the more general attack against both versions of NICE (in [6]) using lattice reduction remains only experimental and relies on two heuristic assumptions. In this paper, we provide an alternative point of view on the lattice attack, which allows to avoid the use of these heuristics and to prove the attack entirely. Both variants of NICE (Real and Imaginary) have originally been described in terms of ideals of quadratic orders, and are based on a morphism between classes of primitive forms of fundamental discriminant p and classes of primitive forms of non-fundamental discriminant N q 2 p. These notions are actually not needed here to understand the lattice attack, therefore we will here give a simple description solely in term of quadratic forms. 4.1

Lifting Quadratic Orders

We summarize some important properties on the relation between the sets Fp and FN of primitives forms of discriminants respectively p and N q 2 p, using the terminology we introduced in the last section. For the cryptographic interest we restrict ourselves to the case where q is an odd prime. The following background theory can be found in [5,4,9]. Integer matrices of determinant q. We deﬁne an equivalence relation modulo SL2 Z between two integer matrices A and B M2 Z by A B M SL2 Z, AM B. The 2 2 integer matrices of determinant q correspond to matrices of rank 1 mod q, they fall into q 1 equivalence class, which are characterized by the (projective) direction from 0, 1, ..., q 1, of their image mod q. Each class contains a unique Hermite normal form: qk 10 Qk , k 0, . . . , q 1 or Q . 01 0q Lift. As we can see in [5, section 7], for each form f of discriminant N pq 2 and each M M2 Z of determinant q, there exists a (non-unique) form g Fp such that f g.M. When M Q , we deﬁne a particular function ϕ (also called lift ) which computes such g Fp from f a, b, c FN such that 2 gcd a, q 1 as follows: ϕ f a, bq2ah , ah q2bhc where h 1 q, . . . , 0 and h b 2a mod q. Note that all the divisions are exact since f is primitive of discriminant N 0 mod q 2 and q is an odd prime. It must be noted that the

Smallest Reduction Matrix of Binary Quadratic Forms

43

lift preserves the ﬁrst coeﬃcient a of the form. It is also clear that ϕ preserves primary normalization, because its action on the roots of f is a translation by h 0, q 1 followed by a division by q, which stabilizes the interval 0, 1 of the largest root. Finally, equivalence of forms is stable by lift !f, f FN , f f " ϕ f ϕ f . The converse is in general false. Given a form g Fp and U its fundamental automorphism, there are exactly q p q primitive forms (in FN ) among

g.Q0 , . . . , g.Qq 1 , g.Q where p q denotes the Legendre symbol. These forms split into q p q sq sets of sq equivalent forms (see [5] theorem 7.4), where sq is the order of U modulo q. The fundamental unit N is equal to the power s p q . These q p q sq diﬀerent classes of equivalence are the only ones to be lifted to the class of g. Reduced cycle. Let g FΔ be a classically-reduced form of discriminant Δ 0, the right neighbour of g is the classical normalization of g.SE. If we note H g the largest normalization of g (by the integer among hg , h g of largest absolute value), then the right neighbour of g is g.SET H g.SE . Successive iterations of the right neighbour enumerates all the reduced forms equivalent to g, and deﬁne the reduced cycle of the class of g. The cardinality of such reduced cycle is in O log Δ where Δ is the fundamental unit. Principal cycle, and q-belt. The principal class of a discriminant Δ 0 is the class containing 1, 1, #. The principal form is the classical-normalization of this form, and the principal cycle ½Δ is the reduced cycle of the principal class. Note that the principal class is the only class containing a form of ﬁrst (or last) coeﬃcient equals to 1. We deﬁne the q-belt of a discriminant N pq 2 as the set of all primary normalized forms q 2 , kq, # of the principal class. Necessarily, k p, 2q p. There are exactly sq 1 forms in the q-belt of N : let g0 be the principal form 1, #, # of FN and f ϕ g0 is (necessarily) the principal form of Fp . Let U be the fundamental automorphism of f , we set by induction k0 and ki the unique integer such that U Qki1 Qki for i 1. Note that Qki U i Qk0 , and that the order of U mod q is precisely sq , therefore the sequence ki is periodic and ksq k0 . Finally, the q-belt of N is the set g1 f.Qk1 , . . . , gk f.Qksq 1 . They are indeed primary-normalized and equivalent by construction. A transformation matrix from gi to gi 1 is by construction Qki1 U Qki1 SL2 Z, because U Qki1 Qki . 4.2

Cryptosystem Real NICE

We now describe the NICE Real encryption and decryption. The public key is a composite integer N pq 2 and the secret key p, q with p and q two distinct primes of the same size, satisﬁes two conditions: – p is a Schinzel prime [19] which is a positive squarefree integer of the form p A2 x2 2Bx C with A, B, C, x Z, A 0 and B 2 4AC dividing 4 gcd A2 , B 2 . Such special primes implies a very low number of reduced

44

A. Bernard and N. Gama

forms in each class, namely there are O log p reduced forms in Fp in each equivalence class ([8] and [22, theorem 5.8, p. 52]). It is therefore practical to enumerate every reduced form equivalent to a given one. With a generic discriminant, the number of reduced forms per cycle would be exponential, around O p) (see [3]). To avoid any confusion, please note that even for a Schinzel prime, the number of classes in Fp remains exponential. – q is such that sq is linear in q. This imply that the number of reduced forms of discriminant N q 2 p in each equivalence class is at least linear in q and upper-bounded by O q log p, which is exponential. The encryption of a message m works as follows: m is embedded into a (usually prime) integer a p 2 which satisﬁes some low-probability pattern, and such 2 that q p is a square modulo a. This integer is expanded into a quadratic form fs a, b , c of discriminant q 2 p (which is not printed). The ciphertext is a random reduced form fc equivalent to fs (there are exponentially many). It can be generated from fs by successive multiplications by random unimodular matrices and reductions. The decryption algorithm lifts the ciphertext in Fp and enumerate all the reduced forms equivalent to ϕ fc , looking for the pattern. Of course, the knowledge of q is needed to compute ϕ. There are only O log p of them. It will necessarily ﬁnd it, because the (unknown) lift of fs fc is an equivalent form ϕ fs a, #, #, whose normalization a, #, # is reduced due to the small size of a, and it satisﬁes the pattern by construction. Due to the small number of reduced forms, it is likely the only one of the small reduced cycle to satisfy the pattern, and the plaintext m is eventually extracted from a. 4.3

Cryptanalysis

The cryptanalysis of NICE Real presented in [6] works as follows. The authors present an algorithm inspired of Coppersmith methods (see [10,17]), which solves in polynomial time the equation au2 buv xv 2 0 mod q 2 in the variables u, v, q where N pq 2 is known and max u, v O N 1 9 . They call this algorithm Homogeneous-Coppersmith in [6]. Their cryptanalysis of NICE Real is: Pick3 a form g of the principal cycle, and try to solve the equation g u, v 0 mod q 2 with Homogeneous-Coppersmith. Repeat this until it ﬁnds a solution u, v, q and return the private key q. The proof of the attack of [6] relies on this heuristic assumption: Assumption 1. The cardinality of the set A g 1 O N 9 and g u, v 0 mod q 2 is linear in sq . 3

½N ,

u, v max u, v

The authors of [6] enumerates the forms sequentially, until it ﬁnds a solvable one. They need an assumption not only on the large number of such forms, but also on their regular repartition on the principal cycle. Randomizing the enumeration avoids to prove the assumption on regular repartition (Heuristic 2 in [6]), which is feasable using the distance introduced in Theorem 3, but is beyond the scope of this paper.

Smallest Reduction Matrix of Binary Quadratic Forms

45

The authors of [6] experimentally verify this assumption. Namely, if g¯k denotes the reduction of the form gk q 2 , #, # of the q-belt by Classical Gauss reduction. The bottom two coeﬃcients of the reduction matrix satisfy g¯k δ, γ q 2 . Homogeneous-Coppersmith experimentally recovers δ, γ for most of the g¯k and even a few of their direct left or right neighbours on the principal cycle. This indicates that the norm of the reduction matrix is in general upper-bounded by O N 1 9 . However we also found rare cases of g¯k where the norm of reduction matrix was by an order greater than N 1 9 , and on which Homogeneous-Coppersmith algorithm cannot ﬁnd any solution. We call these particular forms unbalanced, because they have in general an unusually small coeﬃcient. The main three diﬃculties which prevented the authors of [6] to prove Assumption 1 were to justify that that the proportion of unbalanced forms is negligible among the set of g¯k , that the reduction matrix using Classical Gauss reduction is bounded by O N 1 9 , and that Classical Gauss is injective on a large enough subset of the q-belt, which prevents g¯k from being too small. Our ﬁrst improvement in their analysis is to replace the Classical Gauss reduction algorithm with RedGL2. This allows to square-root the upper-bounds on the reduction matrix as of Theorem 2. Thus we deﬁne gˆk as the reduction by RedGL2 of the q-belt form gk for each k. We ensure that gˆk is classically reduced and that the reduction matrix has determinant 1 using Lemma 1. The ﬁrst point of Theorem 2 implies that the norm of the reduction matrix is in O N 1 9 as soon as the smallest coeﬃcient of gˆk is greater than N 4 9 . We can either prove that this condition is satisﬁed by a large proportion of the gk , or we can also circumvent this limitation by using the second point of Theorem 2, which indicates that the size of the product uv is always upper-bounded by O N 1 6 . We therefore improve the Homogeneous-Coppersmith algorithm so that it also ﬁnds unbalanced solutions: namely, we design a rational variant of Boneh-DurfeeHowgraveGraham algorithm [2] which in particular solves g u, v au2 buv cv 2 0 mod q 2 on u, v, q as soon as the product uv is in O N 2 9 . Our new polynomial attack on Nice Real is the following: Randomly select a form g on the principal cycle ½N , and try to solve g u, v 0 mod q 2 in u, v, q using Rational-BonehDurfeeHowgraveGraham. Repeat until it ﬁnds a solution, and return q. The proof of this attack works in two steps: ﬁrst, we prove (in Theorem 3) that the above-deﬁned gˆk represent a non-negligible proportion of the principal cycle, and second, we prove (in Section 4.4) that Rational-BonehDurfeeHowgraveGraham ﬁnds q from any of the gˆk in polynomial time. Definition 2 (distance). we deﬁne a notion of distance between two equivalent forms f g as dist f, g min log M, M SL2 Z and f.M g . Let f, g, h be three equivalent forms in FΔ , the distance function satisﬁes the following properties: 1. dist f, g dist g, f 0 2. dist f, g 0 f g or f

g.SE

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3. dist f, h dist f, g dist g, h 4. if M SL2 Z satisﬁes f.M g and log M.

M

Δ , then dist f, g

Proof. The ﬁrst three points follow from basic properties of the induced norm, and the fact that only isometries have a unit norm. To prove the fourth statement, let U be the fundamental automorphism of f , the eigenvalues of U are Δ and Δ1 . Any non-trivial automorphism V of f satisﬁes V Δ , because V is a non-zero power of U , and its spectral radius is a positive power of Δ . The matrix M of the fourth point is necessarily the smallest transformation matrix from f to g, otherwise any matrix X SL2 Z such that f.X g and X M would produce a non-trivial automorphism MX 1 of f of too small norm MX 1 Δ , which is impossible. One of the greatest advantage of this distance is the fourth statement, which in general indicates that any polynomial transformation matrix is necessarily the smallest one. This allows to eﬃciently lower-bound a distance. As shown in the proof, it is essential that the group of automorphism is cyclic, the fourth statement would be false on GL2 Z. The authors of [6] used another distance between f, g , which could have been formalized as the smallest k N such that k there exists h1 , . . . , hk such that i1 SET hi transforms f into g or g.SE. Inside the reduced cycle, this corresponds to Shanks distance [20]. Unfortunately, it does not satisfy any equivalent of the fourth point: there is no way to eﬃciently verify that a given distance, as small as it could be, is correct. All the variants we found of this distance, which aims to approximate this statement, based either on the logarithms of the hi or some maximum norms, break the positive deﬁniteness or the triangular inequality. This explains why we do not base our proof on Shanks distance and introduce our own instead. Theorem 3. Given a NICE modulus N pq 2 , the set A gˆk RedGL2 gk , k 1, . . . sq 1 of the reduced of the q-belt has at least K.sq elements for some constant K 0. Proof. We now call Up the fundamental automorphism of the principal form of Fp . We verify that Upj 2 jp p j and that for all i, j, Qki1 Upj Qkij transforms gi into gij . Its norm is bounded by 1q Qki U j Qkij 4q jp p j . Due to point 4, for all j 1, sq 2 2, the distance dist gi , gij log Qki1 Upj Qkij is greater than j log p log 2q . By Theorem 2, the norm 2 of the reduction matrix from a gi to gˆi is upper-bounded by 2 21q N 42q p, and it follows that dist gˆi , gˆij j log p log 3528q 3 p. For this reason, if j log 3528q 3 p log p , then dist gˆi , gˆij 0 and gˆi gˆij . Using the NICE parameters, one has log 3528q 3p log p 3, thus the forms gˆ1 , gˆ4 , gˆ7 , . . . , gˆ3n1 are distinct (with n sq 6.

Smallest Reduction Matrix of Binary Quadratic Forms

4.4

47

Rational Improvement of the Boneh-Durfee-HowgraveGraham’s Algorithm

In this section, we describe our Rational-BonehDurfeeHowgraveGraham algorithm as a variant of Boneh Durfee Howgrave-Graham algorithm [2] solving rational linear equations u v C 0 mod q in the variables u, v, q when a multiple N pq r is known. The description of Rational-BonehDurfeeHowgraveGraham is summarized in Algorithm 2. Among others, it can be used to solve all the equations gˆk u, v au2 buv cv 2 0 mod q 2 of discriminant pq 2 of the previous section, because they are equivalent to u v b 2a 0 mod q. Since the solution we are looking for satisﬁes uv O N 1 6 , the following Theorem 4 proves that Rational-BonehDurfeeHowgraveGraham ﬁnds all solutions uv O N 2 9 , and concludes the proof of our new attack on Nice Real. More generally, given a polynomial P , the technique due to Boneh Durfee Howgrave-Graham transforms the equation P u v 0 mod q, into a lattice L of dimension m and bounded determinant, and whose short vectors are orthogonal to the integer vector S um , um 1 v, ..., uv m 1 , v m . The solutions u and v can be extracted from any of those short lattice vectors. This lattice is described by a basis B, whose rows contain the coeﬃcients of m 1-degree polynomials having u v as a root modulo a power of q. When u and v have approximately the same size (like in Homogeneous-Coppersmith of [6]), the celebrated LLL reduction algorithm on B outputs directly the desired vector orthogonal to S. Otherwise, when u and v are unbalanced, say for instance that u is 1000 times larger than v, one ﬁrst needs to re-balance the lattice by multiplying each i-th column by C i , where C is close to 1000, and only then reduce the basis. The original Boneh-Durfee-HowgraveGraham’s algorithm, which interests in integer solutions (arbitrary u and v 1), follows the above rule: the lattice basis which is actually LLL-reduced is the basis of Homogeneous-Coppersmith where each i-th column has been multiplied by X i , where X is a power of 2 just larger than the solution u. More generally, if we don’t know the relative balance between u and v but only know that the size of uv is n-bits, then we can test the n possible powers of two sequentially within a linear-factor overhead. Besides, we remark that instead of multiplying the columns of the input Homogeneous-Coppersmith basis by 1, 2, 4, ..., 2m, we describe the exact same lattice by multiplying the columns of the LLL-reduced basis, and the second one is almost reduced (LLL terminates in a very few steps). Thus after the reduction of the ﬁrst Homogeneous-Coppersmith basis, one obtains all the other possible balances of u and v for free. Theorem 4. Given any integer N pq r (where p and q are unknown), and a r bound β 14 q log q log N , Algorithm 2 terminates in polynomial time, and ﬁnds a solution (if it exists) of the equation uv c mod q where u, v are unknown integers satisfying uv β. u Proof. Let U, V R2 such

that

U and v 1 log qr . ters m N 0 and t mlog

N

V . We use the same parame-

We denote by Rm X, Y the span of homogeneous polynomials of degree m, and we deﬁne the isomorphism ϕ : Rm X, Y $ Rm1 which computes

48

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Algorithm 2. Rational Boneh-Durfee-HowgraveGraham Input: An integer N N of the form pq r (p and q are unknown), an integer c logq r log N 0, N 1 and a bound β 14 q Output: u, v N3 such that uv c mod q and u v β if it exists 1: Choose the smallest m such that

m1logq r logN

1

1

1

N 2 8r m 1 2

m11

1.5, and set t

. tk

2: Compute the family Pk X, Y N r X cY k Y mk for k 0..m 3: for l 0 to log2β do 4: U 2l ; V β 2l 5: Express (or update) the family Pk k0..m on the monomial k mk Y X , and form a matrix B Mm1 Z U k V mk k0..m 6: LLL-reduce B, and call α0 , . . . , αm the ﬁrst vector αk k 7: for each rational root uv of RX m k0 U k V mk X 0 do 8: if uv β and gcdu cv, N is non-trivial return u, v 9: end for 10: end for k

basis

mk

Y the coordinates of a polynomial on the basis X U k V mk k0..m . For instance, k m k k m k U V ek where ek is the k-th canonical basis vector. Let ϕX Y tk Pk k0..m be the family Pk X, Y N r X cY k Y m k Rm X, Y . By m construction, any integer linear combination R k0 Z Pk satisfy R u, v 0 mod q t and R u, v m 1 ϕ R2 (using Cauchy-Schwartz inequality). We now suppose that ϕ R is a short vector of the lattice generated by the (triangular) basis B ϕ Pk k1,m . By that, we mean ϕ R2 1.08m1 det B 1 m1 . Such a vector can be found by running the LLL algorithm on the lattice basis B (see [16]). The remainder of the proof is just a formal veriﬁcation that when m grows, det B is small enough to guaranty that R u, v q t , and therefore that R u, v 0 (in Z). Since R is homogeneous, this allows to recover u and v.

5

Conclusion

We saw that reduction algorithms are conceptually simpler to study in GL2 Z, because we mostly manipulate only positive matrices, which are easy to bound. The precision of our analysis, in the worst case and also in the average case, allows us to fully prove a lattice-based total-break attack against Nice cryptosystems [12,13], which is unusual in the history of lattice based cryptology. A further lead would be to extend these results on the reduction of the forms in higher dimension. Acknowledgements. We would like to thank Fabien Laguillaumie and Guilhem Castagnos for useful discussions and valuable comments on this paper.

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References 1. Biehl, I., Buchmann, J.: An analysis of the reduction algorithms for binary quadratic forms. In: Voronoi’s Impact on Modern Science, pp. 71–98 (1999) 2. Boneh, D., Durfee, G., Howgrave-Graham, N.A.: Factoring n pr q for large r. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, p. 326. Springer, Heidelberg (1999) 3. Buchmann, J., Thiel, C., Williams, H.: Short representation of quadratic integers. Proc. of CANT 1992, Math. Appl. 325, 159–185 (1995) 4. Buchmann, J., Vollmer, U.: Binary Quadratic Forms An Algorithmic Approach. Springer, Heidelberg (2007) 5. Buell, D.A.: Binary Quadratic Forms Classical Theory and Modern Computations. Springer, Heidelberg (1989) 6. Castagnos, G., Joux, A., Laguillaumie, F., Nguyen, P.Q.: Factoring pq 2 with quadratic forms: Nice cryptanalyses. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 469–486. Springer, Heidelberg (2009) 7. Castagnos, G., Laguillaumie, F.: On the security of cryptosystems with quadratic decryption: The nicest cryptanalysis. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 260–277. Springer, Heidelberg (2010) 8. Cheng, K.H.F., Williams, H.C.: Some results concerning certain periodic continued fractions. Acta Arith. 117, 247–264 (2005) 9. Cohen, H.: A Course in Computational Algebraic Number Theory, 2nd edn. Springer, Heidelberg (1995) 10. Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. of Cryptology 10(4), 233–260 (1997); Revised version of two articles from Eurocrypt 1996 (1996) 11. Gauss, C.F.: Disquisitiones Arithrneticae. PhD thesis (1801) 12. Hartmann, M., Paulus, S., Takagi, T.: NICE - New Ideal Coset Encryption. In: Ko¸c, C.K., ¸ Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 328–339. Springer, Heidelberg (1999) 13. Jacobson, M.J., Scheidler, R., Weimer, D.: An adaptation of the NICE cryptosystem to real quadratic orders. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 191–208. Springer, Heidelberg (2008) 14. Lagarias, J.C.: Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithm 1, 142–186 (1980) 15. Lagrange, J.L.: Recherches d’arithm´etique. Nouveaux M´emoires de l’Acad´emie de Berlin (1773) 16. Lenstra, A.K., Lenstra Jr., H.W., Lov´ asz, L.: Factoring polynomials with rational coeﬃcients. Mathematische Ann. 261, 513–534 (1982) 17. May, A.: Using LLL-reduction for solving RSA and factorization problems: A survey. In: Nguyen, P., Vallee, B. (eds.) The LLL algorithm, survey and Applications, Information Security and Cryptography, pp. 315–348 (2010) 18. Nguyen, P.Q., Stehl´e, D.: Low-dimensional lattice basis reduction revisited (extended abstract). In: Proceedings of ANTS VI. LNCS, Springer, Heidelberg (2004) 19. Schinzel, A.: On some problems of the arithmetical theory of continued fractions. Acta Arithmetica 6, 393–413 (1961) 20. Shanks, D.: The infrastructure of a real quadratic ﬁeld and its applications. In: Proc. NTC 1992, pp. 217–224 (1972) 21. Vallee, B., Vera, A.: Lattice reduction in two dimensions: Analyses under realistic probalistic models. In: Proc. of AofA 2007, DMTCS AH, pp. 181–216 (2007) 22. Weimer, D.: An Adaptation of the NICE Cryptosystem to Real Quadratic Orders, Master’s thesis. PhD thesis, Technische Universitat Darmstadt (2004)

Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields Jean-Fran¸cois Biasse1 and Michael J. Jacobson, Jr.2, 1

´ Ecole Polytechnique, 91128 Palaiseau, France [email protected] 2 Department of Computer Science, University of Calgary 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 [email protected]

Abstract. We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic ﬁeld. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein’s batch smoothness algorithm. We achieve a signiﬁcant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number ﬁeld with a 110decimal digit discriminant.

1

Introduction

Computing invariants of real quadratic ﬁelds, in particular the ideal class group and the regulator, has been of interest since the time of Gauss, and today has a variety of applications. For example, solving the well-known Pell equation is intimately linked to computing the regulator, and integer factorization algorithms have been developed that make use of this invariant. Public-key cryptosystems have also been developed whose security is related to the presumed diﬃculty of these computational tasks. See [16] for details. The fastest algorithm for computing the ideal class group and regulator in practice is a variation of Buchmann’s index-calculus algorithm [6] due to Jacobson [14]. The algorithm on which it is based has subexponential complexity in the size of the discriminant of the ﬁeld. The version in [14] includes several practical enhancements, including the use of self-initialized sieving to generate relations, a single large-prime variant (based on that of Buchmann and D¨ ullman [7] in the case of imaginary quadratic ﬁelds), and a practical version of the required linear algebra. This approach proved to work well, enabling the computation of the ideal class group and regulator of a real quadratic ﬁeld with a 101-decimal digit discriminant [15]. Unfortunately, both the complexity results of Buchmann’s algorithm and the correctness of the output are dependent on the Generalized Riemann Hypothesis (GRH). Nevertheless, for ﬁelds with large discriminants, this approach is the only one that works.

The second author is supported in part by NSERC of Canada.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 50–65, 2010. c Springer-Verlag Berlin Heidelberg 2010

Improvements in Real Quadratic Number Fields

51

Recently, Biasse [4] presented practical improvements to the corresponding algorithm for imaginary quadratic ﬁelds. These included a double large prime variant and improved algorithms for the required linear algebra. The resulting algorithm was indeed faster then the previous state-of-the-art [14], and enabled the computation of the ideal class group of an imaginary quadratic ﬁeld with 110 decimal digit discriminant. In this paper, we describe a number of practical improvements to the indexcalculus algorithm for computing the class group and regulator of a real quadratic ﬁeld. In addition to adaptations of Biasse’s improvements in the imaginary case, we have found some modiﬁcations designed to improve the regulator computation part of the algorithm. We also investigate applying an idea of Bernstein [3] to factor residues produced by the sieve using a batch smoothness test. Extensive computations demonstrating the eﬀectiveness of our improvements are presented, including the computation of class group and regulator of a real quadratic ﬁeld with 110 decimal digit discriminant. This paper is organized as follows. In the next section, we brieﬂy recall the required background of real quadratic ﬁelds, and give an overview of the indexcalculus algorithm using self-initialized sieving. Our improvements to the algorithm are described in Section 3, followed by numerical results in Section 4.

2

Real Quadratic Fields

We present an overview of required concepts related to real quadratic ﬁelds and the index-calculus √ algorithm for computing invariants. For more details, see [16]. Let K = Q( Δ) be the real quadratic ﬁeld of discriminant Δ, where Δ is a positive integer congruent to 0 or 1 modulo 4 with Δ or Δ/4 square-free. The integral closure of Z in K, called the maximal order, is denoted by OΔ . An interesting aspect of real quadratic ﬁelds is that their maximal orders contain inﬁnitely many non-trivial units, i.e., units that are not roots of unity. More precisely, the unit group of OΔ consists of an order 2 torsion subgroup and an inﬁnite cyclic group. The smallest unit greater than 1, denoted by εΔ , is called the fundamental unit. The regulator of OΔ is deﬁned as RΔ = log εΔ . The fractional ideals of K play an important role in the index-calculus algorithm described in this paper. In our setting, a fractional ideal is a rank 2 Z-submodule of K. Any fractional ideal can be represented as √ s b+ Δ a= aZ + Z , d 2 where a, b, s, d ∈ Z and 4a | b2 − Δ. The integers a, s, and d are unique, and b is deﬁned modulo 2a. The ideal a is said to be primitive if s = 1, and da ⊆ OΔ is integral. The norm of a is given by N (a) = as2 /d2 . Ideals can be multiplied using Gauss’s composition formulas for indeﬁnite binary quadratic forms. Ideal norm respects ideal multiplication, and the set

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IΔ forms an inﬁnite abelian group with identity OΔ under this operation. The inverse of a is √ −b + Δ d −1 a = aZ + Z . sa 2 The group IΔ is generated √ by the prime ideals of OΔ , namely those integral ideals of the form pZ + (bp + Δ)/2Z where p is a prime that is split or ramiﬁed in K. As OΔ is a Dedekind domain, the integral part of any fractional ideal can be factored uniquely as a product of prime ideals. To factor a, it suﬃces to factor N (a) and, for each prime p dividing the norm, determine whether the prime ideal p or p−1 divides a according to whether b ≡ bp or −bp modulo 2p. The ideal class group, denoted by ClΔ , is the factor group IΔ /PΔ , where PΔ ⊆ IΔ is the subgroup of principal ideals. The class group is ﬁnite abelian, and its order is called the class number, denoted by hΔ . By computing the class group we mean computing the elementary divisors m1 , . . . , ml with mi+1 | mi for 1 ≤ i < l such that ClΔ ∼ = Z/m1 Z × · · · × Z/ml Z. 2.1

The Index-Calculus Algorithm

Like other index-calculus algorithms, the algorithm for computing the class group and regulator relies on ﬁnding certain smooth quantities, those whose prime divisors are all small in some sense. In the case of quadratic ﬁelds, one searches for smooth principal ideals for which all prime ideal divisors have norm less than a given bound B1 . The set of prime ideals B = {p1 , . . . , pn } with N pi ≤ B1 is called the factor base. A principal ideal (α) = pe11 . . . penn with α ∈ K that factors completely over the factor base yields the relation (e1 , . . . , en , log |α|). The key to the index-calculus algorithm is the fact, proved by Buchmann [6], that the set of all relations forms a sublattice Λ ⊂ Zn × R of determinant hΔ RΔ provided that the prime ideals in the factor base generate ClΔ . This follows, in part, due to the fact that L, the integer component of Λ, is the kernel of the homomorphism from Zn to ClΔ given by pe11 . . . penn for (e1 , . . . , en ) ∈ Zn . If p1 , . . . , pn generate ClΔ , then this homomorphism is surjective, and the homomorphism theorem then implies that Zn /L ∼ = ClΔ . The main idea behind the index-calculus algorithm is to ﬁnd random relations until they generate the entire relation lattice Λ. Let Λ denote the sublattice of Λ generated by the relations that have been computed. To determine whether Λ = Λ, one computes an approximation h∗ of hΔ RΔ such that h∗ < hΔ RΔ < 2h∗ . The value h∗ is obtained by approximating the L-function L(1, χΔ ), where χΔ denotes the Kronecker symbol (Δ/p), and applying the analytic class number formula. If Λ ⊂ Λ, then det(Λ ) is a integer multiple of hΔ RΔ . Thus, Λ = Λ as soon as det(Λ ) < 2h∗ , because hΔ RΔ is the only integer multiple of itself in the interval (h∗ , 2h∗ ).

Improvements in Real Quadratic Number Fields

53

As described in [14], an adaptation of the strategy used in the self-initialized quadratic sieve (SIQS) factoring algorithm is used √ to compute relations. First, compute the ideal a√= pe11 . . . penn = (1/d)[aZ + (b + Δ)/2Z] with N (a) = a/d2 . Let α = (ax + (b + Δ)/2y)/d with x, y ∈ Z be an arbitrary element in a. Then √ √ b− Δ b+ Δ 1 y ax + y = (a/d2 )(ax2 + bxy + cy 2 ) N (α) = 2 ax + d 2 2 where c = (b2 − Δ)/(4a). Because ideal norm is multiplicative, there exists an ideal b with N (b) = ax2 + bxy + cy 2 such that (α) = ab. Thus, ﬁnding x and y such that N (b) factors over the norms of the prime ideals in the factor base yields a relation. Such x and y can be found by sieving the polynomial ϕ(x, y) = ax2 + bxy + cy 2 , and a careful selection of the ideals a yields a generalization of self-initialization, in which the coeﬃcients of the sieving polynomials and their roots modulo the prime ideal norms can be computed quickly. In practice, we use ϕ(x, 1) for sieving, so that the algorithm resembles the SIQS more closely. For more details, see [14] or [16]. The determinant of the relation lattice Λ is computed in two stages. The ﬁrst step is to compute the determinant of the integer part of this sublattice by ﬁnding a basis in Hermite normal form (HNF). Once Λ has full rank, the determinant of this basis is computed as the product of the diagonal elements in a matrix representation of the basis vectors. The group structure is then computed by ﬁnding the Smith normal form of this matrix. The real part of det(Λ ), a multiple of the regulator RΔ , is computed by ﬁrst ﬁnding a basis of the kernel of the matrix consisting of the integer parts of the relations. Every vector (k1 , . . . , km ) ∈ Zm in the kernel corresponds to a multiple of the regulator computed with mRΔ = k1 log |α1 | + · · · + km log |αm |. The “real gcd” of the multiples m1 RΔ , . . . , mn RΔ computed from each basis vector of the kernel, deﬁned as gcd(m1 , . . . , mn )RΔ , is then the real part of det(Λ ). An algorithm of Maurer [21] can be used to compute the real gcd eﬃciently and with guaranteed numerical accuracy given explicit representations of the αi and the kernel vectors. As mentioned in the introduction, the correctness of this algorithm depends on the truth of the Generalized Riemann Hypothesis. In fact, the GRH must be invoked in two places. The ﬁrst is to compute a suﬃciently accurate approximation h∗ of hΔ RΔ via a method due to Bach [2]. Without the GRH, an exponential number of terms in the Euler product used to approximate L(1, χΔ ) must be used (see, for example, [20]). The second is to ensure that the factor base generates ClΔ . Without the GRH, an exponential size factor base is required, whereas by a theorem of Bach [1] the prime ideals of norm less than 6 log(Δ)2 suﬃce. In practice, an even smaller factor base is often used, but in that case, the factor base must be veriﬁed by showing that every remaining prime ideal with norm less than Bach’s bound can be factored over the ideals in the factor base.

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3

Practical Improvements

In this section, we describe our practical improvements for computing the class group structure and the regulator of a the real quadratic ﬁeld. Some of these improvements, such as the double large prime variant and structured Gaussian elimination, were used in [4] for the simpler case of imaginary quadratic number ﬁelds. On the other hand, the batch smoothness test and system solving based methods for computing the regulator had never been implemented in the context of number ﬁelds before. 3.1

Relation Collection

Improving the relation collection phase allows us to speed up every other stage of the algorithm. Indeed, the faster the relations are found, the smaller the factor base can be, thus reducing the dimensions of the relation matrix and the time taken by the linear algebra phase. In addition, the veriﬁcation phase also relies on our ability to ﬁnd relations and therefore beneﬁts from improvements to the relation collection phase. Throughout the rest of the paper, M denotes the relation matrix, the matrix whose rows are the integer parts of the relations. Large prime variants. The large prime variants were developed in the context of integer factorization to speed up the relation collection phase in both the quadratic sieve and the number ﬁeld sieve. A single large prime variant was described by Buchmann and D¨ ullman [7] for computing the class group of an imaginary quadratic ﬁeld, and adapted to the real case by Jacobson [14]. Biasse [4] described how the double large prime strategy could be using in the imaginary case, and obtained a signiﬁcant speed-up. The idea is to keep relations involving one or two extra primes not in the factor base of norm less than B2 ≥ B1 . These relations thus have the form (α) = pe11 . . . penn p and (α) = pe11 . . . penn pp for pi in B, and for p, p of norm less than B2 . We will refer to these types of partial relations as 1-partial relations and 2-partial relations, respectively. Keeping partial relations only involving one large prime is the single large prime variant, whereas keeping those involving one or two is the double large prime variant which was ﬁrst described by Lenstra and Manasse [17]. We do not consider the case of more large primes, but it is a possibility that has been studied in the context of factorization [10]. Partial relations may be identiﬁed as follows. Let m be the remainder of ϕ(x, 1) after the division by all primes p ≤ B1 , and assume that B2 < B12 . If m = 1 then we have a full relation. If m ≤ B2 then we have a 1-partial relation. We can see here that detecting 1-partial relations is almost for free. If we also intend to collect 2-partial relations then we have to consider the following possibilities:

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55

m > B22 ; m is prime and m > B2 ; m is prime and m ≤ B2 ; m is composite and B12 < m ≤ B22 .

In Cases 1 and 2 we discard the relation. In Case 3 we have a 1-partial relation, and in Case 4 we have m = pp where p = N (p) and p = N (p ). Cases 1, 2, and 3 can be checked very easily, but if none are satisﬁed we need to factor m in order to determine whether Case 4 is satisﬁed. We used Milan’s implementation of the SQUFOF algorithm [22] based on the theoretical work of [12] to factor the m values produced. Even though we might have to factor the remainder, partial relations are found much faster than full relations. However, the dimensions of the resulting matrix are much larger, thus preventing us from running the linear algebra phase directly on the resulting relation matrix. In addition, we have to ﬁnd many more relations since we have to produce a full rank matrix. We will see in §3.2 how to reduce the dimensions of the relation matrix using Gaussian elimination techniques. Batch smoothness test. After detecting potential candidates for smooth integers via the SIQS, one has to certify their smoothness. In [4,14], this was done by trial division with the primes in the factor base. The time taken by trial division can be shortened by using Bernstein’s batch smoothness test [3], which uses a product tree structure and modular arithmetic to factor a batch of residues simultaneously in time O b(log b)2 log log b where b is the total number of input bits. Instead of testing the smoothness of every potential candidate as soon as they are discovered, we rather stored them and tested them at the same time using Bernstein’s method as soon their number exceeded a certain limit. This improvement has an eﬀect that is all the more important when the time spent in the trial division is long. In our algorithm, this time mostly depends on the tolerance value T, a parameter used to control the number of candidates yielded by the sieve for smoothness testing. 3.2

Structured Gaussian Elimination

As mentioned in §2.1, in order to determine whether the computed relations generate the entire relation lattice, we need to compute the HNF basis of the sublattice they generate. This can be done by putting the integer components of the relations as rows in a relation matrix, and computing the HNF. The ﬁrst step when using large primes is to compute full relations from all of the partial relations. Traditionally, rows were recombined to give full relations as follows. In the case of 1-partial relations, any pair of relations involving the same large prime p were recombined into a full relation. In the case of 2-partial relations, Lenstra [17] described the construction of a graph whose vertices were the relations and whose edges linked vertices having one large prime in common.

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Finding independent cycles in this graph allows us to recombine partial relations into full relations. In this paper, we instead follow the approach of Cavallar [8], developed for the number ﬁeld sieve, and adapted by the ﬁrst author to the computation of ideal class group structures in imaginary quadratic number ﬁelds [4], which uses Gaussian elimination on columns. The ideas is to eliminate columns using structured Gaussian strategies until the dimensions of the matrix are small enough to allow the computation of the HNF with standard algorithms. Let us recall a few deﬁnitions. First, subtracting two rows is called merging. If two relations corresponding to rows r1 and r2 share the same prime p with coeﬃcients c1 and c2 respectively, then multiplying r1 by c2 and r2 by c1 and merging is called pivoting. Finally, ﬁnding a sequence of pivots leading to the elimination of a column of Hamming weight k is a k-way merge. We aim to reduce the dimensions of the relation matrix by performing kway merges on the columns of weight k = 1, . . . , w in increasing order for a certain bound w. To limit the growth of the density and of the size of the coeﬃcients induced by these operations, we used optimized pivoting strategies. In what follows we describe an algorithm performing k-way merges to minimize the growth of both the density and the size of the coeﬃcients, thus allowing us to go deeper in the elimination process and delay the explosion of the coeﬃcients. As in [4], we deﬁne a cost function C mapping rows onto the integers. The one used in [4] satisﬁed C(r) = 1+c 1, (1) 1≤|ei |≤Q

|ej |>Q

where c and Q are positive numbers, and r = [e1 , . . . , en ] is a row corresponding to (α) = i pei i . This way, the heaviest rows are those which have a high density and large coeﬃcients. In our experiments for this work, we used a diﬀerent cost function, see §4.1. Then, to perform a k-way merge on a given column, we construct a complete graph G of size k such that – the vertices are the rows ri , and – every edge linking ri and rj has weight C(rij ), where rij is obtained by pivoting ri and rj . Finding the best sequence of pivots with respect to the chosen cost function C is equivalent to ﬁnding the minimum spanning tree T of G, and then recombining every row r with its parent starting with the leaves of T . Unlike in [4], we need to keep track of the permutations we apply to the relation matrix, and of the empty columns representing primes of norm less than 6 log2 Δ. This will be required for the regulator computation part of the algorithm described next. 3.3

Regulator Computation

As mentioned in §2.1, the usual way to compute the regulator is to ﬁnd a basis of the kernel of the relation matrix, compute integer multiples of the regulator

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from these basis vectors, and compute their real gcd using Maurer’s algorithm [21]. If det Λ > 2h∗ , then either the class number or regulator computed is too large, and we need to ﬁnd extra relations corresponding to new generators, and new kernel vectors involving them. In this section, we describe a way of taking advantage of the large number of generators involved in the diﬀerent partial relations. Indeed, the dimensions of the relation matrix before the Gaussian elimination stage is much larger than in the base scenario and thus involves more generators. Consequently, given a set of k ≤ dim(ker M ) kernel vectors (uj1 , . . . , ujn )j≤k , the probability that the corresponding elements vj := uj1 log |α1 | + . . . + ujn log |αn | , where αi is the generator of the i-th relation, can be recombined into R is much larger. On the other hand, the dimensions of the matrix prevents us from running a kernel computation directly after the relation collection phase. Thus, rather than attempting to compute the kernel, we use a method similar to that of Vollmer [24] based on solving linear systems. The ﬁrst step of our algorithm consists of putting the matrix in a pseudo-lower triangular form using a permutation obtained during the Gaussian elimination phase. Indeed, as part of this computation we obtain a unimodular matrix U ∈ Zn×n such that ⎛ ⎞ ⎜ ⎜ A ⎜ ⎜ ⎜ ⎜ ⎜ UM = ⎜ ⎜ ⎜ ⎜ ⎜ (∗) ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 (0) ⎟ ⎟. ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎠ (∗) 1 (0)

Thus, solving a linear system of the form xM = b for a vector b ∈ Zm boils down to solving a system of the form x A = b , then doing a trivial descent through the diagonal entries which equal 1 and ﬁnally permuting back the coeﬃcients using U . To solve the small linear systems, we used the algorithm certSolveRedLong from the IML library [9]. It takes a single precision dense representation of A and returns an LLL-reduced solution. Once M is in pseudo-lower triangular form, we draw a set of relations r1 , . . . rd which are not already rows of M , and for each ri , i ≤ d, we solve the system xi A = ri . We then augment M with the rows ri for i ≤ d and the vectors xi with d extra coordinates, which are all set to zero except for the i-th which is set to −1.

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⎛ ⎜ ⎜ M := ⎜ ⎝

⎞ M ri

⎟ ⎟ ⎟ xi := xi 0 . . . 0 −1 0 . . . 0 . ⎠

We clearly have xi M = 0 for i ≤ d, and the xi can be used to ﬁnd a multiple of RΔ as described in §2.1.

4

Numerical Results

In this section, we give numerical results showing the impact of our improvements. For each timing, we specify the architecture used. All the timings were obtained with our code in C++ based on the libraries GMP [11], NTL [23], IML [9] and Linbox [19]. All timings are in CPU seconds. 4.1

Comparative Timings

The state of the art concerning class group and regulator computation was established in [14], where all the timings were obtained with the SPARCStation II architecture. In addition, most of the code used at the time is unavailable now, including the HNF computation algorithm. Thus, providing a meaningful comparison between our methods and those of [14] is diﬃcult. We chose to implement the HNF computation algorithm in a way that resembles the one of [14], but takes advantage of the libraries available today for computing the determinant and the modular HNF. We used this implementation in each diﬀerent scenario. The relation collection phase is easier to compare, since our method relies on SIQS. In the following, we will refer to the base case as the strategy consisting of ﬁnding the relation matrix without using the large prime variants or the smoothness batch test, and calculating the regulator by computing its kernel with the algorithm nullspaceLong from IML library. It diﬀers from the 0 large prime case (0LP) where we use the algorithm described in §3.3 for computing the regulator, along with a relation collection phase that does not use large primes. We also denote the 1 large prime scenario by 1LP, the 2 large primes by 2LP and 2LP Batch when using batch smoothness test. Relation collection phase. In Table 1, we give the time taken to collect all necessary relations. Without large primes, we collected |B| + 100 relations, whereas when we allow large primes we need to collect enough relations to ensure that the number of rows is larger than the number of non-empty columns. We used a 2.4 GHz Opteron with 16GB of memory and took Δ = 4(10n + 3) with 40 ≤ n ≤ 70. For each discriminant, we used the optimal parameters given in [14], including the size of the factor base, even if we tend to reduce this parameter when optimizing the overall time. The only parameter we modiﬁed is the tolerance value for the SIQS, as a higher tolerance value is required for the large prime variations. In each case we took B2 = 12B1 . It is shown in [4] that the ratio B2 /B1 does not have an important impact on the sieving time.

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Table 1. Comparative table of the relation collection time n 0LP 1LP 2LP 2LP Batch 40 0.83 0.48 0.63 0.90 45 6.70 3.10 2.70 2.20 50 23.00 9.50 9.20 6.10 55 56.00 26.00 23.00 15.00 60 202.00 86.00 69.00 41.00 65 1195.00 513.00 354.00 227.00 70 4653.00 1906.00 1049.00 834.00

The timings in Table 1 correspond to the optimal value of the tolerance value in each case, found by trying values between 1.7 and 4, and keeping the optimum for each scenario. For 0LP, the optimal value is between 1.7 and 2.3 whereas it is around 2.3 for 1LP, 2.8 for 2LP and 3.0 for 2LP Batch. The latter case has a higher optimal tolerance value because using the batch smoothness test allows one to spend more time factoring the residues. When using Bernstein’s smoothness test, we took batches of 100 residues. In our experiments, this value did not seem to have an important eﬀect on the relation collection time. We observe in Table 1 that the use of the large prime variants has a strong impact on the relation collection phase, and that using the smoothness batch test strategy yields an additional speed-up of approximately 20% over the double large prime strategy. Structured Gaussian elimination. Structured Gaussian elimination allows us to reduce the time taken by the linear algebra phase by reducing the dimensions of the relation matrix. Our method minimizes the growth of the density and of the size of the coeﬃcients. To illustrate the impact of the algorithm described in §3.2, we monitor in Table 2 the evolution of the dimensions of the matrix, the average Hamming weight of its rows, the extremal values of its coeﬃcients and the time taken for computing its HNF in the case of a relation matrix corresponding to Δ = 4(1060 + 3). We keep track of these values after all i-way merges for some values of i between 5 and 170. The original dimensions of the matrix are 2000 × 1700, and the timings are obtained on a 2.4 Ghz Opteron with 32GB of memory. In [4], the ﬁrst author regularly deleted the rows having the largest coeﬃcients. To do this, we need to create more rows than in the base case. To provide a fair comparison between the two strategies, we used the same relation matrix resulting from a relation collection phase without large primes, and with as few rows as was required to use the same algorithm as in [14]. We therefore had to drop the regular row deletion. We also tuned the cost function to compensate for the resulting growth of the coeﬃcients, using C(r) = 1 + 100 |ej | , 1≤|ei |≤8

instead of (1).

|ej |>8

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The HNF computation consists of taking the GCD of the determinants of two diﬀerent submatrices of the matrix after elimination using Linbox, and using the modular HNF of NTL with this value. Indeed, this GCD (which is likely to be relatively small) is a multiple of hΔ . This method, combined with an elimination strategy due to Havas [13], was used in [14] and implemented in LiDIA [18]. As this implementation is no longer available, we instead refer to the timings of our code, which has the advantage of using the best linear algebra libraries available today. Table 2. Comparative table of elimination strategies Naive Gauss i Row Nb Col Nb Average weight max coeﬀ min coeﬀ HNF time 5 1189 1067 27.9 14 -17 357.9 10 921 799 49.3 22 -19 184.8 30 757 635 112.7 51 -50 106.6 50 718 596 160.1 81 -91 93.7 70 699 577 186.3 116 -104 85.6 90 684 562 205.5 137 -90 79.0 125 664 542 249.0 140 -146 73.8 160 655 533 282.4 167 -155 72.0 170 654 532 286.4 167 -155 222.4 With dedicated elimination strategy i Row Nb Col Nb Average weight max coeﬀ min coeﬀ HNF time 5 1200 1078 26.8 13 -12 368.0 10 928 806 42.6 20 -15 187.2 30 746 624 82.5 33 -27 100.8 50 702 580 107.6 64 -37 84.3 70 672 550 136.6 304 -676 73.4 90 656 534 157.6 1278 -1088 67.5 125 637 515 187.1 3360 -2942 63.4 160 619 497 214.6 5324 -3560 56.9 170 615 493 247.1 36761280 -22009088 192.6

Table 2 shows that the use of our elimination strategy leads to a matrix with smaller dimensions (493 rows with our method, 533 with the naive elimination) and lower density (the average weight of its rows is of 214 with our method and 282 with the naive elimination). These diﬀerences result in an improvement of the time taken by the HNF computation: 56.9 seconds with our method against 72.0 seconds with the naive Gaussian elimination. The regular cancellation of the rows having the largest coeﬃcients over the course of the algorithm would delay the explosion of the coeﬃcient size, but require more rows for the original matrix. This brutal increase in the size of the extremal values of the matrix can be seen in Table 2. At this point these higher values propagate during pivoting operations, and any further column elimination becomes counter-productive.

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Factor base verification. The improvements in the relation collection phase have an impact on the factor base veriﬁcation. The impact of the smoothness batch test is straightforward, whereas the large prime variants act in a more subtle way. Indeed, we create many more relations when using the large prime variants, and the relations created involve primes of larger norm. Therefore, a given prime not in B of norm less than 6 log2 Δ is more likely to appear in a relation, and thus not to need to be veriﬁed. Table 3 shows the impact of the large prime variants and of the batch smoothness test on the veriﬁcation time. We used a 2.4 GHz Opteron with 16GB of memory. We considered discriminants of the form Δ = 4(10n + 3) for n between 40 and 70, and we chose in every case the factor base giving the best results for the base scenario. Table 3. Comparative table of the factor base veriﬁcation time n 40 45 50 55 60 65 70

0LP 17.0 77.0 147.0 308.0 826.0 8176.0 9639.0

1LP 11.0 44.0 85.0 167.0 225.0 1606.0 4133.0

2LP 2LP Batch 11.0 6.2 30.0 18.0 52.0 43.0 134.0 110.0 282.0 274.0 1760.0 1689.0 5777.0 2706.0

Regulator computation. Our method for computing the regulator avoids computing the relation matrix kernel. Instead, we need to solve a few linear systems involving the matrix resulting from the Gaussian elimination. To illustrate the impact of this algorithm, we used the relation matrix obtained in the base case for discriminants of the form 4(10n + 3) for n between 40 and 70. The timings are obtained on a 2.4GHz Opteron with 16GB of memory. In Table 4, the timings corresponding to our system solving approach are taken with seven kernel vectors. However, in most cases only two or three vectors are required to compute the regulator. As most of the time taken by our approach Table 4. Comparative table of regulator computation time n Kernel Computation System Solving 40 15.0 6.2 45 18.0 8.3 50 38.0 20.0 55 257.0 49.0 60 286.0 103.0 65 5009.0 336.0 70 10030.0 643.0

62

J.-F. Biasse and M.J. Jacobson Table 5. Eﬀect on the overall time n 40

45

50

55

60

65

70

strategy base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch base 0LP 1LP 2LP 2LP Batch

|B| relations elimination HNF regulator veriﬁcation total 400 0.8 0.1 3.2 14.6 16.8 35.6 400 0.7 0.1 2.2 6.0 16.6 25.7 300 0.8 0.2 2.5 6.4 13.1 23.1 250 1.7 0.3 4.8 8.7 18.0 33.3 250 0.5 0.2 3.6 6.7 4.4 15.5 500 6.7 0.1 5.1 18.0 77.0 107.0 500 5.9 0.2 4.9 10.0 85.0 106.0 400 4.0 0.4 6.0 11.0 50.0 71.0 350 3.8 0.5 12.0 17.0 36.0 69.0 350 2.6 1.1 9.0 14.0 30.0 57.0 750 23.0 0.3 16.0 38.0 147.0 224.0 700 21.0 0.4 15.0 20.0 147.0 203.0 450 20.0 0.4 10.0 17.0 108.0 155.0 400 14.0 0.8 22.0 23.0 74.0 133.0 400 10.0 0.6 21.0 25.0 62.0 119.0 1200 129.0 1.9 60.0 257.0 308.0 756.0 1300 47.0 0.7 52.0 49.0 265.0 414.0 650 61.0 0.7 28.0 33.0 255.0 378.0 550 40.0 1.1 48.0 48.0 177.0 313.0 550 34.0 1.0 47.0 48.0 141.0 271.0 1700 322.0 2.9 95.0 286.0 830.0 1535.0 1700 187.0 1.3 106.0 103.0 846.0 1244.0 750 309.0 1.0 45.0 64.0 865.0 1284.0 700 143.0 2.1 152.0 137.0 365.0 799.0 700 142.0 1.8 103.0 100.0 309.0 655.0 2700 10757.0 12.0 652.0 5009.0 8176.0 24607.0 2700 1225.0 2.8 489.0 336.0 3676.0 5730.0 1900 1003.0 15.0 318.0 262.0 2984.0 4583.0 1200 753.0 4.7 525.0 398.0 1943.0 3624.0 1000 1030.0 35.0 199.0 219.0 1642.0 3125.0 3700 17255.0 24.0 1869.0 10031.0 9639.0 38818.0 3600 4934.0 19.0 1028.0 644.0 9967.0 16591.0 2500 3066.0 17.0 845.0 646.0 9005.0 13579.0 1700 2414.0 27.0 2054.0 1295.0 4590.0 10379.0 1700 2588.0 20.0 1372.0 934.0 5078.0 9991.0

is spent on system solving, we see that computing fewer kernel vectors would result in an improvement of the timings, at the risk of obtaining a multiple of the regulator. Overall time. We have studied the individual impact of our improvements on each stage of the algorithm. We now present their eﬀect on the overall time taken by the algorithm, including the factor base veriﬁcation time, for discriminants of the form Δ = 4(10n + 3) with 40 ≤ n ≤ 70 on a 2.4 GHz Opteron with 16GB of memory. We used the same parameters as in [14], except for the tolerance

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and the size of the factor base. We notice in Table 5 that the optimal size of the factor base is smaller when we use improvements for the sieving phase. For example the optimal size for the double large prime variant is half the one of the base case scenario. This results in an improvement in the HNF and regulator computation whereas the relation collection time can remain unchanged, or even increase. The tolerance value we chose varies only with the strategy, but not with the size of the discriminant. We chose 2.0 for the base case and 0LP whereas we set it to 2.3 for 1LP, 2.8 for 2LP and 3.0 for 2LP Batch. We eliminated columns of weight up to w = 150 since Table 2 indicates that further elimination is counter-productive. Table 5 shows that there is an overall speed-up of of a factor of 2 for the smallest discriminants and 4 for the largest. The base case with the largest discriminants suﬀers from the necessity of ﬁnding some relations in a more randomized way. This ensures that we can get full rank submatrices of the relation matrix after the Gaussian elimination to compute a small multiple of hΔ . Matrices produced using the large prime variants do not need this extra step, even with the largest discriminants. This naturally aﬀects the sieving time, since we cannot use SIQS for that purpose, but also aﬀects phases relying on linear algebra. Indeed, elimination produces a matrix with larger entries and dimensions. 4.2

Large Example

The improvements we described allow us to compute class groups and regulators of real number ﬁelds with larger discriminants than was previously possible. The key is to parallelize the relation collection and veriﬁcation phase, while the linear algebra has to be performed the usual way. These methods were successfully used in [4] to compute the class group structure of an imaginary quadratic ﬁeld with a 110-digit discriminant. We used a cluster with 260 2.4GHz Xeon cores to compute a relation matrix corresponding to the discriminant Δ110 := 4(10110 +3) in 4 days. We allowed two large primes, used a tolerance value of 3.0, tested batches of 100 residues, took w = 250 and set |B| = 13000 . Then, we used three 2.4 GHz Opterons with 32GB of memory each to compute determinants of full-rank submatrices of the relation matrix after the Gaussian elimination in 1 day, and one 2.4GHz Opteron to compute the HNF modulo the GCD of these determinants in 3 days. We had to ﬁnd 4018 extra relations during the veriﬁcation phase that took 4 days on 96 2.4GHz Xeon cores. We thus obtained that ClΔ110 ∼ (2) = Z/12Z × Z/2Z , and the corresponding regulator is RΔ110 ≈ 70795074091059722608293227655184666748799878533480399.6730200233 .

We estimate that it would take two weeks (4000 relations per day) to complete the relation collection for Δ120 with the same factor base as Δ110 , thus requiring a similar time for the linear algebra.

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Conclusions

Recently, our work has been extended to the problems of principal ideal testing and solving the discrete logarithm problem in the ideal class group [5]. The double large prime variant and improvements to relation generation translated directly to improvements in this context. However, HNF computations are not required for this problem, and linear system solving over Z can be used instead. The numerical results were used to give estimates for discriminant sizes that oﬀer equivalent security to recommended sizes of RSA moduli. Some possibilities for further improvements remain to be investigated. For example, a lattice sieving strategy could be used to sieve ϕ(x, y) instead of ϕ(x, 1). Factor reﬁnement and coprime factorization techniques may be a useful alternative to Bernstein’s batch smoothness test. Multiple large primes have been successfully used for integer factorization and could also be tried in our context. There is also still room for improvement to the linear algebra components. For example, a HNF algorithm that exploits the natural sparseness of the relation matrix, perhaps as a black-box algorithm, would be useful. If such an algorithm were available, we could reconsider using Gaussian elimination techniques since they induce a densiﬁcation of the matrix. We could also study the eﬀect of other dense HNF algorithms in existing linear algebra packages such as KASH, Pari, Sage and especially MAGMA which seems to have the most eﬃcient HNF algorithm for our types of matrices. In that case, we would need the elimination phase regardless of how these algorithms are aﬀected by the density and the size of the coeﬃcients of the matrix. Indeed, we cannot aﬀord manipulating a dense representation of the matrix before the Gaussian elimination phase.

References 1. Bach, E.: Explicit bounds for primality testing and related problems. Math. Comp. 55(191), 355–380 (1990) 2. Bach, E.: Improved approximations for Euler products. In: Number Theory: CMS Proc., vol. 15, pp. 13–28. Amer. Math. Soc., Providence (1995) 3. Bernstein, D.: How to ﬁnd smooth parts of integers. Mathematics of Computation (submited) 4. Biasse, J.-F.: Improvements in the computation of ideal class groups of imaginary quadratic number ﬁelds. In: Advances in Mathematics of Communications (to appear 2010) 5. Biasse, J.-F., Jacobson Jr., M.J., Silvester, A.K.: Security estimates for quadratic ﬁeld based cryptosystems. In: ACISP (to appear 2010) 6. Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds. In: S´eminaire de Th´eorie des Nombres (Paris), pp. 27–41 (1988-1989) 7. Buchmann, J., D¨ ullmann, S.: Distributed class group computation. In: Festschrift aus Anlaß des sechzigsten Geburtstages von Herrn Prof. Dr. G. Hotz, pp. 69–79. Universit¨ at des Saarlandes (1991), Teubner, Stuttgart (1992)

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8. Cavallar, S.: Strategies in ﬁltering in the number ﬁeld sieve. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 209–232. Springer, Heidelberg (2000) 9. Chen, Z., Storjohann, A., Fletcher, C.: IML: Integer Matrix Library. Software (2010), http://www.cs.uwaterloo.ca/~ astorjoh/iml.html 10. Dodson, B., Leyland, P.C., Lenstra, A.K., Muﬀett, A., Wagstaﬀ, S.: MPQS with three large primes. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 446–460. Springer, Heidelberg (2002) 11. GMP, The GNU multiple precision bignum library. Software (2010), http://gmp-lib.org/ 12. Gower, J.E., Wagstaﬀ, S.: Square form factorization. Mathematics of Computation 77, 551–588 (2008) 13. Havas, G., Majewski, B.S.: Integer matrix diagonalization. Journal of Symbolic Computing 24, 399–408 (1997) 14. Jacobson Jr., M.J.: Subexponential class group computation in quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999) 15. Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: The eﬃciency and security of a real quadratic ﬁeld based key exchange protocol. In: Public-Key Cryptography and Computational Number Theory, Warsaw, Poland, pp. 89–112. de Gruyter (2001) 16. Jacobson Jr., M.J., Williams, H.C.: Solving the Pell equation. CMS Books in Mathematics. Springer, Heidelberg (2009) ISBN 978-0-387-84922-5 17. Lenstra, A.K., Manasse, M.S.: Factoring with two large primes (extended abstract). In: Damg˚ ard, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 72–82. Springer, Heidelberg (1991) 18. LiDIA Group, LiDIA: a c++ library for computational number theory. Software, Technische Universit¨ at Darmstadt, Germany (1997), http://www.informatik.tu-darmstadt.de/TI/LiDIA 19. LinBox, Project LinBox: Exact computational linear algebra. Software (2010), http://www.linalg.org/ 20. Louboutin, S.: Computation of class numbers of quadratic number ﬁelds. Math. Comp. 71(240), 1735–1743 (2002) 21. Maurer, M.: Regulator approximation and fundamental unit computation for real quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999) 22. Milan, J.: Tifa. Software (2010), http://www.lix.polytechnique.fr/Labo/Jerome-Milan/tifa/tifa.xhtml 23. Shoup, V.: NTL: A Library for doing Number Theory. Software (2010), http://www-shoup.net/ntl 24. Vollmer, U.: An accelerated Buchmann algorithm for regulator computation in real quadratic ﬁelds. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 148–162. Springer, Heidelberg (2002)

On the Use of the Negation Map in the Pollard Rho Method Joppe W. Bos, Thorsten Kleinjung, and Arjen K. Lenstra Laboratory for Cryptologic Algorithms EPFL, Station 14, CH-1015 Lausanne, Switzerland

Abstract. The negation map can be used to speed up the Pollard rho method to compute discrete logarithms in groups of elliptic curves over ﬁnite ﬁelds. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose eﬀective alternative countermeasures. As a result, fruitless cycles can be resolved, but the best speedup we managed to achieve is√by a factor of only 1.29. Although this is less than the speedup factor of 2 generally reported in the literature, it is supported by practical evidence. Keywords: Pollard’s rho method, fruitless cycles, negation map.

1

Introduction

The diﬃculty of the elliptic curve discrete logarithm problem (ECDLP) underlies the security of cryptographic schemes based on elliptic curves over ﬁnite ﬁelds [11,13]. The best method known to solve ECDLP for curves without special properties is the parallelized [17] Pollard rho method [15]. A common optimization is to halve the search space by identifying a point with its inverse [18,9,7]. Because representatives for the equivalence classes can quickly be computed using the negation√map, this equivalence relation may result in a speedup by a factor of up to 2 when solving ECDLP. For the elliptic curves over binary extension ﬁelds F2t from [12], order t equivalence relations can be used as well, √ resulting in a speedup by a factor of up to 2t [18,9]. Usage of the negation map in the context of the Pollard rho method leads to fruitless cycles, useless cycles trapping the random walks. An analysis of their likelihood of occurrence appeared in [7]. Various methods have been proposed [18,9] to deal with them, all leading to costlier random walks and administrative overhead. The literature suggests that √ the resulting ineﬃciencies are negligible, and that a speedup by a factor of 2 is attainable [1, Section 19.5.5]. We analyze fruitless cycles and the previously published methods to avoid their ill eﬀects and show that current approaches to escape from cycles suﬀer from recurring cycles. These may have contributed to the lack of practical usage of the negation map to solve prime ﬁeld ECDLPs: it was not used for the solutions G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 66–82, 2010. c Springer-Verlag Berlin Heidelberg 2010

On the Use of the Negation Map in the Pollard Rho Method

67

[10,6] of the 79-, 89-, 97- and 109-bit prime ﬁeld Certicom challenges [5]. Neither was it used by the independent current 112-bit prime ﬁeld record [3]. We present and analyze alternative methods to deal with fruitless cycles. All our analyses are supported by experiments. We found that the negation map indeed leads to a speedup, but we have √ not been able to reach more than a factor of 1.29, somewhat short of the 2 that we had hoped for. We also found that the best attainable speedup depends on the platform one uses: for instance, if the Pollard rho method is parallelized in SIMD fashion, it is a challenge to achieve any speedup at all. This has consequences for the applicability of the negation map in large scale prime ﬁeld ECDLP solution attempts. For such efforts, all participating processors must use the same random walk deﬁnition, so one may desire to gear the implementation towards processors with the best performance/price ratio, such as graphics cards (which are SIMT, a SIMD variant). The negation map (while dealing with cycles) slows down random walks in three ways. In the ﬁrst place, on average more elliptic curve group operations are required per step of each walk. This is unavoidable and attempts should be made to minimize the number of additional operations. Secondly, dealing with cycles entails administrative overhead and branching, which cause a non-negligible slowdown when running multiple walks in SIMD-parallel fashion. Finally, the best way to counter the eﬀect of the higher average number of group operations per step is making the walks “more random” by allowing a ﬁner grained decision per step. However, the beneﬁcial eﬀects of this approach are, in most circumstances on current processors, wiped out by cache ineﬃciencies. It will be seen that it is best to strike a balance between the ﬁrst and third of these slowdowns. The second slowdown somewhat aﬀects regular PCs, but is a major obstacle to the negation map in SIMD environments. This paper is organized as follows. Section 2 recalls background on ECDLP, the Pollard rho method and fruitless cycles. Section 3 introduces recurring cycles and presents and analyzes new methods to deal with them. Section 4 compares the various cycle reduction, detection, and escape methods in practice.

2 2.1

Preliminaries The Elliptic Curve Discrete Logarithm Problem

Let Fp denote a ﬁnite ﬁeld of odd prime characteristic p. Any a, b ∈ Fp with 4a3 + 27b2 = 0 deﬁne an elliptic curve Ea,b over Fp . The additively written group of points Ea,b (Fp ) of Ea,b over Fp is deﬁned as the zero point o along with the set of pairs (x, y) ∈ Fp × Fp that satisfy the shortened Weierstrass equation y 2 = x3 + ax + b. Let p, a, b and g ∈ Ea,b (Fp ) of prime order q be such that the index [Ea,b (Fp ) : g] is small. For h ∈ g, the ECDLP is to ﬁnd an integer m such that mg = h. For curves without special properties, solving ECDLP is √ believed to require an eﬀort on the order of q. Pollard’s rho method achieves this run time, while requiring more or less constant memory.

68

2.2

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Pollard’s Rho Method

If objects are selected truly at random and with replacement from q objects, the conditional probability at step n + 1 of ﬁnding the ﬁrst duplicate (or collision) is nq (if n < q). Via straightforward arguments this leads to πq/2 for the expected number of steps until the ﬁrst collision. If random objects are selected as ug + vh ∈ g for random integer multipliers u, v, a collision corresponds to u, v, u ¯, v¯ such that ug + vh = u ¯g + v¯h. Unless v¯ ≡ v mod q, the value m = u−¯ u mod q solves the discrete logarithm problem. The expected number of steps v ¯−v of this idealized version of Pollard’s rho method [15] is πq/2. r-adding and r+s-mixed walks. Pollard’s rho method uses an approximation of a truly random walk in g. Let, for a small integer r, an index function : g → [0, r − 1] induce an r-partition g = ∪r−1 i=0 Gi of g, where Gi = {x : x ∈ g, (x) = i} and all Gi have cardinality close to qr . For random integers ui , vi , elements fi = ui g + vi h ∈ g are precomputed for 0 ≤ i < r. Starting at a random but known multiple of g, the successor of a point p of the walk is deﬁned as p + f(p) ∈ g. It is easy to keep track of the u, v such that p = ug + vh. Such an r-adding walk results in anexpected number of steps until a collision occurs that is somewhat larger than πq/2, as shown by Brent and Pollard [4] i and expanded upon in [2]. Assume that is perfectly random. Let pi = #G q . A point in the walk is said to belong to class i if its predecessor upon its ﬁrst occurrence belongs to Gi . If the nth point belongs to Gj (with probability pj ) and the (n + 1)st point produces the ﬁrst collision, the collision point cannot be of class j (this happens with probability pj ), since then the collision would have occurred in step n. Therefore, the probability that the ﬁrst collision occurs at step n + 1 is r−1 n (1 − p2j ). q j=0 With q =

1−

q r−1 j=0

p2j

this is

n q .

We get via the same arguments referred to above

πq = 2

2(1 −

πq r−1 j=0

p2j )

(1)

for the expected number of steps until the ﬁrst collision. Pollard [15] uses r = 3, f0 = h, and f2 = g, but replaces the i = 1 case by the doubling 2p. Teske [16] shows that a larger r, such as r = 20, leads to better performance on average, conform the analysis, even if none of the choices does an explicit doubling, as Pollard’s i = 1 case. Inclusion of doublings leads to r + s-mixed walks: with : g → [0, r + s − 1] q partitioning g into r + s parts of cardinality close to r+s , the next point equals p + f(p) if 0 ≤ (p) < r, but 2p if (p) ≥ r. Pollard’s walk is a 2 + 1-mixed walk. The analysis above applies again, assuming that we consider the doublings as one class, hit with probability pD . Experiments by Teske show that best performance is achieved for rs between 14 and 12 but that apart from the case r = 3 mixed

On the Use of the Negation Map in the Pollard Rho Method

69

walks are not signiﬁcantly better. The analysis and our own experiments, as reported below, suggest that the optimal ratio rs is close to zero. Per step the occurrence probability of the event p = fi (and thus a chance to solve the discrete logarithm problem) is negligible compared to the probability of a birthday collision. So, for r-adding walks doublings most likely will not occur. Parallelized random walks. Parallelization of Pollard’s rho method does not consist of running any number of random walks in parallel, until one of √ them collides: on M processors the expected speedup would be by a factor of M , so √ overall it would require M more processing power than a single processor. The proper way to parallelize Pollard’s rho method is presented in [17]. It achieves an M -fold speedup on M processors, thus requiring the same overall processing 1 power as a single process, but in M th of the time. Diﬀerent processes must be able to eﬃciently recognize if, probably at diﬀerent points in time, their walks collide. To achieve this, each process generates a single random walk, each from its own random starting point, but all using the same index function and the same fi ’s. As soon as a walk hits upon a distinguished point, this point is reported. The idea is that when two walks collide – without noticing it – they will keep taking the same steps (because they use the same walk deﬁnition) and will thus both ultimately reach the same distinguished point. This will be noticed when the colliding distinguished point is reported. The discrete logarithm can then be computed from the two, hopefully distinct, pairs of integer multipliers u, v that correspond to the same distinguished point. A distinguished point must be easy to recognize, occur with low enough probability to make it possible to store them all and to eﬃciently ﬁnd collisions, but occur often enough for every walk to hit one. The distinguishing property could be that k speciﬁc bits of the point’s x-coordinate are zero, in which case walks may hit a distinguished point once every 2k steps. The parallelized version of Pollard’s rho method requires a unique, and thus aﬃne, point representation to make the walks well-deﬁned and to recognize distinguished points. The fastest suitable type of elliptic curve group arithmetic uses the aﬃne Weierstrass point representation. Per group operation, it requires a (usually expensive) modular inversion. Its cost is amortized among the walks running in parallel per processor, at the cost of three modular multiplications per step per walk, using Montgomery’s simultaneous inversion [14]. Point doubling requires an extra modular squaring compared to regular non-doubling point addition. This makes doubling on average about 76 times slower than regular addition when parallelized walks and simultaneous inversion are used. Using automorphisms. Following [18], deﬁne an equivalence relation ∼ on g by p ∼ −p for p ∈ g and, instead of searching g of size q, search g/∼ of size about q2 . Denoting the equivalence class containing p and −p by ∼p, it may be represented by the element with y-coordinate of least absolute value. It is trivial to calculate since −(x, y) = (x, −y) for (x, √y) ∈ g. Thus, using this negation map one would expect to save a factor of 2 in the number of steps. √ For r-adding and r + s-mixed walks the speedup by a factor of 2 is slightly too pessimistic. Let the deﬁnitions of pi , pD , and of class i be as above. Assume

70

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Table 1. Number of steps required by the Pollard rho method in random elliptic curve groups of 31-bit q over prime ﬁelds of random 31-bit prime characteristic p, prime order divided by πq/2 or by πq/4 (without or with the negation map). Lowest and highest averages are over 10 measurements. Each measurement calculates the average number of steps taken until a collision occurs, over 100 000 collision searches where for each search a prime p and an elliptic curve over Fp are randomly selected until the order q of the group of points is prime. Overall average is the average of the 10 averages (thus, the average over one million searches). Expression (1) and (2) columns are the quotients as 1 expected based on expressions (1) (with pi = 1r for 0 ≤ i < r) and (2) (with pi = r+s s for 0 ≤ i < r and pD = r+s ), respectively. Those expressions are for q → ∞ and indeed for larger (smaller) q they give a better (worse) ﬁt. Without negation map Averages Expression lowest overall highest (1) 8-adding 1.079 1.083 1.085 1.069 16-adding 1.032 1.037 1.040 1.033 32-adding 1.014 1.018 1.019 1.016 16 + 4-mixed 1.041 1.043 1.044 1.043 16 + 8-mixed 1.075 1.078 1.081 1.078

With negation map Averages Expression lowest overall highest (2) 1.035 1.039 1.042 1.033 1.015 1.017 1.020 1.016 1.007 1.009 1.011 1.008 1.036 1.038 1.040 1.031 1.075 1.077 1.079 1.069

that the nth point belongs to Gj and that the (n + 1)st point produces the ﬁrst collision while hitting the representative p, directly or after negation. If this step is a doubling then the analysis is as above. This happens with probability p2D . Otherwise, we only exclude the case that, as a result of just the addition, the two predecessors hit the same point (p or −p). This happens with probability Therefore, the probability that the ﬁrst collision occurs at step n + 1 is

p2j 2 .

r−1 2 pj 2n (1 − p2D − ). q 2 j=0

As above we get

πq 4(1 −

p2D

−

1 2

r−1 j=0

p2j )

(2)

for the expected number of steps until√the ﬁrst collision. For the same parameter values this expression is more than 2 smaller than Expression (1). However, usage of the negation map requires modiﬁcations to the iteration function due to the occurrence of fruitless cycles. This disadvantage of the negation map was already pointed out in [9,18]. It is the focus of this article. The group g may admit other trivially computable maps. For Koblitz curves the √ Frobenius automorphism of a degree t binary extension ﬁeld leads to a further t-fold speedup. This does not apply to the case considered here. Small scale experiments. We checked the accuracy of predictions based on expressions (1) and (2). The results, for 31-bit primes q, are listed in Table 1.

On the Use of the Negation Map in the Pollard Rho Method

71

With all averages larger than 1, both r-adding and r + s-mixed walks on average perform worse than truly random walks. For most walks with the negation map the averages are lower than their negation-less counterparts, indicating√that the reduction factor in the expected number of steps is indeed larger than 2. This does not imply a speedup by the same factor, because to obtain the ﬁgures costly fruitless cycle detection methods had to be used. It can be seen that r + s-mixed walks are disadvantageous if s > r4 . 2.3

Fruitless Cycles

Straightforward application of the negation map to Pollard’s rho method with r-adding or r + s-mixed walks does not work due to fruitless cycles. This section describes the current state-of-the-art of dealing with those cycles. Length 2 cycles. If a random walk step goes from p to −p − fi (with probability 12 , for some i) and −p − fi ∈ Gi (with probability 1r ), then the next point after −p − fi is p again (with probability 1), thereby cancelling the eﬀect of the previous step. It follows that a fruitless 2-cycle starts from a random point with 1 probability 2r , cf. [7, Proposition 31]. This 2-cycle is denoted as (i,−)

(i,−)

p −→ −(p + fi ) −→ p. Here “(i, s)” with s ∈ {−, +} indicates that addition constant fi is added to a point p after which the result is left as is (s = +) or negated (s = −) to ﬁnd the correct representative (p + fi if s = +, or −p − fi if s = −). Any walk with two consecutive steps “(i, −)” is trapped in an inﬁnite loop. Because this happens 1 with probability 2r , all walks can be expected to end up in fruitless cycles after a moderate number of steps when the negation map is used with r-adding walks. Looking ahead to reduce 2-cycles. To reduce the occurrence of 2-cycles, Wiener and Zuccherato propose to use a more costly iteration function that results in a lower probability that two successive points belong to the same partition [18]. This can be achieved by using the ﬁrst i of (p), (p) + 1, . . ., (p) + r − 1 such that i mod r = (∼ (p + fi )), if such an index exists (here and in the sequel indices i in fi are understood to be taken modulo r). Thus, deﬁne the next point as f (p) with f : g → g deﬁned by E(p) if j = (∼(p + fj )) for 0 ≤ j < r f (p) = ∼(p + fi ) with i ≥ (p) minimal s.t. (∼(p + fi )) = i mod r. The function E : g → g may restart the walk at a new random initial point. The latter is expected to happen once every rr steps and will therefore not aﬀect the The expected cost per step of the walk is increased by a factor of r eﬃciency. 1 1 1 i=0 r i , which lies between 1 + r and 1 + r−1 . Dealing with fruitless cycles in general. Although the look-ahead technique reduces the frequency of 2-cycles, they may still occur [18]. This is elaborated upon in Section 3. Even so, it is well known that just addressing 2-cycles does

72

J.W. Bos, T. Kleinjung, and A.K. Lenstra 4.5e+06 4e+06

steps / second

3.5e+06 3e+06 2.5e+06 2e+06 1.5e+06 1e+06 500000 0

2

4

6

8

10

log2 (r)

12

14

16

18

Fig. 1. Total number of steps per second as a function of r, taken by 200 parallel r-adding walks sharing the modular inversion and not using the negation map, for Pollard’s rho method applied to a 131-bit prime ECDLP

not solve the problem of fruitless cycles, because longer cycles will occur as well. Reducing their occurrence requires additional overhead on top of what is already incurred to reduce 2-cycles. Given that fruitless cycles are unavoidable, they must be eﬀectively dealt with when they occur. In [9] a general approach is proposed to detect cycles and to escape from them: after α steps record a length β sequence of successive points and compare the next point to these β points. If a cycle is detected a cycle representative p is chosen deterministically from which the cycle is escaped. One may add f(p)+c for a ﬁxed c ∈ [2, r − 1] (the choice c = 1 is bad as it could lead to an immediate cycle recurrence). Instead one may add a distinct precomputed value f that does not depend on the escape-point, or one may add f(p) from a distinct list of r precomputed values f0 , f1 , . . . , fr−1 . In the next section we discuss fruitless cycles in greater detail and propose alternative methods that avoid problems that the method from [9] may run into.

3

Improved Fruitless Cycle Handling

The probability to enter a fruitless cycle decreases with increasing r [7]. This does not imply that it suﬃces to take r large enough to make the probability sufﬁciently low. Fig. 1 depicts the eﬀect of increasing r-values on the performance of an r-adding walk, measured as number of steps per second. The performance deterioration can be attributed to the increasing rate of cache misses during retrieval of the addition constants fi . The eﬀect varies between processors, implementations, and elliptic curves. It is worsened for more contrived walks, such as those using the negation map where cycle reduction, detection and escape methods are unavoidable. Unless the expected overall number of steps (of or√ der q) is too small to be of interest, r cannot be chosen large enough to both

On the Use of the Negation Map in the Pollard Rho Method

(i,−)

(i,−) −p−fi =q

p (i,−)

(i−1, ..)

(∼(p+fi−1 )) = i−1

73

(i−1, ..)

(∼(q+fi−1 )) = i−1

−p−fi =q

p (i−1, ..)

(i,−)

¯ p= ∼(p+fi−1 )

¯q = ∼(q+fi−1 )

(j, ..) (∼(¯p +fj )) ∈ {i−1, j}

(i−1, ..)

(k, ..) (∼(¯q +fk )) ∈ {i−1, k}

Fig. 2. 2-cycles caused by 2-cycle reduction (left) and 4-cycle reduction. The dotted steps are prevented.

avoid fruitless cycles and achieve adequate performance. Therefore, in this section we concentrate on other ways to deal with fruitless cycles. We ﬁrst discuss short-cycle reduction techniques, next discuss cycle detection methods and analyze their behavior, and ﬁnally propose alternative methods. 3.1

Short Fruitless Cycle Reduction

2-cycles. Unfortunately, the look-ahead technique to reduce 2-cycles presented above introduces new 2-cycles. The dotted lines in the left example in Fig. 2 are the steps taken by the regular iteration function, the new cycle is depicted by the solid lines which are the steps taken as a result of f (p) and f (q). This new cycle occurs with probability 2r13 . It is the most likely 2-cycle introduced by the look-ahead technique. Lemma 1. The probability to enter a fruitless 2-cycle when looking ahead to reduce 2-cycles while using an r-adding walk is 1 2r

r−1 2

1 (rr−1 − 1)2 1 1 . = = + O i 2r−1 2 3 r 2r (r − 1) 2r r4 i=1

Proof. With i as in the deﬁnition of f , the probability is r−c that i ≥ (p) + c for 0 ≤ c < r (considering the case E(p) as i = ∞), hence i = (p) + c with 1 probability r−1 r rc . We compute the probability of entering a cycle consisting of points p and q starting at p. Let j = (p) and k = (q), and let the steps from p to q and back be adding fj+c and fk+d , respectively. This implies that j + c ≡ k + d mod r and that the step from p to q involves a negation. From the deﬁnition of f it follows

74

J.W. Bos, T. Kleinjung, and A.K. Lenstra

(∼(˜ p + fk )) ∈ {i, k} (k, ..) ˜ p =∼(p + fi ) (i, ..) p

(∼(˜q + fn ) ∈ {j, n} (n, ..) ∼(−p − fj+1 + fj ) = ˜q (j + 1,−)

(i + 1,+)

(j, ..) −p − fj+1 (i + 1,+) −p − fi+1 − fj+1

p + fi+1 (j, ..)

(j + 1,−)

¯ p =∼(p + fi+1 + fj ) (l, ..) (∼(¯ p + fl )) ∈ {j, l}

(i, ..) ∼(−p − fi+1 − fj+1 + fi ) = ¯q (m, ..) (∼ (¯q + fm )) ∈ {i, m}

Fig. 3. A 4-cycle when the 4-cycle reduction method is used

that (q) ≡ j + c mod r, thus d = 0 and by symmetry c = 0. Since j is given and k is determined by j, c and d, the probabilities must be summed over all possible c and d. The probability for a c, d pair is the product of the following probabilities: • • •

r−1 1 r rc 1 2 1 r−1

•

1 rd

for the ﬁrst step being c; for the sign; for (∼(p + fj+c )) = k (we know already that (∼(p + fj+c )) ≡ j + c ≡ k mod r); for the second step being d (since (∼(q + fk+d )) ≡ k + d mod r).

This results in the probability

r−1 r−1 1 1 1 . 2r c=1 rc rd

d=1

We conclude that, even when the look-ahead technique is used, 2-cycles are still too likely to occur for relevant values of q and r. Some of the new 2-cycles are prevented by other short-cycle reduction methods, but the remaining ones must be dealt with using detection and escape methods. This is discussed below. 4-cycles. Unless the addition constants fi have been chosen poorly, 3-cycles do not occur as a direct result of the negation map, so that 4-cycles are the next type of short cycles to be considered. Excluding again that the fi have unlikely properties, a fruitless 4-cycle without proper sub-cycle is of the form (i,+)

(j,−)

(i,+)

(j,−)

p −→ p + fi −→ −p − fi − fj −→ −p − fj −→ p. The cycle may be entered at any of its four points. Hence, a fruitless 4-cycle starts from a random point with probability r−1 4r 3 . This is a lower bound for the probability of occurrence of 4-cycles when looking ahead to reduce 2-cycles.

On the Use of the Negation Map in the Pollard Rho Method

75

An extension of the 2-cycle reduction method looks ahead to the ﬁrst two successors of a point, thereby reducing the frequency of 2-cycles and 4-cycles, while still being deterministic: ⎧ E(p) if j ∈ {(q), (∼(q + f(q) ))} or (q) = (∼(q + f(q) )) ⎪ ⎪ ⎨ where q =∼(p + fj ), for 0 ≤ j < r, g(p) = q =∼(p + fi ) with i ≥ (p) minimal s.t. ⎪ ⎪ ⎩ i mod r = (q) = (∼(q + f(q) )) = i mod r. Compared to f (p), the probability that E is called increases from ( 1r )r to at least ( 2r )r because (∼(q + f(q) )) ∈ {j mod r, (q)} with probability 2r for each j. This times slower than the standard one, because iteration function is at least r+4 r with probability 2r at least two additional group operations need to be carried 1 2 out, an eﬀect that is slightly alleviated by a factor of ( r−1 r ) since the image of g is a subset of g of cardinality approximately r−1 q. The value ∼(q+f(q)) can be r stored for use in the next iteration. Usage of g reduces the occurrence of 4-cycles, and also prevents some of the 2-cycles newly introduced by the 2-cycle reduction method (such as the one depicted on the left in Fig. 2). But g introduces new types of 2-cycles and 4-cycles as well, both of which do indeed occur in practice. A newly introduced 2-cycle is shown in the right example in Fig. 2. There the 2 points ¯ p and ¯ q are ∈ Gi−1 ∪ Gi . This 2-cycle occurs with probability 2(r−2) (r−1)r 4 , which is therefore a lower bound for the probability of 2-cycles when using the 4-cycle reduction method. Fig. 3 depicts an example of a newly introduced 4cycle: the points reached via dotted lines belong to a partition diﬀerent from their predecessors. The probability that such a 4-cycle starts from a random 4 (r−1) point is at least 4(r−2) . r 11 We have not been able to design or to ﬁnd in the literature short-cycle reduction methods that do not introduce other (lower probability) short cycles. We therefore turn our attention to cycle detection and escape methods. 3.2

Cycle Detection and Escape

Recurring cycles. The cycle detection and escape method from [9] described in Section 2.3, does not prevent recurrence to the same cycle. When using f(p)+c to escape (we ﬁxed c = 4 as it worked as well as any other choice = 1), Fig. 4 depicts how the (wavy) escape from the (solid) 4-cycle recurs to the 4-cycle via one of the dotted possibilities. The probability of recurrence depends on the escape method and on which point in the cycle the walk recurs to. With f(p)+c 1 as escape, immediate recurrence to the escape point happens with probability 2r 1 when no cycle reduction is used, recurrence happens with probability at least 2r2 2

with 2-cycle reduction, and with probability at least (r−2) with 4-cycle and thus r4 2-cycle reduction. Similar recurrences occur, with lower probabilities, when f or f(p) are used to escape. Lemma 2. Lower bounds for the probabilities to enter 2-cycles or 4-cycles or to recur to cycles for three diﬀerent cycle escape methods are listed in Table 2

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J.W. Bos, T. Kleinjung, and A.K. Lenstra

−p − fi − fj (j, −)

(i, +) −p − fj

p + fi (k, −)

p

(i, +)

(k, +) (j, −)

−p − fk − fj

(k, +)

−p − fi − fk

(i, −)

(j, −)

p + fk

Fig. 4. Escaping from a fruitless 4-cycle, and recurring to it (i = j = k = i)

if no cycle reduction, or 2-cycle reduction (f ), or 4-cycle reduction (g) is used, along with a lower bound for the slowdown factor caused by f or g. Proof. The proofs for many entries of Table 2 were given earlier. We prove the entries in rows four and ﬁve. Let p be the escape point and let q be the point it escapes to. Using f or f(p) one can recur to the escape point p by entering another cycle at q and escaping from it at q again. This new cycle could be a 2-cycle. For this to happen the ﬁrst escape step to q has to involve a negation (probability 12 ), a 2-cycle has to be entered at q (probabilities in ﬁrst row, but see below), the escape point of this 2-cycle has to be q (probability 12 ), and, in the case of fi , the partition that q belongs to has to be the same as the one p belongs to (probability 1r ). In the case of 4-cycle reduction the probability to enter a 2-cycle at q is slightly lower since we do not have the information that (∼(q + f(q) )) = (q); a calculation analogous to the one done at the end of Section 3.1 produces the values listed in the table. 6-cycles. With proper fi and no sub-cycle, a common 6-cycle is of the form (i,+)

(j,−)

(k,+)

(i,+)

(j,−)

(k,+)

p −→ p+fi −→ −p−fi −fj −→ −p−fi −fj +fk −→ −p−fj +fk −→ p−fk −→ p (i = j = k = i) where with appropriate sign changes steps four and ﬁve may be swapped. It may be entered at any of its six points and occurs, when using 4-cycle reduction, with probability 4r13 + O( r14 ). A lower bound to recur to it follows by multiplying this probability with the recurring probabilities from Table 2.

On the Use of the Negation Map in the Pollard Rho Method

77

Table 2. Summary of eﬀect of cycle reduction, detection, and escape methods. With the exception of the two bold entries, all ﬁgures are lower bounds. Probability to enter

Cycle reduction method:

none

2-cycle 4-cycle

⎧ ⎪ ⎨ f(p)+c Probability to recur to escape point using f ⎪ ⎩ f(p) Slowdown factor of iteration function

3.3

2-cycle

4-cycle

1 2r r−1 4r3

1 2r 3 r−1 4r 3

2(r−2)2 (r−1)r 4 4(r−2)4 (r−1) r 11

1 2r 1 8r 1 8r 2

1 2r 2 1 8r 3 1 8r 4

(r−2)2 r4 (r−2)2 2r 5 (r−2)2 2r 6

n/a

r+1 r

r+4 r

Alternative Approaches

The purpose √ of using the negation map is to obtain a speedup, hopefully by a factor of 2. From Fig. 1 it follows that large r-values cannot be used. From Table 2 it follows that for small r-values and relevant q-values fruitless cycles are likely to occur and recur. Medium r-values look the most promising, but are not compatible with all environments. Since fruitless cycle occurrence and recurrence cannot be rooted out, alternative methods are needed if we want to make the negation map useful. In this section several possibilities are oﬀered. Heuristic. A cycle with at least one doubling is most likely not fruitless. Proof. Let p = ug + vh be a point on the cycle. The subsequent points are obtained by adding one of the fi or by doubling, and negating if needed, thus are up to sign linear combinations of the fi and a power-of-two multiple of p. If c ≥ 1 is the number of doublings in the cycle, we get a relation of the form p = ±2c p +

r−1

ci fi = ±2c p +

i=0

(1 ∓ 2c )u −

r−1 i=0

r−1 i=0

ci u i

ci u i g +

r−1

g+

ci vi h

and thus

i=0

(1 ∓ 2c )v −

r−1

ci vi

h = 0,

i=0

r−1 where ci ∈ Z. Since 1 ∓ 2c = 0, the expression (1 ∓ 2c )u − i=0 ci ui is most likely not divisible by the group order. This also holds if {fi : 0 ≤ i < r} is enlarged with f or with {fi : 0 ≤ i < r}. This concludes our heuristic argument. Cycle reduction by doubling. The regular structure required for cycles is caused by repeated addition and subtraction using the same set of constants. This structure would be broken eﬀectively by using an occasional doubling, i.e., a mixed walk. If such walks are used, the heuristics suggest that cycles occur

78

J.W. Bos, T. Kleinjung, and A.K. Lenstra

only between two doublings. If the doubling frequency is suﬃciently high, only short cycles would have to be dealt with. As borne out by expressions (1) and (2) when using the idealized values pi = 1 s r+s for 0 ≤ i < r and pD = r+s for r > 0, and as supported by the experiments reported in Table 1, an r + s-mixed walk with s > 1 always displays noticeably less random behavior than a well-partitioned r -adding walk for any r > r. Nevertheless, using properly tuned r + s-mixed walks may be a way to address the cycle problem while avoiding impractically large r-values. However, r + s-mixed walks have disadvantages caused by the underlying arithmetic. Given the relative speeds of addition and doubling, an r + s-mixed walk is r+7s/6 times slower than an r-adding walk. In a SIMD environment r+s where many walks are processed simultaneously, per step a fraction of about r r+s of the walks will do an addition, whereas the others do a doubling. If the addition and doubling code diﬀer, as is the case for the aﬃne Weierstrass representation, the two types of steps cannot be executed simultaneously. Thus, in such environments, to avoid a slowdown by a factor of more than 2 one needs to swap walks to make all parallel step-operations identical (at non-negligible overhead), or one has to settle for a suboptimal aﬃne point representation that allows identical code. SIMD-application of the negation map and the possibility of another point representation are subjects for further study. Doubling based cycle reduction and escape. Taking into account that doubling should not be used too frequently, usage could be limited to cycle reduction or escape. This would not solve the SIMD-issue, but the relative ineﬃciency and non-randomness would be addressed. If doublings are used to escape from fruitless cycles, they would not recur, as that would contradict the heuristics. Cycle reduction using doubling replaces f (p) and g(p) by f¯(p) and g¯(p), respectively, where ∼(p + f(p) ) if (p) = (∼(p + f(p) )), ¯ f (p) = ∼(2p) otherwise, g¯(p) =

q =∼(p + f(p) ) if (q) = (p) = (∼(q + f(q) )) = (q), ∼(2p) otherwise.

It follows from the heuristics that these functions avoid recurring fruitless cycles. Alternative cycle detection. Because shorter cycles are more frequent, a potentially interesting modiﬁcation of the cycle detection method from [9] (described at the end of Section 2.3) would be to occasionally compare a point to its kth successor, where k is the least common multiple of all even short cycle lengths that one wants to catch. Detecting, for instance, cycles up to length 1 12 requires only 120 th comparison per step. This can be done in several steps, recording every 12th point to catch 4- and 6-cycles, recording every 10th of these recorded points to catch 8- and 10-cycles, etc. It can be combined with the regular method with large α and β to catch longer cycles infrequently. However, if a cycle has been detected the k points need to be recorded as before, so an escape point can be chosen deterministically. This argues against

On the Use of the Negation Map in the Pollard Rho Method

79

using large k. It also suggests that an improvement can be expected only if cycles occur with low probability, and therefore that the improvement will be marginal at best (cf. α and β choices in Section 4). For this reason we did not conduct extensive experiments with this method.

4

Comparison

We implemented and compared on a traditional non-SIMD platform all previously published and newly proposed methods to deal with fruitless cycles when using the negation map. Here we report on our ﬁndings. It quickly turned out that the cycle detection methods from [9] when combined with doubling based cycle reduction and escape, are considerably more eﬃcient than r+s-mixed walks with their on average slower steps and less random behavior. Mixed walks are therefore not further discussed. Experiments with the alternative cycle detection method were quickly abandoned as well. For each combination of iteration function, escape method, and r-value a search was conducted to determine the α and β to be used for the cycle detection method from [9]. Using a heuristic argument that for β = 2k with k much smaller than r, cycles of length ≥ β occur with probability on the order of (k−1)! , values (2r)k for k that make this probability low enough resulted in good initial values for the search for close to optimal α and β. To give some examples, for “f , e,” as explained in Table 3 we used α = 31 and β = 20 for r = 16, α = 3264 and β = 12 for r = 128, and α = 52 418 and β = 10 for r = 256. For “f¯, ¯e” and the same r-values we used the same β-values but replaced the α-values by 1 618, 838 848, and 53 687 081, respectively. Each of the benchmarks presented in Table 3 was run on a single core of an AMD Phenom 2.2GHz 4-core processor, with each of the four cores processing a diﬀerent combination. A 10-bit distinguishing property was used to get a signiﬁcant amount of data in a reasonable amount of time. This somewhat aﬀects the performance, but not the cycle behavior as walks continue after hitting a distinguished point. The ﬁgures in millions as given in the table are thus an underestimate for the actual per-core yield in units when a more realistic 30-bit distinguishing property would be used (since 230 /210 = 220 ≈ 106 ). In order to be able to compare the long term yield ﬁgures, the expected number of steps must be taken into account using expressions 1 and 2. As a 1 2 result, the yields are corrected by a factor of ( r−1 r ) for the iteration functions 1 2 that do not use the negation map, and by a factor of ( 2r−1 r ) for the others, with 1 r an extra factor of ( r−1 ) 2 for g and g¯. After this correction, the best iteration function without the negation map is the one with r = 64. Comparing that one with each iteration function that uses the negation map, thus boosting the 63 12 63 12 latter’s yield ratio by a factor of C = (( 2r−1 or C = (( 2r−1 r )/( 64 )) r−1 )/( 64 )) for g and g¯, leads to the long term speedup ﬁgure given in Table 3. Note that the correction factor√C depends on the iteration function, and is close to and for some r larger than 2.

80

J.W. Bos, T. Kleinjung, and A.K. Lenstra

Table 3. For the (iteration function, escape method, r-value) combinations speciﬁed, the non-italics entries list the long term yield (millions of distinguished points, found during the second half hour) and the long term speedup over the best r-value (r = 64) without the negation map, taking into account the correction factor C as explained in the text. Cycle detection and subsequent escape by adding f(p)+4 , f , f(p) and by doubling is indicated by “e,” “e ,” “e ” and by “¯e,” respectively. The iteration functions f (2-cycle reduction), g (4-cycle and 2-cycle reduction), f¯ (2-cycle reduction using doubling), and g¯ (4-cycle and 2-cycle reduction using doubling) are as in sections 2.3, 3.1 and 3.3. The yields are for 256 parallel walks (sharing the inversion) for a 131-bit ECDLP with a 131-bit prime order group. The yields during the ﬁrst half hour are almost consistently higher, considerably so for poorly performing combinations. They are not meaningful and are thus not listed. The italics entries are A above D, followed 9 −A) , as explained in the text. by the maximal achievable speedup factor of C(10 109 +D/6 †: This applies to “no reduction, no escape,” “just f ,” “just f¯,” “just e,” and “just e .” r = 16

r = 32

Without negation map 7.29: 0.98 7.28: 0.99 With negation map 0.00: 0.00 0.00: 0.00 † 0.00: 0.00 0.00: 0.00 just g 0.00: 0.00 0.00: 0.00 just g ¯ just e 0.00: 0.00 0.00: 0.00 3.34: 0.64 4.89: 0.95 just ¯ e 0.00: 0.00 0.00: 0.00 f, e 9 .4e8 6 .6e8 0 .0e0 }0 .48 0 .0e0 }0 .08 0.00: 0.00 3.24: 0.63 f , e 8 .0e7 3 .9e8 0 .0e0 }0 .86 0 .0e0 }1 .30 0.00: 0.00 5.34: 1.04 f , e 6 .0e7 1 .3e8 0 .0e0 }1 .22 0 .0e0 }1 .33 3.71: 0.72 6.36: 1.24 f, ¯ e 9 .2e7 6 .8e7 9 .9e5 }1 .27 2 .8e5 }1 .32 0.00: 0.00 0.01: 0.00 g, e 8 .7e8 3 .7e8 0 .0e0 }0 .19 0 .0e0 }0 .91 0.00: 0.00 0.01: 0.00 g, e 7 .8e8 3 .0e8 0 .0e0 }1 .00 0 .0e0 }0 .32 0.00: 0.00 1.09: 0.21 g, e 7 .6e8 1 .2e8 0 .0e0 }1 .27 0 .0e0 }0 .34 0.76: 0.15 5.91: 1.17 g, ¯ e 1 .7e8 3 .3e8 1 .6e5 }0 .97 6 .0e4 }1 .19 0.00: 0.00 0.00: 0.00 f¯, e 8 .7e8 4 .3e8 2 .4e6 }0 .18 1 .7e7 }0 .80 0.01: 0.0 4.24: 0.82 f¯, e 6 .8e7 2 .6e8 4 .3e7 }1 .03 2 .9e7 }1 .31 1.34: 0.26 5.80: 1.13 f¯, e 8 .9e7 5 .3e7 5 .2e7 }1 .27 2 .9e7 }1 .33 5.58: 1.06 6.14: 1.18 f¯, ¯ e 6 .1e7 3 .7e7 4 .2e7 }1 .31 3 .0e7 }1 .36 2.56: 0.51 5.80: 1.15 g ¯, e 1 .4e8 7 .9e7 9 .9e7 }1 .23 5 .6e7 }1 .31 4.74: 0.94 5.88: 1.16 g ¯, e 7 .8e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31 4.72: 0.94 5.80: 1.15 g ¯, e 7 .7e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31 4.83: 0.96 5.87: 1.16 g ¯, ¯ e 7 .9e7 1 .2e8 1 .0e8 }1 .25 5 .6e7 }1 .31

r = 64

r = 128

r = 256

r = 512

7.27: 1.00

7.19: 0.99

6.97: 0.96

6.78: 0.94

0.00: 0.00 0.00: 0.00 0.00: 0.00 0.00: 0.00 5.85: 1.14 1.52: 0.30 1 .0e8 0 .0e0 }1 .28 6.04: 1.18 4 .6e7 0 .0e0 }1 .35 6.21: 1.21 4 .2e7 0 .0e0 }1 .36 6.50: 1.27 4 .2e7 6 .5e4 }1 .36 4.89: 0.96 6 .6e7 0 .0e0 }1 .34 5.32: 1.05 6 .0e7 0 .0e0 }1 .35 5.37: 1.13 6 .0e7 0 .0e0 }1 .35 6.02: 1.18 8 .1e7 8 .1e3 }1 .32 2.70: 0.53 5 .4e7 1 .5e7 }1 .34 6.32: 1.23 3 .9e7 1 .5e7 }1 .36 6.23: 1.22 3 .9e7 1 .5e7 }1 .36 6.34: 1.23 1 .8e7 1 .5e7 }1 .39 6.02: 1.18 5 .1e7 2 .9e7 }1 .35 6.14: 1.21 5 .3e7 2 .9e7 }1 .35 6.08: 1.20 5 .3e7 2 .9e7 }1 .35 6.09: 1.20 5 .2e7 2 .9e7 }1 .35

0.00: 0.00 0.00: 0.00 0.75: 0.15 0.61: 0.12 6.10: 1.19 5.93: 1.16 3 .6e7 0 .0e0 }1 .37 6.41: 1.25 3 .3e7 0 .0e0 }1 .38 6.30: 1.23 3 .3e7 0 .0e0 }1 .38 6.57: 1.29 3 .3e7 1 .5e4 }1 .38 6.22: 1.22 4 .2e7 0 .0e0 }1 .37 6.26: 1.23 4 .1e7 0 .0e0 }1 .37 6.08: 1.20 4 .2e7 0 .0e0 }1 .37 6.25: 1.23 5 .4e7 1 .0e3 }1 .35 5.96: 1.16 1 .1e7 7 .7e6 }1 .41 6.43: 1.26 3 .2e7 7 .6e6 }1 .38 6.21: 1.22 3 .6e7 7 .5e6 }1 .37 6.42: 1.25 1 .1e7 7 .7e6 }1 .41 6.09: 1.20 4 .1e7 1 .5e7 }1 .37 6.28: 1.23 3 .9e7 1 .5e7 }1 .37 6.05: 1.19 3 .8e7 1 .5e7 }1 .37 6.16: 1.21 4 .0e7 1 .5e7 }1 .37

0.00: 0.00 0.04: 0.01 4.90: 0.96 4.94: 0.97 6.28: 1.23 6.47: 1.27 2 .9e7 0 .0e0 }1 .38 6.29: 1.23 2 .9e7 0 .0e0 }1 .38 6.20: 1.21 2 .9e7 0 .0e0 }1 .38 6.47: 1.27 2 .9e7 3 .8e3 }1 .38 6.23: 1.22 3 .3e7 0 .0e0 }1 .38 6.25: 1.23 3 .0e7 0 .0e0 }1 .38 6.06: 1.19 3 .5e7 0 .0e0 }1 .38 6.13: 1.20 4 .0e7 1 .2e2 }1 .37 6.34: 1.24 1 .0e7 3 .9e6 }1 .41 6.33: 1.24 2 .8e7 3 .8e6 }1 .38 6.15: 1.20 2 .8e7 3 .8e6 }1 .38 6.27: 1.23 1 .0e7 3 .9e6 }1 .41 6.19: 1.21 2 .6e7 7 .6e6 }1 .39 6.05: 1.19 2 .6e7 7 .6e6 }1 .39 5.91: 1.16 1 .8e7 7 .6e6 }1 .40 6.09: 1.20 2 .6e7 7 .6e6 }1 .39

0.00: 0.00 3.59: 0.70 5.90: 1.16 5.73: 1.12 6.18: 1.21 6.36: 1.25 2 .5e7 0 .0e0 }1 .39 6.21: 1.22 2 .6e7 0 .0e0 }1 .39 5.99: 1.17 2 .7e7 0 .0e0 }1 .39 6.30: 1.25 2 .7e7 9 .7e2 }1 .39 6.05: 1.19 1 .3e7 0 .0e0 }1 .41 6.11: 1.20 5 .5e7 0 .0e0 }1 .35 5.86: 1.15 4 .3e7 0 .0e0 }1 .37 6.00: 1.18 2 .7e7 9 .0e0 }1 .39 6.20: 1.21 1 .4e7 1 .9e6 }1 .40 6.20: 1.22 2 .7e7 1 .9e6 }1 .39 6.00: 1.18 2 .6e7 1 .9e6 }1 .39 6.07: 1.19 1 .4e7 1 .9e6 }1 .40 5.74: 1.13 7 .7e6 3 .9e6 }1 .41 5.80: 1.14 7 .7e6 3 .9e6 }1 .41 5.67: 1.11 7 .7e6 3 .9e6 }1 .41 5.70: 1.12 7 .7e6 3 .9e6 }1 .41

On the Use of the Negation Map in the Pollard Rho Method

81

Non-doubling 2-cycle reduction (f ) with doubling-based cycle escape (¯e) and r = 128 performed best, with an overall speedup by a factor of 1.29: although fewer distinguished points are found than for the best case without the negation map (r = 64), there is a considerable overall gain because fewer distinguished points (by a factor of C, for the relevant C) should suﬃce. For r = 16 most iteration functions with the negation map perform poorly. √ We measured to what extent our failure to achieve a speedup by a factor of 2 can be blamed on cycle detection and escape and other overheads, and which part is due to the higher average cost of the iteration function. For most combinations in Table 3 we counted the number S of useful steps performed when doing 109 group operations, while keeping track of the number D of doublings among them. Here a step is useful if it is not taken as part of a fruitless cycle, so all D doublings are useful. Without the negation map, S would be 109 and D = 0; this is the basis for the comparison. With the negation map, A = 109 − S is counted as the number of additional additions due to cycle reductions or fruitless cycles. The inherent slowdown of that iteration function is then 1 + A+D/6 , so that it S 9

−A) CS = C(10 can achieve a speedup by a factor of at most S+A+D/6 109 +D/6 , with C as deﬁned above. Based on Table 3 and Fig. 1, we√conclude that our failure to better approach the optimal speedup by a factor of 2 is due to an onset of cache eﬀects combined with various overheads. The italics ﬁgures from Table 3 make us believe that improvements may be obtained when using better implementations.

Previous results. The only publication that we know that presents practical data about Pollard’s rho method used with the negation map is [8]. Only relatively small ECDLPs were solved (42- and 43-bit prime ﬁelds) and small r-values were avoided. The adverse cycle behavior that we witnessed can therefore not be expected and we doubt if the results reported are signiﬁcant for the sizes that we consider. Only mixed walks were used, and an overall speedup by a factor of about 1.35 was reported. Cycle escaping was done by jumping to the sum of all points in a cycle, which cannot be expected to work in general because the sum may depend just on the addition constants.

5

Conclusion

With judicious application of doubling, usage of the negation map to solve ECDLPs over prime ﬁelds using Pollard’s rho method can indeed be recommended. In the best of circumstances that √ we have been able to create, however, the speedup falls short of the hoped for 2, but is with 1.29 still considerable. This conclusion does not apply to SIMD-environments where occasional doublings cause considerable delays. Alternative point representations need to be considered to assess the usefulness of the negation map for SIMD platforms, in particular because such platforms are becoming popular again. Acknowledgements. This work was supported by the Swiss National Science Foundation under grant numbers 200021-119776 and 206021-117409 and

82

J.W. Bos, T. Kleinjung, and A.K. Lenstra

by EPFL DIT. We gratefully acknowledge useful suggestions by Marcelo E. Kaihara and very insightful comments by the ANTS reviewers.

References 1. Avanzi, R.M., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC (2006) 2. Bailey, D.V., et al.: Breaking ECC2K-130. In: Cryptology ePrint Archive, Report 2009/541 (2009), http://eprint.iacr.org/ 3. Bos, J.W., Kaihara, M.E., Montgomery, P.L.: Pollard rho on the PlayStation 3. In: Workshop record of SHARCS 2009, pp. 35–50 (2009), http://www.hyperelliptic.org/tanja/SHARCS/record2.pdf 4. Brent, R.P., Pollard, J.M.: Factorization of the eighth Fermat number. Math. Comp. 36(154), 627–630 (1981) 5. Certicom. Certicom ECC Challenge (1997), http://www.certicom.com/images/pdfs/cert_ecc_challenge.pdf 6. Certicom. Press release: Certicom announces elliptic curve cryptosystem (ECC) challenge winner (2002), http://www.certicom.com/index.php/2002-press-releases/ 38-2002-press-releases/340-notre-dame-mathematician-solveseccp-109-encryption-key-problem-issued-in-1997 7. Duursma, I.M., Gaudry, P., Morain, F.: Speeding up the discrete log computation on curves with automorphisms. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 103–121. Springer, Heidelberg (1999) 8. Escott, A.E., Sager, J.C., Selkirk, A.P.L., Tsapakidis, D.: Attacking elliptic curve cryptosystems using the parallel Pollard rho method. CryptoBytes Technical Newsletter 4(2), 15–19 (1999), ftp.rsasecurity.com/pub/cryptobytes/crypto4n2.pdf 9. Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Improving the parallelized Pollard lambda search on anomalous binary curves. Math. Comp. 69(232), 1699–1705 (2000) 10. Harley, R.: Elliptic curve discrete logarithms project, http://pauillac.inria.fr/~harley/ 11. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987) 12. Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992) 13. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986) 14. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48, 243–264 (1987) 15. Pollard, J.M.: Monte Carlo methods for index computation (mod p). Math. Comp. 32, 918–924 (1978) 16. Teske, E.: On random walks for Pollard’s rho method. Math. Comp. 70(234), 809– 825 (2001) 17. van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. Journal of Cryptology 12(1), 1–28 (1999) 18. Wiener, M.J., Zuccherato, R.J.: Faster attacks on elliptic curve cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999)

An O(M (n) log n) Algorithm for the Jacobi Symbol Richard P. Brent1 and Paul Zimmermann2 1 2

Australian National University, Canberra, Australia INRIA Nancy - Grand Est, Villers-l`es-Nancy, France

Abstract. The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M (n) log n), using Sch¨ onhage’s fast continued fraction algorithm combined with an identity due to Gauss. We give a diﬀerent O(M (n) log n) algorithm based on the binary recursive gcd algorithm of Stehl´e and Zimmermann. Our implementation — which to our knowledge is the ﬁrst to run in time O(M (n) log n) — is faster than GMP’s quadratic implementation for inputs larger than about 10000 decimal digits.

1

Introduction

We want to compute the Jacobi symbol1 (b|a) for n-bit integers a and b, where a is odd positive. We give three algorithms based on the 2-adic gcd from Stehl´e and Zimmermann [13]. First we give an algorithm whose worst-case time bound is 3 ); we call this the cubic algorithm although this is pessimistic O(M (n)n2 ) = O(n since the algorithm is quadratic on average as shown in [5], and probably also in the worst case. We then show how to reduce the worst-case to O(M (n)n) = 2 ) by combining sequences of “ugly” iterations (deﬁned in Section 1.1) into O(n one “harmless” iteration. Finally, we obtain an algorithm with worst-case time O(M (n) log n). This is, up to a constant factor, the same as the time bound for the best known algorithm, apparently never published in full, but sketched in Bach [1] and in more detail in Bach and Shallit [2] (with credit to Bachmann [3]). The latter algorithm makes use of the Knuth-Sch¨ onhage fast continued fraction algorithm [9] and an identity of Gauss [6]. Although this algorithm has been attributed to Sch¨ onhage, Sch¨ onhage himself gives a diﬀerent O(M (n) log n) algorithm [10,15] which does not depend on the identity of Gauss. The algorithm is mentioned in Sch¨ onhage’s book [11, §7.2.3], but no details are given there. With our algorithm it is not necessary to compute the full continued fraction or to use the identity of Gauss for the Jacobi symbol. Thus, it provides an alternative that may be easier to implement. 1

Notation: we write the Jacobi symbol (b|a), since this is easier to typeset and as less ambiguous than the more usual ab . Also, M (n) is the time to multiply n-bit (n)) means O(f (n)(log f (n))c ) for some constant c ≥ 0. numbers, and O(f

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 83–95, 2010. c Springer-Verlag Berlin Heidelberg 2010

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It is possible to modify some of the other fast GCD algorithms considered by M¨ oller [8] to compute the Jacobi symbol, but we do not consider such possibilities here. At best they give a small constant factor speedup over our algorithm. We recall the main identities satisﬁed by the Jacobi symbol: (bc|a) = (b|a)(c|a); 2 (2|a) = (−1)(a −1)/8 ; (b|a) = (−1)(a−1)(b−1)/4 (a|b) for a, b odd; and (b|a) = 0 if (a, b) = 1. Note that all our algorithms compute (b|a) with b even positive and a odd positive. For the more general case where b is any integer, we can reduce to b even and positive using (b|a) = (−1)(a−1)/2 (−b|a) if b is negative, and (b|a) = (b+a|a) if b is odd. We ﬁrst describe a cubic algorithm to compute the Jacobi symbol. The quadratic algorithm in Section 2 is based on this cubic algorithm, and the subquadratic algorithm in Section 3 uses the same ideas as the quadratic algorithm but with an asymptotically fast recursive implementation. For a ∈ Z, the notation ν(a) denotes the 2-adic valuation ν2 (a) of a, that is the maximum k such that 2k |a, or +∞ if a = 0. 1.1

Binary Division with Positive Quotient

Throughout the paper we use the binary division with positive quotient deﬁned by Algorithm 1.1. Compared to the “centered division” of [13], it returns a quotient in [1, 2j+1 − 1] instead of in [1 − 2j , 2j − 1]. Note that the quotient q is always odd. Algorithm 1.1. BinaryDividePos Input: a, b ∈ N with ν(a) = 0 < ν(b) = j Output: q and r = a + qb/2j such that 0 < q < 2j+1 , ν(b) < ν(r) 1: q ← −a/(b/2j ) mod 2j+1 q is odd and positive 2: return q, r = a + qb/2j .

With this binary division, we deﬁne Algorithm CubicBinaryJacobi, where the fact that the quotient q is positive ensures that all a, b terms computed remain positive, and a remains odd, thus (b|a) remains well-deﬁned.2 Theorem 1. Algorithm CubicBinaryJacobi is correct (assuming it terminates). Proof. We prove that the following invariant holds during the algorithm, if a0 , b0 are the initial values of a, b: (b0 |a0 ) = (−1)s (b|a). This is true before we enter the while-loop, since s = 0, a = a0 , and b = b0 . For each step in the while loop, we divide b by 2j , swap a and b = b/2j , replace a 2

M¨ oller says in [8]: “if one tries to use positive quotients 0 < q < 2k+1 , the [binary gcd] algorithm no longer terminates”. However, with a modiﬁed stopping criterion as in Algorithm CubicBinaryJacobi, the algorithm terminates (we prove this below).

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Algorithm 1.2. CubicBinaryJacobi Input: a, b ∈ N with ν(a) = 0 < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: b ← b/2j 4: (q, r) ← BinaryDividePos(a, b) 2 5: s ← (s + j(a2 − 1)/8 + (a − 1)(b − 1)/4 + j(b − 1)/8) mod 2 j 6: (a, b) ← (b , r/2 ), j ← ν(b) 7: if a = 1 then return (−1)s else return 0

by r = a + qb , and divide r by 2j . The Jacobi symbol is modiﬁed by a factor 2 (−1)j(a −1)/8 for the division of b by 2j , by a factor (−1)(a−1)(b −1)/4 for the 2 interchange of a and b , and by a factor (−1)j(b −1)/8 for the division of r by 2j . At the end of the loop, we have gcd(a0 , b0 ) = a; if a = 1, since (b|1) = 1, we have (b0 |a0 ) = (−1)s , otherwise (b0 |a0 ) = 0. Lemma 1. The quantity a + 2b is non-increasing in Algorithm CubicBinaryJacobi. Proof. At each iteration of the “while” loop, a becomes b/2j , and b becomes (a + qb/2j )/2j . In matrix notation 0 1/2j a a ← . (1) b b 1/2j q/22j Therefore a + 2b becomes b 2a b a + qb/2j = j + (1 + 2q/2j ) j . +2 j j 2 2 2 2

(2)

Since j ≥ 1, the ﬁrst term is bounded by a. In the second term, q ≤ 2j+1 − 1, thus the second term is bounded by (5/2j − 2/22j )b, which is bounded by 9b/8 for j ≥ 2, and equals 2b for j = 1. If j ≥ 2, then a + 2b is multiplied by a factor at most 9/16. If j = q = 1 then a + 2b decreases, but by a factor which could be arbitrarily close to 1. The only case where a + 2b does not decrease is when j = 1 and q = 3; in this case a + 2b is unchanged. This motivates us to deﬁne three classes of iterations: good, bad, and ugly. Let us say that we have a good iteration when j ≥ 2, a bad iteration when j = q = 1, and an ugly iteration when j = 1 and q = 3. Since q is odd and 1 ≤ q ≤ 2j+1 − 1, this covers all possibilities. For a bad iteration, (a, b) becomes (b/2, a/2 + b/4), and for an ugly iteration, (a, b) becomes (b/2, a/2 + 3b/4). We denote the matrices corresponding to good, bad and ugly iterations by G, B and U respectively. Thus

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G = Gj,q =

0 1/2j 1/2j q/4j

, B=

0 1/2 1/2 1/4

, U=

0 1/2 1/2 3/4

.

The eﬀect of m successive ugly iterations is easily seen to be given by the matrix 1 1 + 4(−1/4)m 2 − 2(−1/4)m m U = . (3) 5 2 − 2(−1/4)m 4 + (−1/4)m Assume we start from (a, b) = (a0 , b0 ), and after m > 0 successive ugly iterations we get values (am , bm ). Then, from Equation (3), 5am = (a + 2b) + 2(2a − b)(−1/4)m , m

5bm = 2(a + 2b) − (2a − b)(−1/4) .

(4) (5)

We can not have 2a0 = b0 or the algorithm would have terminated. However, am must be an integer. This gives an upper bound on m. For a0 , b0 of n bits, the number of successive ugly iterations is bounded by n/2 + O(1) (a precise statement is made in Lemma 2). If there were no bad iterations, this would prove that for n-bit inputs the number of iterations is O(n2 ), since each sequence of ugly iterations would be followed by at least one good iteration. Bad iterations can be handled by a more complicated argument which we omit, since they will be considered in detail in §2 when we discuss the complexity of the quadratic algorithm (see the proof of Theorem 2). Since the number of iterations is O(n2 ) from Theorem 2, and each iteration costs time O(M (n)), the overall time for Algorithm CubicBinaryJacobi is 3 ). Note that this worst-case bound is almost certainly too O(n2 M (n)) = O(n pessimistic (see §4).

2

A Provably Quadratic Algorithm

Suppose we have a sequence of m > 0 ugly iterations. It is possible to combine the m ugly iterations into one harmless iteration which is not much more expensive than a normal (good or bad) iteration. Also, it is possible to predict the maximal such m in advance. Using this trick, we reduce the number of iterations (good, 2 ). Without loss bad and harmless) to O(n) and their cost to O(M (n)n) = O(n of generality, suppose that we start from (a0 , b0 ) = (a, b). Lemma 2. If μ = ν(a−b/2), then we have exactly μ/2 ugly iterations starting from (a, b), followed by a good iteration if μ is even, and by a bad iteration if μ is odd. Proof. We prove the lemma by induction on μ. If μ = 0, a − b/2 is odd, but a is odd, so b/2 is even, which yields j ≥ 2 in BinaryDividePos, thus a, b yield a good iteration. If μ = 1, a − b/2 is even, which implies that b/2 is odd, thus we have j = 1. If we had q = 3 in BinaryDividePos, this would mean that

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a + 3(b/2) = 0 mod 4, or equivalently a − b/2 = 0 mod 4, which is incompatible with μ = 1. Thus we have q = 1, and a bad iteration. Now assume μ ≥ 2. The ﬁrst iteration is ugly since 4 divides a − b/2, which implies that b/2 is odd. Thus j = 1, and a − b/2 = 0 mod 4 implies that q = 3. After one ugly iteration (a, b) becomes (b/2, a/2 + 3b/4), thus a − b/2 becomes −(a − b/2)/4, and the 2-valuation of a − b/2 decreases by 2. From the above, we see that, for a sequence of m ugly iterations, a0 , a1 , . . . , am satisfy the three-term recurrence 4ai+1 − 3ai − ai−1 = 0 for 0 < i < m, and similarly for b0 , b1 , . . . , bm . It follows that ai = a mod 4, and similarly bi = b mod 4, for 1 ≤ i < m. We can modify Algorithm CubicBinaryJacobi to consolidate m consecutive ugly iterations into one harmless iteration, using the expressions (4)–(5) for am and bm (we give an optimised evaluation below). It remains to modify step 5 of CubicBinaryJacobi to take account of the m updates to s. Since j = 1 for each ugly iteration, we have to increment s by an amount 2 a2i − 1 bi − 1 ai − 1 bi − 1 δ= + + mod 2, 8 8 2 2 0≤i<m

where we write bi for bi /2. However, ai+1 = bi for 0 ≤ i < m, so the terms involving division by 8 “collapse” mod 2, leaving just the ﬁrst and last terms. The terms involving two divisions by 2 are all equal to (a − 1)/2 · (b − 1)/2 mod 2, using the observation that ai mod 4 is constant for 0 ≤ i < m. Thus 2 a0 − 1 a 1 − 1 a0 − 1 a2m − 1 + +m mod 2. δ= 8 8 2 2 One further simpliﬁcation is possible. Since a0 = a1 mod 4, and a0 is odd, we can replace a1 by a0 in the last term, and use the fact that x2 = x mod 2 to obtain 2 a0 − 1 a0 − 1 a2m − 1 + +m mod 2. (6) δ= 8 8 2 We can economise the computation of am and bm from (4)–(5) by ﬁrst computing d = a − b , m = ν(d) div 2, c = (d − (−1)m (d/4m ))/5, where the divisions by 4m and by 5 are exact; then am = a − 4c, bm = b + 2c. From these observations, it is easy to modify Algorithm CubicBinaryJacobi to obtain Algorithm QuadraticBinaryJacobi. In this algorithm, steps 7–11 implement a harmless iteration equivalent to m > 0 consecutive ugly iterations; steps 13–14 implement bad and good iterations, and the remaining steps are common to both. Step 5 of Algorithm CubicBinaryJacobi is split into three steps 4, 13 and 15. In the case of a harmless iteration, the computation of δ satisfying (6) is implicit in steps 4, 10 and 15.

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Algorithm 2.1. QuadraticBinaryJacobi Input: a, b ∈ N with ν(a) = 0 < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: b ← b/2j 4: s ← (s + j(a2 − 1)/8) mod 2 5: (q, r) ← BinaryDividePos(a, b) 6: if (j, q) = (1, 3) then 7: d ← a − b 8: m ← ν(d) div 2 9: c ← (d − (−1)m d/4m )/5 10: s ← (s + m(a − 1)/2) mod 2 11: (a, b) ← (a − 4c, b + 2c) 12: else 13: s ← (s + (a − 1)(b − 1)/4) mod 2 14: (a, b) ← (b , r/2j ) 15: s ← (s + j(a2 − 1)/8) mod 2, j ← ν(b) 16: if a = 1 then return (−1)s else return 0

harmless iteration

good or bad iteration

Theorem 2. Algorithm QuadraticBinaryJacobi is correct and terminates after O(n) iterations of the “while” loop (steps 2–15) if the inputs are positive integers of at most n bits, with 0 = ν(a) < ν(b). Proof. Correctness follows from the equivalence to Algorithm CubicBinaryJacobi. To prove that convergence takes O(n) iterations, we show that a + 2b is multiplied by a factor at most 5/8 in each block of three iterations. This is true if the block includes at least one good iteration, so we need only consider harmless and bad iterations. Two harmless iterations do not occur in succession, so the block must include either (harmless, bad) or (bad, bad). In the ﬁrst case, the corresponding matrix is BU m = BU · U m−1 for some m > 0. We saw in §1.1 that the matrix U leaves a + 2b unchanged, so U m−1 also leaves a + 2b unchanged, and we need only consider the eﬀect of BU . Suppose that (a, b) is transformed into ( a, b) by BU . Thus a a 1/4 3/8 a . b = BU b = 1/8 7/16 b We see that a + 2b =

a 5b 5 + ≤ (a + 2b). 2 4 8

The case of two successive bad iterations is similar – just replace BU by B 2 in the above, and deduce that a + 2b ≤ (a + 2b)/2. We conclude that the number of iterations of the while loop is at most cn + O(1), where c = 3/ log2 (8/5) ≈ 4.4243.

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Remarks 1. A more complicated argument along similar lines can reduce the constant c to √ 2/ log2 (1/ρ(BU )) = 2/ log2 ((11 − 57)/2) ≈ 2.5424. Here ρ denotes the spectral radius: ρ(A) = limk→∞ ||Ak ||1/k . 2. In practice QuadraticBinaryJacobi is not much (if any) faster than CubicBinaryJacobi. Its advantage is simply the better worst-case time bound. A heuristic argument suggests that on average only 1/4 of the iterations of CubicBinaryJacobi are ugly. 3. Our implementations of CubicBinaryJacobi and QuadraticBinaryJacobi are slower than GMP’s O(n2 ) algorithm (which is based on Stein’s binary gcd, as in Shallit and Sorenson [12]). However, in the next section we use the ideas of our QuadraticBinaryJacobi algorithm to get an O(M (n) log n) algorithm. We do not see how to modify the algorithm of Shallit and Sorenson to do this.3

3

An O(M (n) log n) Algorithm

Algorithm HalfBinaryJacobi below is a modiﬁcation of Algorithm Half-GB-gcd from [13]. (Algorithm Half-GB-gcd is a subquadratic right-to-left gcd algorithm; for more on the general structure of subquadratic gcd algorithms, we refer the reader to M¨ oller [8].) The main diﬀerences between Half-GB-gcd and our algorithm are the following: 1. binary division with positive (not centered) quotient is used; 2. the algorithm returns an integer s such that if a, b are the inputs, c, d the output values deﬁned by Theorem 3, then (b|a) = (−1)s (d|c); 3. at steps 4 and 27, we reduce mod 22k1 +2 (resp. 22k2 +2 ) instead of mod 22k1 +1 (resp. 22k2 +1 ), so that we have enough information to correctly update s0 at steps 10, 17, 21 and 25; 4. we have to “cut” some harmless iterations in two (step 15). Remarks. The matrix Q occurring at step 19 is just 22m U m , where U m is given by Equation (3). Similarly, the matrix Q occurring at step 23 is 22j0 Gj0 ,q . In practice, steps 13–20 can be omitted (so the algorithm becomes a fast version of CubicBinaryJacobi) – this variant is simpler and slightly faster on average. We now state our main theorem. Its proof is based on comparing the GB sequence of a, b and that of a1 , b1 , where a1 = a mod 22k1 +2 and b1 = b mod 22k1 +2 . The GB — which stands for Generalized Binary division, see [13] — sequence of a, b is the sequence of remainders we obtain by applying the binary division iteratively. Two GB sequences match if they produce the same binary quotients qi . 3

In Algorithm Binary Jacobi of [12], it is necessary to know the sign of a − n (b − a in our notation) to decide whether to perform an interchange. This makes it diﬃcult to construct an recursive O(M (n) log n) algorithm along the lines of Algorithm HalfBinaryJacobi.

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Algorithm 3.1. HalfBinaryJacobi Input: a ∈ N, b ∈ N ∪ {0} with 0 = ν(a) < ν(b), and k ∈ N Output: two integers s, j and a 2 × 2 matrix R 1: if ν(b) > k then 10 2: Return 0, 0, 01 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28:

b = 0 is possible

k1 ← k/2 a1 ← a mod 22k1 +2 , b1 ← b mod 22k1 +2 s1 , j1 , R ← HalfBinaryJacobi(a1 , b1 , k1 ) a ← 2−2j1 (R1,1 a + R1,2 b), b ← 2−2j1 (R2,1 a + R2,2 b) j0 ← ν(b ) if j0 + j1 > k then Return s1 , j1 , R 2 s0 ← j0 (a − 1)/8 mod 2 q, r ← BinaryDividePos(a , b ) b ← b /2j0 if (j0 , q) = (1, 3) then d ← a − b m ← min(ν(d) div 2, k − j1 ) c ← (d − (−1)m d/4m )/5 s0 ← s0 + m(a − 1)/2 mod 2 harmless iteration (a2 , b2) ← (a − 4c, 2(b + c)) (4m + 4(−1)m )/5 2(4m − (−1)m )/5 Q← 2(4m − (−1)m )/5 (4m+1 + (−1)m )/5 else s0 ← s0 + (a − 1)(b − 1)/4 mod 2 r/2j0 ) good or bad iteration (a2 , b2) ← (b , j0 0 2 Q← 2j0 q m ← j0 s0 ← s0 + j0 (a22 − 1)/8 mod 2 k2 ← k − (m + j1 ) s2 , j2 , S ← HalfBinaryJacobi(a2 mod 22k2 +2 , b2 mod 22k2 +2 , k2 ) Return (s0 + s1 + s2 ) mod 2, j1 + j2 + m, S × Q × R

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Theorem 3. Let a, b, k be the inputs of Algorithm HalfBinaryJacobi, and s, j, R the corresponding outputs. If dc = 2−2j R ab , then: (b|a) = (−1)s (d|c)

and

ν(2j c) ≤ k < ν(2j d).

Proof (outline). We prove the theorem by induction on the parameter k. The key ingredient is that if we reduce a, b mod 22k1 +1 in step 4, then the GB sequence of a1 , b1 matches that of a, b, for the terms computed by the recursive call at step 5. This is a consequence of [13, Lemma 7] (which also holds for binary division with positive quotient). It follows that in all the binary divisions with inputs ai , bi in that recursive call, ai and bi /2ji match modulo 2ji +1 the corresponding values that would be obtained from the full inputs a, b (otherwise the corresponding binary quotient qi would be wrong). Since here we reduce a, b mod 22k1 +2 instead of mod 22k1 +1 , ai and bi /2ji now match modulo 2ji +2 — instead of modulo 2ji +1 — the values that would be obtained from the full inputs a, b, where 2ji +2 ≥ 8 since ji ≥ 1. At step 10, s0 depends only on j0 mod 2 and a mod 8, at step 17 it depends on m mod 2 and a mod 4, and at step 21 on a mod 4 and b mod 4. Since a and b at step 21 correspond to some ai and bi /2ji , it follows that a and b agree mod 8 with the values that would be computed from the full inputs, and thus the correction s0 is correct. This proves by induction that (b|a) = (−1)s (d|c). Now we prove that ν(2j c) ≤ k < ν(2j d). If there is no harmless iteration, it is a consequence of the proof of Theorem 1 in [13]. In case there is a harmless iteration, ﬁrst assume that m = ν(d) div 2 at step 15. The new values a2 , b2 at step 18 correspond to m successive ugly iterations, which yield j = j1 + m ≤ k. Thus ν(2j a2 ) ≤ k: we did not go too far, and since we are computing the same sequence of quotients as Algorithm QuadraticBinaryJacobi, the result follows. Now if k − j1 < ν(d) div 2, we would go too far if we performed ν(d) div 2 ugly iterations, since it would give j0 := ν(d) div 2 > k −j1 , thus j := j1 +j0 > k, and ν(2j a2 ) would exceed k. This is the reason why we “cut” the harmless iteration at m = k − j1 (step 15). The other invariants are unchanged. Finally we can present our O(M (n) log n) Algorithm FastBinaryJacobi, which computes the Jacobi symbol by calling Algorithm HalfBinaryJacobi. The general structure is similar to that described in [8] for several asymptotically fast GCD algorithms. Daireaux, Maume-Deschamps and Vall´ee [5] prove that, for the positive binary division, the average increase of the most signiﬁcant bits is 0.65 bits/iteration (which partly cancels an average decrease of two least signiﬁcant bits per iteration); compare this with only 0.05 bits/iteration on average for the centered division.4

4

We have computed more accurate values of these constants: 0.651993 and 0.048857 respectively.

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Algorithm 3.2. FastBinaryJacobi Input: a, b ∈ N with 0 = ν(a) < ν(b) Output: Jacobi symbol (b|a) 1: s ← 0, j ← ν(b) 2: while 2j a = b do 3: k ← max(ν(b), (b) div 3) 4: s , j, R ← HalfBinaryJacobi(a, b, k) 5: s ← (s + s ) mod 2 6: (a, b) ← 2−2j (R1,1 a + R1,2 b, R2,1 a + R2,2 b), 7: if a = 1 then return (−1)s else return 0

4

(b) is length of b in bits

j ← ν(b)

Experimental Results

We have implemented the diﬀerent algorithms in C (using 64-bit integers) and in GMP (using multiple-precision integers), as well as in Maple/Magma (for testing purposes). For max(a, b) < 226 the maximum number of iterations of Algorithm CubicBinaryJacobi is 64, with a = 15548029 and b = 66067306. The number of iterations seems to be O(n) for a, b < 2n : see Table 1. This is plausible because, from heuristic probabilistic arguments, we expect about half of the iterations to be good, and experiments conﬁrm this. For example, if we consider all admissible a, b < 220 , the cumulated number of iterations is 3.585×1012 for 238 calls, i.e., an average of 13.04 iterations per call (max 48); the cumulated number of good, bad and ugly iterations is 51.78%, 25.47%, and 22.75% respectively. For a, b < 260 , a random sample of 108 pairs (a, b) gave 42.72 iterations per call (max 89), with 50.54%, 25.14%, and 24.31% for good, bad and ugly respectively. These ratios seem to be converging to the heuristically expected 1/2 = 50%, 1/4 = 25%, and 1/4 = 25%. When we consider all admissible a, b < 220 , the maximum number of iterations of QuadraticBinaryJacobi is 37 when a = 933531, b = 869894, the cumulated number of iterations is 3.405 × 1012 (12.39 per call), the cumulated number of good, bad and harmless iterations is 54.51%, 26.82%, and 18.67% respectively. For a, b < 260 , a random sample of 108 pairs (a, b) gave 40.21 iterations per call (max 76), with 53.70%, 26.71%, and 19.59% for good, bad and harmless respectively. These ratios seem to be converging to the heuristically expected 8/15 = 53.33%, 4/15 = 26.67%, and 1/5 = 20%. We have also compared the time and average number of iterations for huge numbers, using the fast gcd algorithm in GMP, say gcd — which implements the algorithm from [8] — and an implementation of the algorithm from [13], say bgcd. For inputs of one million 64-bit words, gcd takes about 45.8s on a 2.83Ghz Core 2, while bgcd takes about 48.3s and 32,800,000 iterations: this is in accordance with the fact proven in [5] that each step of the binary gcd discards on average two least signiﬁcant bits, and adds on average about 0.05 most signiﬁcant bits. Our algorithm bjacobi (based on Algorithms 3.1–3.2) takes about 83.1s

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1e+06 mpz_jacobi FastBinaryJacobi 100000 10000 1000 100 10 1 0.1 0.01 0.001 0.0001 1

10

100

1000

10000

100000

Fig. 1. Comparison of GMP 4.3.1 mpz jacobi routine with our FastBinaryJacobi implementation in log-log scale. The x-axis is in 64-bit words, the y-axis in milliseconds on a 2.83Ghz Core 2. Table 1. Worst cases for CubicBinaryJacobi(b|a), max(a, b) < 2n n iterations example (a, b) 5 6 (7, 30) 10 19 (549, 802) 15 34 (23449, 19250) 20 48 (656227, 352966) 21 51 (1596811, 1493782)

n iterations example (a, b) 22 53 (2214985, 2781506) 23 55 (1383497, 8292658) 24 58 (2236963, 12862534) 25 62 (28662247, 30847950) 26 64 (15548029, 66067306)

and 47,500,000 iterations (for a version with steps 13–20 of Algorithm 3.1 omitted in the basecase routine), which agrees with the theoretical drift of 0.651993 bits per iteration. The break-even point between the O(n2 ) implementation of the Jacobi symbol in GMP 4.3.1 and our O(M (n) log n) implementation is about 535 words, that is about 34, 240 bits or about 10, 300 decimal digits (see Fig. 1).

5

Concluding Remarks

Weilert [15] says: “We are not able to use a GCD calculation in Z[i] similar to the binary GCD algorithm · · · because we do not get a corresponding quotient

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sequence in an obvious manner ”. In a sense we ﬁlled that gap for the computation of the Jacobi symbol, because we showed how it can be computed using a binary GCD algorithm without the need for a quotient sequence. We showed how to compute the Jacobi symbol with an asymptotically fast time bound, using such a binary GCD algorithm. Our implementation is faster than a good O(n2 ) implementation for numbers with bitsize n > 35000. Our subquadratic implementation is available from http://www.loria.fr/~zimmerma/ software/#jacobi. Binary division with a centered quotient does not seem to give a subquadratic algorithm; however we can use it with the “cubic” algorithm (which then becomes provably quadratic) since then we control the sign of a, b. For a better quadratic algorithm, we can choose the quotient q so that abq < 0, by replacing q by q − 2j+1 if necessary: experimentally, this gains on average 2.194231 bits per iteration, compared to 1.951143 for the centered quotient, and 1.348008 for the positive quotient. In comparison, Stein’s “binary” algorithm gains on average 1.416488 bits per iteration [4, §7][7, §4.5.2]. Acknowledgement. We thank Steven Galbraith who asked us about the existence of an O(M (n) log n) algorithm for the Jacobi symbol, Arnold Sch¨ onhage for his comments and a pointer to the work of his former student Andr´e Weilert, Damien Stehl´e who suggested adapting the binary gcd algorithm, and Marco Bodrato and Niels M¨oller for testing our implementation. We also thank the two anonymous reviewers, especially the one who actually implemented our new algorithm in Magma! We thank INRIA for its support of the ANC “´equipe associ´ee”. The ﬁrst author acknowledges the support of the Australian Research Council.

References 1. Bach, E.: A note on square roots in ﬁnite ﬁelds. IEEE Trans. on Information Theory 36(6), 1494–1498 (1990) 2. Bach, E., Shallit, J.O.: Algorithmic Number Theory: Eﬃcient Algorithms, vol. 1. MIT Press, Cambridge (1996) (Solution to problem 5.52) 3. Bachmann, P.: Niedere Zahlentheorie, Teubner, Leipzig, vol. 1 (1902); Reprinted by Chelsea, New York (1968) 4. Brent, R.P.: Twenty years’ analysis of the binary Euclidean algorithm. In: Davies, J., Roscoe, A.W., Woodcock, J. (eds.) Millennial Perspectives in Computer Science: Proceedings of the 1999 Oxford - Microsoft Symposium in honour of Professor Sir Antony Hoare, Palgrave, New York, pp. 41–53 (2000), http://wwwmaths.anu.edu.au/~brent/pub/pub183.html 5. Daireaux, B., Maume-Deschamps, V., Vall´ee, B.: The Lyapunov tortoise and the dyadic hare. In: Proceedings of the 2005 International Conference on Analysis of Algorithms, DMTCS Proc. AD, pp. 71–94 (2005), http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/81 6. Gauss, C.F.: Theorematis fundamentalis in doctrina de residuis quadraticis, demonstrationes et ampliatones novæ. Comm. Soc. Reg. Sci. Gottingensis Rec. 4 (presented February 10, 1817) (1818); Reprinted in Carl Friedrich Gauss Werke, Bd. 2: H¨ ohere Arithmetik, G¨ ottingen, pp. 47–64 (1876)

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7. Knuth, D.E.: The Art of Computer Programming. In: Seminumerical Algorithms, 3rd edn., vol. 2, Addison-Wesley, Reading (1997) 8. M¨ oller, N.: On Sch¨ onhage’s algorithm and subquadratic integer GCD computation. Mathematics of Computation 77(261), 589–607 (2008) 9. Sch¨ onhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971) 10. Sch¨ onhage, A.: Personal communication by email (December 2009) 11. Sch¨ onhage, A., Grotefeld, A.F.W., Vetter, E.: Fast Algorithms: A Multitape Turing Machine Implementation. BI-Wissenschaftsverlag, Mannheim (1994) 12. Shallit, J., Sorenson, J.: A binary algorithm for the Jacobi symbol. ACM SIGSAM Bulletin 27(1), 4–11 (1993), http://euclid.butler.edu/~sorenson/papers/binjac.ps 13. Stehl´e, D., Zimmermann, P.: A binary recursive gcd algorithm. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 411–425. Springer, Heidelberg (2004) 14. Vall´ee, B.: A unifying framework for the analysis of a class of Euclidean algorithms. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 343–354. Springer, Heidelberg (2000) 15. Weilert, A.: Fast Computation of the Biquadratic Residue Symbol. Journal of Number Theory 96, 133–151 (2002)

New Families of ECM Curves for Cunningham Numbers ´ Eric Brier1 and Christophe Clavier2,3

2

1 Ingenico S.A. 1, rue Claude Chappe, B.P. 346, 07530 Guilherand-Granges, France [email protected] Institut d’Ing´enierie Informatique de Limoges (3iL) 43, rue Sainte Anne F-87000 Limoges [email protected] 3 Universit´e de Limoges – XLIM D´epartement de Math´ematiques et Informatique 83, rue d’Isle F-87000 Limoges [email protected]

Abstract. In this paper we study structures related to torsion of elliptic curves deﬁned over number ﬁelds. The aim is to build families of elliptic curves more eﬃcient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational Z/4Z × Z/4Z torsion and positive rank over the ﬁeld Q(ζ8 ) and a family of elliptic curves with rational Z/6Z × Z/3Z torsion and positive rank over the ﬁeld Q(ζ3 ). These families have been used in ﬁnding new prime factors for the numbers 2972 + 1 and 21048 + 1. Along the way, we classify and give a parameterization of modular curves for some torsion subgroups.

1

Introduction

The Elliptic Curve Method (ECM in short) is a factoring algorithm, whose complexity depends on the size of the smallest prime factor instead of the size of the number to be factored. It can be seen as a variation of the p − 1 method. The idea is to build an elliptic curve over the ring Z/N Z with a point P on it and to compute the scalar multiplication M · P . Since N is not a prime, the elliptic curve is not deﬁned over a ﬁeld. However, computations are done as if we were working on a ﬁeld and if something fails, this means that a non-trivial factor of N has been found. The number M is chosen to be the product of powers of small primes and thus, a prime factor p is found as soon as the order of the elliptic curve reduced modulo p is smooth. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 96–109, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Many improvements of the ECM are described in the literature. We will focus on an improvement consisting in choosing the elliptic curve as the reduction modulo N of an elliptic curve deﬁned over the ﬁeld Q with a non-trivial torsion group and positive rank. The torsion group of an elliptic curve is the group of elements of ﬁnite order and the rank is the number of generators of the torsionfree part of the group. As soon as small prime factors have been removed from N , the torsion group is preserved in most cases by the modulo N reduction of the curve, which helps to make the order of the curve smooth. The positive rank is needed to set the starting point P of the algorithm. Possible torsion groups for elliptic curves deﬁned over Q are in ﬁnite number, with maximal order 16. For each possible torsion group, at least a family of elliptic curves with positive rank has been found. The idea we follow in this paper is to use a number ﬁeld K for which reduction modulo N can be made explicit and to build over K an elliptic curve with positive rank and a torsion subgroup as large as possible. Let us give an example : if the number to be factored is of the form N = u2 + 1, we can make use of the ﬁeld K = Q(i) with mapping i → u. The numbers of the Cunningham Project (i.e. numbers of the form am ±1) allow to use m-th roots of unity. It will be interesting to focus on cyclotomic ﬁelds or on their subﬁelds. It is important to note that all quadratic extensions of Q lie in cyclotomic ﬁelds. The paper is organized as follows. Section 2 introduces the necessary notions about modular curves and classiﬁes torsion subgroups that can be of any interest for ECM integer factoring. Section 3 is devoted to construction of parameterized elliptic curves with given torsion subgroup over some cyclotomic extensions of the ﬁeld of rationals. Section 4 focuses on the search for inﬁnite subfamilies of elliptic curves having nonzero rank, which is mandatory to ECM usage. Section 5 rephrases previous sections results in the context of ECM and gives some instances of new prime factors of Cunningham Project numbers discovered thanks to the work presented here. Finally, section 6 concludes and suggests some research areas to go further.

2

Elliptic Curve Torsion and Modular Curves

An elliptic curve E deﬁned over a number ﬁeld K turns out to be a commutative group. The Mordell-Weil theorem states that this group is ﬁnitely generated and can be written as: E(K) ∼ = T ⊗ Zr where the integer r is called rank and T is the so called torsion group, which consists in elements of ﬁnite order. Furthermore, T is isomorphic to Z/m1 Z × Z/m2 Z with the constraints that m2 divides m1 and the m2 -th roots of unity all lie in the ﬁeld K. Whereas it is conjectured that the rank is not constrained, the torsion group can take only ﬁnitely many diﬀerent shapes over the ﬁeld of rationals:

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Theorem 1 (Mazur). The torsion group T of an elliptic curve defined over the field Q is isomorphic to one of the following groups: Z/mZ with 1 ≤ m ≤ 10 or m = 12 Z/2mZ × Z/2Z with 1 ≤ m ≤ 4 This theorem is eﬀective in the sense that for each of these cases, it is possible to give equations of elliptic curves. These parameterizations come from modular curves. Over the ﬁeld C of complex numbers, there is a one-to-one correspondance between isomorphism classes of elliptic curves and the Riemann Surface X(1), which is the quotient H∗ /SL2 (Z), where H∗ is the compactiﬁed Poincar´e halfplane. For any subgroup Γ of SL2 (Z), the quotient surface H∗ /Γ is called a modular curve. Extending notations of [3], we deﬁne the following subgroups of SL2 (Z): a b ∈ SL2 (Z), a ≡ d ≡ 1 mod m, b ≡ c ≡ 0 mod m Γ (m) = c d Γ1 (m) = Γ0 (m) =

a b c d a b c d

∈ SL2 (Z), a ≡ d ≡ 1 mod m, c ≡ 0 mod m

∈ SL2 (Z), c ≡ 0 mod m

and the quotients: X(m) = H∗ /Γ (m) X1 (m) = H∗ /Γ1 (m) X0 (m) = H∗ /Γ0 (m) X1 (m1 , m2 ) = H∗ /(Γ1 (m1 ) ∩ Γ (m2 )) when m2 |m1 A point on the surface X(m) corresponds to an elliptic curve together with a basis for its [m]-torsion subgroup, up to isomorphism. A point on the surface X1 (m) corresponds, up to isomorphism, to an elliptic curve together with a [m]torsion point. A point on the surface X0 (m) corresponds, up to isomorphism, to an elliptic curve together with a cyclic torsion subgroup of order m. A point on the surface X(m1 , m2 ) corresponds to an elliptic curve with a [m1 ]-torsion point and an independent [m2 ]-torsion point. Though these notions make use of complex number and analytical tools, the modular curves can also be represented as algebraic curves. An algebraic model of X1 (m) can be found over Q and the correspondance with an elliptic curve and a [m]-torsion point on it is algebraic and deﬁned over Q. The curve X(m) involves the full [m]-torsion subgroup and, due to existence of Weil pairing, m-th roots of unity are involved. The rational models and correspondance for X(m) (resp. X1 (m1 , m2 )) are deﬁned over the cyclotomic ﬁeld Q(ζm ) (resp. Q(ζm2 )). The modular curves associated to torsion

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subgroups in Mazur’s theorem are genus 0 algebraic curves. This explains why it is possible to give parametric Weierstrass equations. The Elliptic Curve Method needs a non-bounded number of elliptic curves to compute with. Since algebraic curves of genus greater than 2 have only ﬁnitely many rational points over a given number ﬁeld, we will focus only on torsion structures for which the associated modular curve has genus 0 or 1. Computing the genus of an algebraic curve is not an easy task in the general case but the task is easy with a computer for modular curves of rather small level. A theorem from Shimura states that the genus of the modular curve X1 (p) for a prime p ≥ 5 is given by : g=

(p − 5)(p − 7) . 24

This implies that the only primes for which the genus of X1 (p) is 0 or 1 are {2, 3, 5, 7, 11}. When n|m, there is a surjective mapping X1 (m) → X1 (n), and thus the genus of X1 (m) is at least the genus of X1 (n). Computing the genus of X1 (m) for rather small values of m being easy, we can increase the power of these primes until the genus is strictly greater than 1 and we get that the only prime powers for which the genus of X1 (pe ) is 0 or 1 are {2, 4, 8, 3, 9, 5, 7, 11}. Now, combining this ﬁnite set, it is possible to check the following proposition with a ﬁnite amount of work: Proposition 1. The integers m such that X1 (m) is of genus 0 or 1 are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15} When m2 |m1 , there is a surjective mapping X1 (m1 , m2 ) → X1 (m1 ), and thus the genus of X1 (m1 , m2 ) is at least the genus of X1 (m1 ). This implies that if the genus of the modular curve X1 (m1 , m2 ) is 0 or 1, the number m1 is in the list given in proposition 1. Building on this, for any m1 in this list, we can check if the genus of X1 (m1 , m2 ) is 0 or 1 for all divisors m2 of m1 . The result is given in next proposition. Proposition 2. The torsion groups for which the associated modular curve is of genus 0 are: Z/2Z, Z/3Z, Z/4Z, Z/5Z, Z/6Z, Z/7Z Z/8Z, Z/9Z Z/10Z Z/12Z

Z/2Z × Z/2Z Z/3Z × Z/3Z Z/4Z × Z/2Z, Z/5Z × Z/5Z Z/6Z × Z/2Z, Z/8Z × Z/2Z

Z/4Z × Z/4Z Z/6Z × Z/3Z

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The torsion groups for which the associated modular curve is of genus 1 are: Z/6Z × Z/6Z Z/8Z × Z/4Z Z/9Z × Z/3Z Z/10Z × Z/2Z Z/11Z Z/12Z × Z/2Z Z/14Z Z/15Z

3

Parameterization of Elliptic Curves with Given Torsion Structure

When the base ﬁeld is Q, several papers (e.g. [8] and [4]) describe the construction of elliptic curves with prescribed torsion groups. We will study cases that need to work over extensions. 3.1

Construction of Z/3Z × Z/3Z

To study torsion points, one can use the division polynomials, whose roots are the abscises of torsion points. Since we wish all 3-torsion points to be rational, we start by imposing two rational roots x1 and x2 to the polynomial ϕ3 (x) = 3x4 + 6ax2 + 12bx − a2 The system ϕ3 (x1 ) = ϕ3 (x2 ) = 0 considered as equations in the variables a and b has roots if and only if −3x1 x2 is a square. A convenient parameterization is x1 = 6ξ x2 = −2ρ2 ξ and the corresponding parameters are a = −12ξ 2 ρ(ρ2 − 3ρ + 3) b = 2ξ 3 (ρ2 − 3)(ρ4 − 6ρ3 + 18ρ2 − 18ρ + 9) At this stage, we introduce two linear factors in ϕ3 . The remaining quadratic factor of ϕ3 has discriminant equal to −3(ρ − 1)2 (ρ − 3)2 . We need −3 to be a square, which is natural since the Weil pairing introduces cubic roots of unity. We thus have x-coordinates of point of order 3 rational. We now turn on to y-coordinates. Substitutions of x1 and x2 in x3 + ax + b yield y12 = 2ξ 3 (ρ − 3)2 (ρ2 + 3)2 and y12 = −6ξ 3 (ρ − 1)2 (ρ2 + 3)2 . To obtain squares, we set ξ = 2λ2 and for convenience ρ = 1 − τ . In conclusion, an elliptic curve in short Weierstrass form has rational 3-torsion over Q(ζ3 ) if and only if its parameters can be written as: a = 48λ4 (τ 3 − 1) b = 16λ6 (τ 6 − 20τ 3 − 8)

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101

Construction of Z/6Z × Z/3Z

Given as an input the results of previous section, we now have to ensure that x3 + ax + b has a linear factor to get a point of order 2. In a ﬁrst step, we set x = ξλ2 to get rid of the homogeneity parameter λ. We consider then x3 + ax + b as a equation in ξ and τ 3 , which is quadratic relatively to the unknown τ 3 . The discriminant of this quadratic equation is −(ξ − 12)3 . It is natural to set ξ = 12 − ν 2 . We now have: x3 + ax + b = λ6 (6ν 2 + ν 3 − 4τ 3 − 32)(6ν 2 − ν 3 − 4τ 3 − 32) Both factors diﬀer only in a sign change for ν. We will keep the ﬁrst factor, which is a cubic in ν and τ . Since the underlying modular curve has genus 0, this curve must have a singularity. We easily ﬁnd that the point (ν = −4, τ = 0) is singular and to reduce the degree of the curve, we set ν = μτ − 4. After replacement and factorization, we have a degree one equation in τ . To keep consistency in notations and to avoid denominators, we rename μ as 1/τ and modify the scaling factor λ. In conclusion, an elliptic curve in short Weierstrass form has rational 3-torsion and a point of order 2 over Q(ζ3 ) if and only if its parameters can be written as: a = −3λ4 (τ 12 − 8τ 9 + 240τ 6 − 464τ 3 + 16) b = −2λ6 (τ 18 − 12τ 15 − 480τ 12 + 3080τ 9 − 12072τ 6 + 4128τ 3 + 64) 3.3

Modular Curve for Z/6Z × Z/6Z

We know that the modular curve X(6) has genus 1. In this section, we will give a very simple model for this elliptic curve. Let us start with the equation for Z/6Z× Z/3Z torsion subgroup. The polynomial x3 + ax + b has by construction a linear and a quadratic factor. The discriminant of the quadratic factor is −9(8τ 3 − 1)3 . From this we derive the following model: X(6) : s2 = t3 + 1 3.4

Modular Curve for Z/9Z × Z/3Z

We start from parameterization of curves with full 3-torsion. One can note that the parameter is involved only to the third power, we thus note σ = τ 3 and will work in a ﬁrst stage only with σ. We introduce the polynomial χ9 whose roots are the sums of x-coordinates of points in cyclic subgroups of order 9 and whose degree is 12: χ9 (z) = z 12 + 792az 10 + 47520bz 9 + ... − 3543478272a6 We can de-homogenize this polynomial by setting λ = 1 and, since a and b are polynomials in σ, we get a polynomial equation in z and σ having a quadratic factor in σ. This factor has a root iﬀ z − 48 is six times a square. We set

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z = 6ζ 2 + 48 and get σ = (ζ 3 + 6ζ 2 + 12ζ + 72)/8. We can now factor the division polynomial ϕ9 (x) and obtain an equation of degree 3 in x and 6 in ζ. The solution x = 12 and ζ = −2 being a singularity, we set x = 12 + ξ(ζ + 2)2 and obtain the relation ζ = −2

ξ 3 + 3ξ 2 − 6ξ + 1 ξ 3 − 3ξ 2 + 1

We are guaranteed that a point of order 9 has rational x-coordinate, its happens that the y-coordinate is also rational. It is now time to remember that σ must be a cube. Elliptic curve with torsion group of type Z/9Z × Z/3Z have same parameters as for Z/3Z × Z/3Z, provided that τ3 =

8(ξ 2 − ξ + 1)3 (ξ 3 − 6ξ 2 + 3ξ + 1) (ξ 3 − 3ξ 2 + 1)3

Some algebraic manipulations turn the equation σ 3 = ξ 3 − 6ξ 2 + 3ξ + 1 into the elliptic model: X1 (9, 3) : s2 = t3 + 16 3.5

Construction of Z/4Z × Z/4Z

In short Weierstrass form, points of order 2 are points whose y-coordinate is 0. It follows that the general form of curve with Z/2Z torsion is: y 2 = (x − u)(x2 + ux + v) For the same reasons the general form of curve with Z/2Z × Z/2Z torsion is: y 2 = (x − u)(x − v)(x + u + v) On this elliptic curve, a point P = (x, y) can be written P = 2Q iﬀ the numbers x − u, x − v and x + u + v are squares, see [2, Theorem 4.2 page 85]. Thus, if we require that all 4-torsion are rational, all 2-torsion points must be doubles and we ask for 0, ±(u − v), ±(2u + v) and ±(u + 2v) being squares. One can note that −1 has to be a square, which is not a surprise: if 4-torsion is rational, the Weil pairing will produce fourth roots of unity, i.e. square roots of −1. We ﬁrst impose 2u + v and 2v + u to be squares. To do so, we invert the system: 2u + v = r2 u = (2r2 − s2 )/3 ⇐⇒ 2 v = (2s2 − r2 )/3 u + 2v = s Then, it remains to ensure that u − v is also a square. The factorization of u − v is (r − s)(r + s). It is convenient to write r = μ + ν and s = μ − ν. We get u − v = 4μν, which must be a square. We can set μ = τ 2 ν. Last, to get rid of denominators, we set ν = 3λ. In conclusion, an elliptic curve in short Weierstrass form has rational 4-torsion over Q(ζ4 ) if and only if its parameters can be written as: a = −27λ4 (τ 8 + 14τ 4 + 1) b = 54λ6 (τ 12 − 33τ 8 − 33τ 4 + 1)

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3.6

103

Modular Curve for Z/8Z × Z/4Z

To obtain a point of order 8, one of the points of order 4 must be expressed as the doubling of a rational point. We take for instance one of the points with x = 3τ 4 − 15. Diﬀerences with x-coordinates of 2-torsion points must be squares, these diﬀerences factor as: −18 (τ 2 + 1) 18 (τ 2 − 1) 9 (τ 4 − 1) We can easily impose the second expression to be a square by setting τ = (κ2 + 2)/(κ2 − 2) Then, the two other expressions are squares iﬀ κ4 + 4 is a square. In the equation σ 2 = κ4 + 4, we apply the change of variables σ = s2 /t2 − 2t and κ = −s/t and get the model: X1 (8, 4) : s2 = t3 − t 3.7

Construction of Z/5Z × Z/5Z

To reach full rational 5-torsion, we begin with two rational cyclic subgroups of order 5. Let χ5 denote the polynomial, whose roots are the sums of x-coordinates of points over the 6 cyclic subgroups of order 5: χ5 (z) = z 6 + 20az 4 + 160bz 3 − 80a2 z 2 − 128abz − 80b2 We note z1 and z2 two roots of χ5 and to take beneﬁt of symmetry use the transformation z1 = u + v and z2 = u − v. We consider the system χ5 (z1 ) = χ5 (z2 ) = 0 as equations in a and b and eliminate the unknown a, obtaining a quartic in b with parameters u and v. It is then convenient to set b = (u2 − v 2 )β to reduce degrees in u and v. This quartic presents a strong singularity when v = 0 and β = u/4, which leads us to set β = (u/4 + γv/8). The result is still a quartic in γ but the degree in v fell down to 2 and the discriminant of this quadratic equation in v is a square iﬀ 9 − 5γ 2 is ﬁve times a square. We use conic parameterization techniques to obtain: γ=

6(μ2 + μ − 1) 5(μ2 + 1)

Now v can be expressed as the product of u and a rational function of μ. We unroll substitutions to get the value of b and come back to equations χ5 (z1 ) = χ5 (z2 ) = 0. They have a common linear factor in a and we now have values for a and b. Knowing that χ5 has two rational roots, we can strengthen our wishes and factor the division polynomial ϕ5 . No surprise that we get two quadratic factors, whose discriminants are squares if and only if μ2 + 1 and 5(μ2 + 1) are squares.

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We remember that we are working over the ﬁeld of ﬁfth roots of unity, in which 5 is a square. We just have to set μ=

2τ τ2 − 1

Now that x-coordinates for 5-torsion points are rational, we choose the value of homogeneity parameter u to have y-coordinates rationals u = −6λ2 (τ 2 + 1)(τ 4 − 2τ 3 − 6τ 2 + 2τ + 1)(2τ 4 + τ 3 + 3τ 2 − τ + 2) In conclusion, an elliptic curve in short Weierstrass form has full rational 5-torsion over Q(ζ5 ) if and only if its parameters can be written as: a = −27λ4 (τ 20 + 228τ 15 + 494τ 10 − 228τ 5 + 1) b = 54λ6 (τ 30 − 522τ 25 − 10005τ 20 − 10005τ 10 + 522τ 5 + 1)

4 4.1

Construction of Elliptic Curve with Large Prescribed Torsion and Positive Rank Description of the Method

For an elliptic curve being useful for the Elliptic Curve Method, its rank has to be non-zero. This means that we still have to produce sub-families of curves with an extra rational point. When the modular curve is of genus 1, we did not ﬁnd any method because we are lacking of freedom on the parameters. This section is devoted to the method we use to produce sub-families with positive rank in the case of a parameterization by P1 (K). In this case, the parameters a and b are, up to the scaling factor λ, polynomials in K(τ ) and we can take x to be also a polynomial x = λ2 ξ(τ ). Then x3 + ax + b becomes itself λ6 times a polynomial. The polynomial ξ being ﬁxed, we can look for values of τ , which turns x3 + ax + b into a square. This approach is equivalent to looking for rational points on hyperelliptic curves of rather high genus and will yield only ﬁnitely many curves. Our method consists in choosing the polynomial ξ in such a way that x3 + ax + b contains as much as possible of square factors. We note a = λ4 α(τ ), b = λ6 β(τ ) and σ(τ ) = ξ(τ )3 + α(τ )ξ(τ ) + β(τ ). For readability, we will omit the parameter τ for polynomial and all derivatives will be taken relatively to τ . We wish to have square factors, i.e. relations of type σ ≡ 0 mod (τ − τ0 )2 . In most cases, this relation imposes to deﬁne ξ modulo (τ − τ0 )2 . Since increasing the degree of ξ will in the end increase the degree of σ, we try to obtain this relation with a constraint only on ξ modulo (τ − τ0 ). Let us compute derivatives: σ = (3ξ 2 + α)ξ + (ξα + β ) To avoid constraints modulo (τ − τ0 )2 , we must keep freedom on ξ , which leads to 3ξ 2 + α = 0. Combining this relation with ξ 3 + αξ + β, we get the criterion

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Δ = 4α3 + 27β 2 = 0 and the value for ξ = −3β/2α. Now, we have to check that ξα + β = 0. Under the previous conditions, this is equivalent to Δ = 0. The values τ0 that will be of interest will thus be multiple roots of the discriminant Δ. To have a maximum number of degrees of freedom, for each of these roots we try to impose conditions on ξ modulo (τ − τ0 )e and check whether we get σ ≡ 0 mod (τ − τ0 )2e . The last step is to combine multiple roots using the Chinese Remainder Theorem in K[τ ]. For each possible τ0 , we ﬁx ξ modulo some power (τ − τ0 )e , the exponent e being less than the maximum ”useful” exponent. We obtain candidates for ξ and for each candidate we factor σ. Since we wish to have σ being a square, we write σ = σ12 σ0 with σ0 square-free. If ξ does not correspond to torsion points and if the degree of σ0 is less than 5, we can parameterize by a curve of genus 0 or 1. If the auxiliary curve is of genus one (i.e. an elliptic curve) and if we can exhibit a point, we build an inﬁnite family of elliptic curves with given torsion and rank at least one. 4.2

Results for Z/4Z × Z/4Z

Taking the values for a and b given in section 3.5, we ﬁrst factor the discriminant Δ = −24 312 τ 4 (τ − 1)4 (τ + 1)4 (τ 2 + 1)4 The values of interest for τ are {0, 1, −1, ι, −ι}. We then check that each of them can be used up to the second power. The number of candidates we can generate for ξ is 35 . To simplify exploration of all these candidates, we compute once for all a polynomial Ξ that satisﬁes all modular conditions Ξ = 9τ 8 − 24τ 4 + 3 take its remainder modulo the polynomial τ e0 (τ − 1)e1 (τ + 1)e−1 (τ − ι)eι (τ − ι)e−ι We get values σ0 of degree 0 that are of no interest since they correspond to torsion points. We get no values of degree 1, 16 diﬀerent values of degree 2, 32 of degree 3 and 62 of degree 4. The simplest value of σ0 is 36 (τ 2 − 3), which corresponds to ξ = 9τ 6 − 15τ 4 − 9τ 2 + 3. To turn σ0 into a square, one can set τ=

ν2 + 3 and λ = 8ν 3 2ν

Unrolling substitutions, we have ⎧ a = −432ν 4 (ν 16 + 24ν 14 + 476ν 12 + 4200ν 10 + 18022ν 8 ⎪ ⎪ ⎪ ⎪ +37800ν 6 + 38556ν 4 + 17496ν 2 + 6561) ⎪ ⎪ ⎨ b = 3456ν 6 (ν 24 + 36ν 22 + 66ν 20 − 6732ν 18 − 101409ν 16 − 707256ν 14 ⎪ ⎪ ⎪ ⎪ −2772260ν 12 − 6365304ν 10 − 8214129ν 8 − 4907628ν 6 ⎪ ⎪ ⎩ +433026ν 4 + 2125764ν 2 + 531441)

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106

The point of inﬁnite order is given by ⎧ ⎨ x = 3 (3ν 12 + 34ν 10 + 117ν 8 + 316ν 6 + 1053ν 4 + 2754ν 2 + 2187) ⎩

y = 27 (ν 2 − 3)(ν 2 + 1)(ν 2 + 9)(ν 6 + 5ν 4 + 15ν 2 + 27)2

The choice of parameters giving such a torsion group when −1 is a square has also been studied to speed-up factorisation in [9]. 4.3

Results for Z/6Z × Z/3Z

Following the same steps, we start from formulae given in section 3.2 and factor the discriminant: Δ = −28 36 τ 3 (τ 3 + 1)6 (τ 3 − 8)3 The values of interest for τ0 are {−1, −ζ3, −ζ32 , 0, 2, 2ζ3 , 2ζ32 }. Only the ﬁrst 3 values can be used up to the second power, the four last ones being of interest only to the ﬁrst power. The solution for all modular constraints is Ξ = 2τ 9 − 9τ 6 − 42τ 3 − 4 The 432 possible candidates for ξ yield 32 cases where σ0 is of degree 4. Among them, one of the simplest corresponds to σ0 = −3τ (5τ 3 + 32) with ξ = −13τ 6 − 44τ 3 − 4. The elliptic curve ρ2 = −3τ (5τ 3 + 32) has nonzero rank over Q, a point of inﬁnite order being (−1, 9). The points of this auxiliary elliptic curve parameterize an inﬁnite family of elliptic curve having nonzero rank over Q and a torsion group containing Z/6Z × Z/3Z over Q(ζ3 ) 4.4

Results for Z/5Z × Z/5Z

Once again, we start by factoring: Δ = −28 312 τ 5 (τ 10 − 11τ 5 − 1)5 The eleven values of interest for τ0 can all be used up to the second power and the polynomial compatible with all constraints is Ξ=−

1 (252τ 20 − 5508τ 15 + 29019τ 10 + 7686τ 5 + 75) 25

Unfortunately, the 311 possible candidates for ξ all give σ0 polynomials of degree ﬁve or more, except for those corresponding to 5-torsion points. We also noticed that α is of degree 20 and β of degree 30. If we restrict ourselves to polynomials of degree 10 for ξ, the degree of σ will not exceed 30. In the case the leading coeﬃcient of ξ is −3, the degree of σ falls down to 28. One can see this as using the value τ0 = ∞. This is compatible with 10 modular constraints and we also tried the 24068 candidates built this way, with no success. Remark: To speed up computations and avoid to compute in a quartic extension of Q, we instead performed this computations in the ﬁeld F32621 , which contains ﬁfth roots of unity. For sure, if a solution had been found, we would have needed to perform actual computations in Q(ζ5 ).

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4.5

107

Half Way to Z/8Z × Z/4Z

As the modular curve X1 (8, 4) is of genus one, we lack freedom on curve parameters to ensure in addition a non zero rank. We will try to cover a part of the path from X1 (4, 4) to X1 (8, 4). We elaborate on the results of section 4.2 and will use the same parameterization, with a dedicated choice of values for the parameter ν. As in section 3.5, we use the characterization of points P that can be written P = [2]Q with Q a point with rational coordinates. Among the twelve points of order 4, there is one that needs the quantities ν(ν 2 + 3) and (ν 2 + 1)(ν 2 + 9) to be squares. Would both be squares, we would get a point of order 8. We limit ourselves to the ﬁrst condition only. At this stage, it is quite natural to consider the elliptic curve μ2 = ν(ν 2 + 3). The rank of this auxiliary curve over the ﬁeld Q is one and an inﬁnite subgroup is generated by the point P0 = (1, 2). Each of the points [k]P0 with k ∈ N yields a value of ν to be plugged into the formulæ of section 4.2. We thus get a inﬁnite family of elliptic curves with nonzero rank, Z/4Z×Z/4Z torsion over Q(ζ4 ), and better chances to get Z/8Z × Z/4Z torsion over the same number ﬁeld.

5

Application to Factoring

One can see an ECM implementation as a black box taking as inputs: A number N to be factored Elliptic curve paramaters a and b Coordinates of a point P on the curve modulo N and computing the scalar multiplication M · P on this curve for a smooth large integer M, expecting the result being at inﬁnity for some prime factor of N . In most implementations, projective coordinates are used and if M · P is at inﬁnity modulo a prime factor, this factor can be retrieved by a simple GCD between the number to be factored and the third coordinate. For full explanations on implementations and improvements of ECM, see [10], [1] and [7]. For the torsion groups Z/4Z×Z/4Z and Z/6Z×Z/3Z, we found curves having parameters and a point of inﬁnite order deﬁned over Q. These curves can be used for any number to be factored N . However, the beneﬁt of torsion is attained only when one knowns that suitable roots of unity exist in the ﬁnite ﬁelds deﬁned by prime factors of N . For order 16 torsion groups, numbers of the form a4n −b4n or a2n + b2n satisfy these conditions. The torsion group of order 18 can be used on numbers of the form a3n ± b3n . The suggested extension towards torsion group of order 32 can be used for numbers of the form a8n − b8n or a4n + b4n . To implement results of section 4.2, the parameter ν can be chosen at random or iteratively on integers. To implement results of section 4.3, things are slightly

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less simple, since an auxiliary elliptic curve has to be used. In this case one has to select an integer k randomly or in sequence, compute a scalar multiplication on the auxiliary elliptic curve to get the inputs of ECM. We adapted our ECM implementation in order to use these new families of elliptic curves. Making use of results on Z/4Z × Z/4Z torsion we found several factors of Cunningham numbers. Among them, one can mention the larger one: 5546025484206613872527377154544456740766039233|21048 + 1 We won’t give here full details of the factorization since they do not correspond to notations of these paper, this factor having been found in an early stage of development of this paper. We also implemented the variant with Z/6Z×Z/3Z torsion. Among the factors we found, the larger to mention is 1581214773543289355763694808184205062516817|2972 + 1 This factor has been discovered using the input parameters: ⎧ a = 29826081614523423723477944537088124780779 ⎪ ⎪ ⎨ b = 129980809632665349776106077981744185363149 ⎪ x = 479946793455925131408573042432160264988537 ⎪ ⎩ y = 341223966666174229961942234304018968605682

mod p mod p mod p mod p

The order of the curve modulo p factors as: #E(Fp ) = 2 × 32 × 29 × 241 × 691 × 5279 × 20353 × 252589 × 1489097 × 2258261 × 199312079

6

Conclusion

We exhibited two torsion groups, that can be used for ECM factoring, of orders 16 and 18. Classical implementations make use of the torsion group Z/8Z×Z/2Z that can be used for all numbers but of slightly smaller order. It would be really interesting to have a precise analysis of complexity improvements obtained by using torsion groups, as well as partial construction of torsion structure as in section 4.5. In the case of torsion group of order 25, we did not succeed in constructing elliptic curves having nonzero rank. This by no way means that no such curves exist. Solving this issue would result in speciﬁc implementations for numbers of the form a5n ± 1 with the larger available torsion group. Some torsion groups correspond to a modular curve of genus one. The obstruction in using them for ECM is the lack of freedom to build curve with nonzero rank: to build a curve with this torsion, one only have to select a multiple of a generator on this modular curve. Several approaches could improve the situation: being able to construct a large number of curves with nonzero rank

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by using rank computation software or being able to construct a point on the curve modulo N after the curve has been generated. Last, while inﬁnite families of curves are needed for ECM factoring of integers, individual curves providing large torsion groups over some number ﬁelds could be used during the sieving phase of the special number ﬁeld sieve (see [5] and [6]). Though further research is needed to hunt for interesting individual curves, we quote one preliminary result: the choice of ν = 1 in section 4.2 ensures a torsion subgroup of order 32 over the ﬁelds Q(ζ24 ) and Q(ζ40 ) and of order 64 over the ﬁeld Q(ζ120 ).

References 1. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1991) 2. Knapp, A.W.: Elliptic Curves. Princeton University Press, Princeton (1992) 3. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97. Springer, Heidelberg (1993) 4. Kubert, D.S.: Universal bounds on the torsion of elliptic curves. In: Proceedings of the London Mathematical Society, pp. 193–237 (1976) 5. Lenstra, A.K., Lenstra, H.W.: The Development of the Number Field Sieve. LNM, vol. 1554. Springer, Heidelberg (1993) 6. Lenstra, A.K., Lenstra, H.W., Manasse, M.S., Pollard, J.M.: The Factorization of the Ninth Fermat Number. In: Mathematics of Computation, vol. 61. American Mathematical Society, Providence (1993) 7. Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987) 8. Mazur, B.: Rational isogenies of prime degree. Invent. Math., 129–162 (1978) 9. Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987) 10. Zimmermann, P., Dodson, B.: Twenty Years of ECM. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 525–542. Springer, Heidelberg (2006)

Visualizing Elements of Sha[3] in Genus 2 Jacobians Nils Bruin and Sander R. Dahmen Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada [email protected], [email protected]

Abstract. Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H 1 (Gal(k/k), E) → H 1 (Gal(k/k), A). However, the abelian surface in Mazur’s construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the ShafarevichTate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

1

Introduction

Let E be an elliptic curve over a ﬁeld k with separable closure k. We write H 1 (k, E[3]) := H 1 (Gal(k/k), E[3](k)) for the ﬁrst galois cohomology group taking values in the 3-torsion of E (the notation H i (k, A) is used similarly for other group schemes A/k later in this paper). We are primarily concerned with the question which δ ∈ H 1 (k, E[3]) are visible in the jacobian of a genus 2 curve. Mazur deﬁnes visibility in the following way. Let 0 → E → A → B → 0 be a short exact sequence of abelian varieties over k. By taking galois cohomology, we obtain the exact sequence A(k)

/ B(k)

/ H 1 (k, E)

φ

/ H 1 (k, A) .

(1.1)

Elements of the kernel of φ are said to be visible in A. Mazur chose this term because a model of the principal homogeneous space corresponding to an element ξ ∈ H 1 (k, E) that is visible in A can be obtained as a ﬁber of A over a point in B(k) (this can readily be seen from (1.1)). By extension, we say that δ ∈ H 1 (k, E[n]) is visible in A if the image of δ under the natural homomorphism H 1 (k, E[n]) → H 1 (k, E) is visible in A. Let us restrict to the case that k is a number ﬁeld for the rest of this section. Inspired by some surprising experimental data [4], Mazur [5] proved, that for any element ξ in the Shafarevich-Tate group X(E/k) of order three, there exists an abelian variety A over k such that ξ is visible in A. The abelian variety that

Research of both authors supported by NSERC.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 110–125, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Mazur constructs is almost never principally polarizable over k and hence is almost never a jacobian of a genus 2 curve. In the present paper, we show that any element from X(E/k)[3] is in fact visible in the jacobian of a genus 2 curve. Moreover, we describe how to get an explicit model of such a genus 2 curve.

2

Torsors and Theta Groups

Throughout this section let n > 1 be an integer, let k be a perfect ﬁeld of characteristic not dividing n and let E denote an elliptic curve over k. In [2], many equivalent interpretations are given for the group H 1 (k, E[n]). For our purposes, we need two classes of objects. The ﬁrst is most closely related with descent in general and our question in particular. We consider E-torsors under E[n](k) and, following [2], call them n-coverings. Definition 1. An n-covering π : C → E of an elliptic curve E is an unramified covering over k that is galois and irreducible over k, with Autk (C/E) E[n](k). Two n-coverings π1 : C1 → E, π2 : C2 → E are called isomorphic if there exists a k-morphism φ : C1 → C2 such that π1 = π2 ◦ φ. Over k, all n-coverings are isomorphic to the trivial n-covering, the multiplicationby-n map [n] : E → E. Proposition 1 ([2, Proposition 1.14]). The k-isomorphism classes of n-coverings of E are classified by H 1 (k, E[n]). For δ ∈ H 1 (k, E[n]) we denote by Cδ the curve in the covering Cδ → E corresponding to δ. We remark that δ ∈ H 1 (k, E[n]) has trivial image in H 1 (k, E) if and only if Cδ has a k-rational point. We write O for the identity on E. The complete linear system |n·O| determines a morphism E → Pn−1 , where the translation action of E[n] extends to a linear action on Pn−1 . This gives a projective representation E[n] → PGLn . The lift of this representation to GLn gives rise to a group ΘE , which ﬁts in the following diagram. 1

/ Gm

1

/ Gm

αE

/ ΘE / GLn

βE

/ E[n]

/1

/ PGLn

/1

(2.1)

The group E[n](k) carries additional structure. It also has the Weil pairing eE , which is a non-degenerate alternating galois covariant pairing taking values in the n-th roots of unity eE : E[n](k) × E[n](k) → μn (k). The commutator of ΘE corresponds to the Weil pairing, meaning that for x, y ∈ ΘE we have xyx−1 y −1 = αE (eE (βE (x), βE (y))).

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Definition 2. A theta group for E[n] is a central extension of group schemes β

α

1 → Gm → Θ → E[n] → 1 such that the Weil-pairing on E[n] corresponds to the commutator, i.e. for x, y ∈ Θ we have xyx−1 y −1 = α(eE (β(x), β(y))). Two theta groups 1 → Gm → Θi → E[n] → 1,

i = 1, 2

are called isomorphic if there exists a group scheme isomorphism φ : Θ1 → Θ2 over k making the following diagram commutative. 1

/ Gm

/ Θ1

/ Gm

/ Θ2

/ E[n]

/1

/ E[n]

/1

φ

1

Over k, all theta-groups are isomorphic to ΘE as central extensions; see [2, Lemma 1.30]. Proposition 2. ([2, Proposition 1.31]). Let E[n] be the n-torsion subscheme of an elliptic curve E over a field k, equipped with its Weil pairing. The isomorphism classes of theta-groups for E[n] over k are classified by H 1 (k, E[n]). The theta group associated to δ ∈ H 1 (k, E[n]) may allow for a matrix representation Θ → GLn that ﬁts in a diagram like (2.1). This is measured by the obstruction map Ob introduced in [6] and [2]. This map can be obtained by taking non-abelian galois cohomology of the deﬁning sequence of ΘE : Ob

· · · −→ H 1 (k, ΘE ) −→ H 1 (k, E[n]) −→ H 2 (k, Gm ) = Br(k) −→ · · · . Note that, except in some trivial cases, Ob is not a group homomorphism. The map Ob also has an interpretation in terms of n-coverings. Let C → E be an n-covering associated to δ ∈ H 1 (k, E[n]). We have that Ob(δ) = 0 if and only if C admits a model C → Pn−1 with Autk (C/E) = E[n](k) acting linearly, in which case C is k-isomorphic to E as a curve and the covering C → E is simply a translation composed with multiplication-by-n. Remark 1. Note that if k is a number ﬁeld, then any element in Br(k) that restricts to the trivial element in Br(kv ) in all completions kv of k, is trivial itself. It follows that Ob is trivial on the n-Selmer group S (n) (E/k).

Visualizing Elements of Sha[3] in Genus 2 Jacobians

3

113

Visibility in Surfaces

Let E1 be an elliptic curve over a perfect ﬁeld k of characteristic distinct from 3. In what follows, we will consider δ ∈ H 1 (k, E1 [3]) with Ob(δ) = 0. A possible way of constructing an abelian surface A such that δ is visible in A starts by taking a suitable elliptic curve E2 /k together with a k-group scheme isomorphism λ : E1 [3] → E2 [3]. Let Δ ⊂ E1 × E2 be the graph of λ so that Δ(k) = {(P, λ(P )) : P ∈ E1 [3](k)}. Let A := (E1 × E2 )/Δ and write φ : E1 × E2 → A for the corresponding isogeny. Since Δ ⊂ E1 [3] × E2 [3], we have another isogeny φ : A → E1 × E2 such that φ

φ ◦ φ = 3. We write p∗ for the composition E1 → (E1 × E2 ) → A and p∗ for the φ

composition A → (E1 × E2 ) → E1 and q ∗ , q∗ for the corresponding morphisms concerning E2 . It is straightforward to verify that p∗ , q ∗ are embeddings, that φ = p∗ − q ∗ (where the projections are understood), and that φ = p∗ × q∗ . We combine the galois cohomology of the short exact sequences p∗

q∗

q∗

p∗

0 → E1 → A → E2 → 0, 0 → E2 → A → E1 → 0, and 3

0 → Ei [3] → Ei → Ei → 0 for i = 1, 2 to obtain the big (symmetric) commutative diagram with exact rows and columns E2 (k) 3

E2 (k) α

E1 (k)

3

/ E1 (k)

/ H 1 (k, Δ)

p∗

/ E1 (k)

/ H 1 (k, E2 )

p∗

A(k)

q∗

/ A(k) q∗

E2 (k) / H 1 (k, E1 ) / H 1 (k, A)

where we note that H 1 (k, Δ) H 1 (k, E1 [3]) H 1 (k, E2 [3]). We see that δ is visible in A precisely if δ ∈ H 1 (k, E1 [3]) = H 1 (k, Δ) lies in the image of α, i.e., if the curve Cλ(δ) corresponding to λ(δ) ∈ H 1 (k, E2 [3]) has a rational point. We summarize these observations, which are due to Mazur. Lemma 1. Let E1 be an elliptic curve over a perfect field k of characteristic distinct from 3 and let δ ∈ H 1 (k, E[3]) with Ob(δ) = 0. Suppose that there exists an elliptic curve E2 /k and a k-group scheme isomorphism λ : E1 [3] → E2 [3] such that the curve Cλ(δ) corresponding to λ(δ) has a k-rational point. Then δ is visible in the abelian surface (E1 × E2 )/Δ where Δ denotes the graph of λ.

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Mazur also observed, in the case of a number ﬁeld k, that if δ ∈ S (3) (E/k), then Cδ admits a plane cubic model. Furthermore, there is a pencil of cubics through the 9 ﬂexes of Cδ , and each non-singular member corresponds to a 3-covering Ct → Et , where Et [3] E[3] and Ct → Et represents δ. It is therefore easy to ﬁnd a t such that Ct has a rational point; simply pick a rational point and solve for t. To reﬁne the construction, one can ask Question 1. Can one make δ ∈ H 1 (k, E[3]) visible in the jacobian of a genus 2 curve? Note that E1 ×E2 is principally polarized via the product polarization. This gives rise to a Weil pairing on (E1 × E2 )[3], corresponding to the product pairing. If A is a jacobian, then A must be principally polarized over k. One way this could happen is if the isogeny φ : E1 × E2 → A gives rise to a principal polarization. This would be the case if the kernel Δ is a maximal isotropic subgroup of E1 [3]× E2 [3] with respect to the product pairing. That means that λ : E1 [3] → E2 [3] must be an anti-isometry, i.e. for all P, Q ∈ E1 [3] we must have eE2 (λ(P ), λ(Q)) = eE1 (P, Q)−1 . Note that the original cubic C is a member of the pencil that Mazur constructs, so in his construction λ is actually an isometry, i.e. it preserves the Weil-pairing. Below we consider a pencil of cubics that leads to an anti-isometry λ.

4

Anti-isometric Pencils

Let k be a perfect ﬁeld of characteristic distinct from 2, 3. Following [7], we associate to a ternary cubic form F ∈ k[x, y, z] three more ternary cubic forms. Namely, the Hessian of F ∂F 2 ∂F 2 ∂F 2 ∂x∂y ∂x∂z 1 ∂x∂x 2 2 2 ∂F ∂F ∂F , H(F ) := − ∂y∂x ∂y∂z 2 ∂F 2 ∂y∂y 2 ∂F ∂F 2 ∂z∂x ∂z∂y ∂z∂z

the Caylean of F

∂F (0, z, −y) ∂x 1 ∂F P (F ) := − ∂x (−z, 0, x) xyz ∂F ∂x (y, −x, 0)

∂F ∂y (0, z, −y) ∂F ∂y (−z, 0, x) ∂F ∂y (y, −x, 0)

∂F ∂z (0, z, −y) ∂F ∂z (−z, 0, x) ∂F ∂z (y, −x, 0)

and a ternary cubic form denoted Q(F ), for which we refer to [7, Section 11.2]. For most cases one can take Q(F ) to be H(P (F )) or P (H(F )), but there are some exceptional cases where P (F ), Q(F ) span an appropriate pencil and P (F ), H(P (F )) do not. The left action of GL3 on k 3 induces a right action of GL3 on ternary cubic forms (or, more generally, on k[x, y, z]). For a ternary cubic form F and an M ∈ GL3 we denote this action simply by F ◦M . The signiﬁcance

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of the three associated ternary cubic forms lies in the fact that H(F ) depends covariantly on F (of weight 2) and P (F ) and Q(F ) depend contravariantly on F (of weights 4 and 6 respectively). This means that for every ternary cubic form F and every M ∈ GL3 we have, with d := det M that H(F ◦ M ) = d2 H(F ) ◦ M P (F ◦ M ) = d4 P (F ) ◦ M −T Q(F ◦ M ) = d6 Q(F ) ◦ M −T , where M −T denotes the inverse transpose of M . Now consider a smooth cubic curve C in P2 given by the zero locus of a ternary cubic form F . Then C has exactly 9 diﬀerent ﬂex points Φ, which all lie on the (not necessarily smooth) curve given by H(F ) = 0. The smoothness of C guarantees that F and H(F ) will be linearly independent over k. Hence Φ can be described as the intersection F = H(F ) = 0. We call Φ the flex scheme of C. At least one of P (F ) and Q(F ) turns out to be nonsingular (still assuming that C is nonsingular) and the intersection P (F ) = Q(F ) = 0 equals the ﬂex points Φ∗ of the nonsingular cubics among P (F ) and Q(F ) (if, say, P (F ) is nonsingular, then Φ∗ can of course also be written as P (F ) = H(P (F )) = 0). We can consider the pencil of cubics through Φ, explicitly given by C(s:t) : sF (x, y, z) + tH(F )(x, y, z) = 0.

(4.1)

Classical invariant theory tells us the following. This pencil has exactly 4 singular members and all other members have ﬂex scheme equal to Φ. Conversely, any nonsingular cubic with ﬂex scheme Φ occurs in this pencil. Furthermore, both P (sF + tH(F )) and Q(sF + tH(F )) are linear combinations of P (F ) and Q(F ). This shows that the ﬂex scheme Φ∗ is independent of the choice of C through Φ and only depends on Φ. We call Φ∗ the dual ﬂex scheme of Φ and we will justify this name below. As a simple, but important example we take F := x3 + y 3 + z 3 . Then we compute H(F ) = −108xyz,

P (F ) = −54xyz,

Q(F ) = 324(x3 + y 3 + z 3 ).

Now deﬁne Φ0 to be the ﬂex scheme of F = 0, i.e. Φ0 := {[x : y : z] ∈ P2 : x3 + y 3 + z 3 = xyz = 0}.

(4.2)

Then we see that the ﬂex scheme given by P (F ) = Q(F ) = 0 (which is the ﬂex scheme of Q(F ) = 0) equals Φ0 , i.e. Φ∗0 = Φ0 . The pencil of cubics through Φ0 (note that 108 = 0 in k), which is given by s(x3 + y 3 + z 3 ) = txyz,

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is a model over k for the universal elliptic curve over the (genus zero) modular curve X(3); see [5, p. 225]. Geometrically all ﬂex schemes are linear transformations of each other. In particular, for any ﬂex scheme Φ there exists an M ∈ GL3 (k) such that Φ = M Φ0 . This shows that the pencil (4.1) associates to a general ﬂex scheme Φ is a twist of the universal elliptic curve over X(3). The contravariance of P and Q implies that the assignment Φ → Φ∗ has the contravariance property that for any ﬂex scheme Φ and M ∈ GL3 (M Φ)∗ = M −T Φ∗ .

(4.3)

We also note that this implies that the assignment Φ → Φ∗∗ := (Φ∗ )∗ is covariant in the sense that for any ﬂex scheme Φ and M ∈ GL3 we have (M Φ)∗∗ = M Φ∗∗ . Writing Φ = M Φ0 and using (Φ0 )∗∗ = Φ∗0 = Φ0 we now get Φ∗∗ = (M Φ0 )∗∗ = M Φ∗∗ 0 = M Φ0 = Φ. This justiﬁes calling Φ∗ the dual ﬂex scheme of Φ. Remark 2. In the discussion above it was convenient to consider just one projective plane P2 . A more canonical way would be to consider a projective plane P2 with coordinates x, y, z (for a point) and the dual projective plane, denoted (P2 )∗ , where the point with coordinates u, v, w describes the line ux+vy +wz = 0. Now let C be a smooth cubic curve in P2 given by the zero locus of the ternary cubic form F (x, y, z) with ﬂex scheme Φ. The 9 tangent lines through Φ determine 9 points in (P2 )∗ . Generically, these 9 points in (P2 )∗ will not be the ﬂex points of a smooth cubic curve, hence generically there will a unique cubic curve going through these points. This curve in (P2 )∗ is exactly given by the zero locus of the Caylean, i.e. P (F )(u, v, w) = 0; see also [10, pp.151,190–191]. Moreover, if the characteristic of k is zero, then it turns out that this cubic curve is nonsingular if and only if the j-invariant of C is nonzero. To any ﬂex scheme Φ we associate a group Θ(Φ) ⊂ GL3 as follows. Choose a nonsingular cubic curve C through Φ and let E be its jacobian. After identifying E and C as curves over k, we get an action of E[3] on C, which extends to a linear action on P2 . This determines an embedding χ : E[3] → PGL3 . Obviously, the image χ(E[3]) only depends on Φ. We deﬁne Θ(Φ) to be the inverse image of χ(E[3]) in GL3 . Actually Θ(Φ) can be deﬁned just in terms of Φ, without choosing C, since it turns out that χ(E[3]) consists exactly of the linear transformations that preserve Φ. (One way of quickly ﬁnding these linear transformations explicitly is by using the fact that, for any two distinct points of Φ, the line through these two points intersects Φ in a unique third point.) The construction gives rise to the theta group 1 → Gm → Θ(Φ) → E[3] → 1. Note that the isomorphism class of this theta group may still depend on the choice of identiﬁcation of C with E. This corresponds to the choice of an isomorphism between Θ(Φ)/Gm and E[3]. If Φ is deﬁned over k, then E[3] and

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Θ(Φ) are also deﬁned over k and the element in H 1 (k, E[3]) corresponding to this theta group is the same as the element corresponding to the 3-covering C → C/E[3] E for any nonsingular cubic curve C through Φ. The construction also shows that for any M ∈ GL3 we have Θ(M Φ) = M Θ(Φ)M −1 .

(4.4)

Proposition 3. Let Φ1 ⊂ P2 be a flex scheme and let Φ2 := Φ∗1 be the dual flex scheme. For i = 1, 2 let Ci be a smooth plane cubic with flex scheme Φi , denote its jacobian by Ei and consider an induced theta group 1

/ Gm

αi

/ Θ(Φi )

βi

/ Ei [3]

/1.

(4.5)

Then the outer automorphism (−T ) : GL3 → GL3 given by M → M −T , yields an isomorphism Θ(Φ1 ) → Θ(Φ2 ). There exists an anti-isometry λ : E1 [3] → E2 [3] making the following diagram commutative. 1

/ Gm

α1

x→x−1

1

/ Gm

/ Θ(Φ1 )

β1

(−T )

α2

/ Θ(Φ2 )

β2

/ E1 [3]

/1

(4.6)

λ

/ E2 [3]

/1

In particular, let δi ∈ H 1 (k, Ei [3]) correspond to the theta group (4.5). Then under the isomorphism H 1 (k, E1 [3]) H 1 (k, E2 [3]) induced by λ, the cocycle δ1 maps to δ2 . Proof. Once the isomorphism Θ(Φ1 ) → Θ(Φ2 ) given by M → M −T is established, the existence of an isomorphism λ : E1 [3] → E2 [3] making the diagram (4.6) commutative, follows immediately. That λ must be an anti-isometry can readily be seen as follows. Let P, Q ∈ E1 [3] and choose x, y ∈ Θ(Φ1 ) such that P = β1 (x) and Q = β1 (y). Then α2 (eE2 (λ(P ), λ(Q))) = α2 (eE2 (β2 (x−T ), β2 (y −T ))) = x−T y −T xT y T = (xyx−1 y −1 )−T = α1 (eE1 (β1 (x), β1 (y)))−T = α1 (eE1 (P, Q)−1 ). The last statement of the proposition is also immediate, so we are left with ∼ establishing (−T ) : Θ(Φ1 ) → Θ(Φ2 ). It suﬃces to show that for a ﬂex scheme 2 −T Φ ⊂ P we have Θ(Φ) = Θ(Φ∗ ). Write Φ = M Φ0 for some M ∈ GL3 with Φ0 given by (4.2). Then a straightforward calculation shows that Θ(Φ0 )−T = Θ(Φ0 ). We also know that Φ∗0 = Φ0 , so we get Θ(Φ0 )−T = Θ(Φ∗0 ). Together with (4.3) and (4.4) we ﬁnally obtain,

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Θ(Φ)−T = Θ(M Φ0 )−T = M −T Θ(Φ0 )−T M T = M −T Θ(Φ∗0 )(M −T )−1 = Θ(M −T Φ∗0 ) = Θ((M Φ0 )∗ ) = Θ(Φ∗ ).

Remark 3. The construction above of the dual ﬂex scheme Φ∗ of a ﬂex scheme Φ involved choosing a smooth cubic going through Φ. Without using theta groups, it was not obvious from this construction that the degree 9 ´etale algebra k(Φ) is isomorphic to k(Φ∗ ). However, there exists a nice explicit geometric construction of the dual ﬂex scheme that remedies these shortcomings of the earlier construction. Given a ﬂex scheme Φ, we proceed as follows. We label its 9 points over k with P1 , . . . , P9 . There are 4 sets of 3 lines, (corresponding to the 4 singular members of the pencil of cubics through φ) containing these points. We label the line that contains Pi , Pj , Pk with l{i,j,k} . One can label the points such that the subscripts are {1, 2, 3} {1, 4, 7} {1, 5, 9} {1, 6, 8} {4, 5, 6} , {2, 5, 8} , {2, 6, 7} , {2, 4, 9} , {7, 8, 9} {3, 6, 9} {3, 4, 8} {3, 5, 7} Naturally, two diﬀerent lines l{i1 ,j1 ,k1 } , l{i2 ,j2 ,k2 } meet in a unique point. If for example i1 = i2 , then the intersection point is Pi1 . If the two sets {i1 , j1 , k1 } {i2 , j2 , k2 } are disjoint, then the two lines meet in a point outside Φ. We name this point L{i3 ,j3 ,k3 } , where {i1 , j1 , k1 , i2 , j2 , k2 , i3 , j3 , k3 } = {1, . . . , 9}. As it turns out, the four points that have i in their label all lie on a line pi . It is also straightforward to check that the pi together with the L{i,j,k} form a conﬁguration in (P2 )∗ that is completely dual to the Pi with the l{i,j,k} . The pi form the k points of a ﬂex scheme in (P2 )∗ , which is justiﬁably a ﬂex scheme Φ∗ dual to Φ, and its construction immediately implies the contravariance property (M Φ)∗ = M −T Φ∗ . We can easily verify that the two constructions of Φ∗ coincide for one ﬂex scheme, for instance Φ0 . The general result then follows because any ﬂex scheme can be expressed as M Φ0 for some M ∈ GL3 (k). Since the action of Gal(k/k) on {P1 , . . . , P9 } must act via collinearity-preserving permutations, we see that if σ(Pi ) = Pσ(i) , then σ(pi ) = pσ(i) . Hence, we see that the k-points of Φ and its dual have the same Galois action and hence k(Φ) is isomorphic as a k-algebra to k(Φ∗ ).

5

Recovering the Genus 2 Curve

Let k be a ﬁeld and let E1 , E2 be two elliptic curves over k with an anti-isometry λ : E1 [3] → E2 [3] and denote by Δ the graph of λ as before. Recall that E1 × E2 is

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principally polarized via the product polarization and that the induced polarization on A := (E1 × E2 )/Δ is also principal in this case. It is a classical fact that if A is not geometrically isomorphic to a product of elliptic curves, then A (together with its principal polarization) is isomorphic to the jacobian of a genus 2 curve C. Let us assume from now on that E1 and E2 are non-isogenous. In [8] it is shown that in this case A is always isomorphic over k to the jacobian of a genus 2 curve C/k. This is enough to get our main theoretical result. Theorem 4. Let E be an elliptic curve over a number field k and let ξ ∈ X(E/k)[3]. Then ξ is visible in the jacobian of a genus 2 curve C/k. Proof. Let δ ∈ S (3) (E/k) be a cocycle representing ξ. By Proposition 2, there is a 3-covering Cδ → E corresponding to δ. According to Remark 1, we have that Ob(δ) = 0 and hence that Cδ ⊂ P2 . Let Φ ⊂ P2 be its ﬂex scheme. The construction in Section 4 gives us a pencil of cubics through Φ∗ , so we can easily pick a non-singular one with a rational point. It follows from Proposition 3 that such a curve is of the form Cλ(δ) for some elliptic curve E2 and some anti-isometry λ : E[3] → E2 [3]. This places us in the situation of Lemma 1, so δ is visible in an abelian surface A = (E × E2 )/Δ. We have ensured that λ is an anti-isometry, which implies that the surface is principally polarized. As long as we make sure that E, E2 are non-isogenous (and this is easy given the freedom we have in choosing Cλ (δ)) it follows that A is a jacobian.

Remark 4. We could of course state a more general result about visibility of elements δ ∈ H 1 (k, E[3]) with Ob(δ) = 0 for an elliptic curves E over a perfect ﬁeld k of characteristic distinct from 2 or 3. Note however that if k is too small, there might not be enough non-isogenous elliptic curves available. The exclusion of ﬁelds of characteristic 3 is a serious one, the exclusion of non-perfect ﬁelds less so. Most of what we are saying could be generalized to the non-perfect case, basically because for an elliptic curve over any ﬁeld of characteristic distinct from 3, the multiplication by 3 map is separable. The exclusion of ﬁelds of characteristic 2 stems from the fact that the necessary invariant theory in this case is not readily available. We continue with the construction of the genus 2 curve C. Deﬁne the divisor Θ := 01 × E2 + E1 × 02 on E1 × E2 , which gives a principal polarization on E1 × E2 . Next, consider the set D of eﬀective divisors on E1 × E2 over k which are linearly equivalent to 3Θ and invariant under Δ. Also consider the set C of eﬀective divisors C on A over k whose pull-back to E1 × E2 are linearly equivalent to 3Θ and which satisfy (C · C) = 2. Frey and Kani show that there exist unique curves D ∈ D and C ∈ C deﬁned over k which are invariant under multiplication by −1. Furthermore, because E1 and E2 are not isogenous, D and C are irreducible smooth curves of genus 10 and 2 respectively and the natural map D → C is unramiﬁed of degree 9. If k is a perfect ﬁeld of characteristic distinct from 2 or 3, the curves D and C can be explicitly constructed as follows. Embed E1 in P2 , given by, say

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F (x, y, z) = 0, for a ternary cubic F/k (such an F is readily obtained if E1 is given by a Weierstrass model). Express E2 as G := sP (F ) + tQ(F ) = 0 for some s, t ∈ k. This way, we obtain an embedding of E1 × E2 in P2 × P2 given by F (x, y, z) = G(u, v, w) = 0. Moreover, by appealing to Proposition 3 we obtain that the curve on this surface given by xu + yv + zw = 0 must be the curve D. The genus 2 curve C is the image of D in (E1 × E2 )/Δ. E1 × E2O OOO OOO OOO O' [3]×[3] (E1 × E2 )/Δ o ooo o o oo wooo E1 × E2 The map [3] × [3] is much more accessible, though. We claim that the subgroup of E1 [3] × E2 [3] under which D is invariant is equal to Δ. Hence, we can ﬁnd a (singular) model of C as a curve on E1 × E2 by computing ([3] × [3])(D). This can easily be done via interpolation, as explained in the next section by means of an example. As for our claim above, suppose that D is invariant under some σ ∈ E1 [3] × E2 [3] with σ ∈ / Δ. Without loss of generality we may assume that σ = (P, 0E2 ) ∈ E1 [3] × E2 [3] with P = 0E1 . Denote by M ∈ PGL3 (k) the linear action corresponding to translation by P . Now for all ([x : y : z], [u : v : w]) on D we have (x, y, z)(u, v, t)T = (x, y, z)M T (u, v, w)T = 0. This yields (u, v, t) = (x, y, z) × (x, y, z)M T , where × denotes the standard cross product. This association actually deﬁnes a birational transformation φ : P2 → P2 (a Cremona transformation with singular points corresponding to the eigenspaces of M ). Note that φ is deﬁned on all the [x : y : z] on E1 , so the image of E1 under φ is an irreducible curve birational to E1 . Together with the assumption that E1 and E2 are not isogenous, we get that this image intersects E2 in only ﬁnitely many points, so D is not invariant under σ.

6

Examples

Following the ﬁrst example in [4, Table 1], consider the elliptic curve 681b1 (in Cremona’s notation), given by the minimal Weierstrass equation E1 : y 2 + xy = x3 + x2 − 1154x − 15345. It turns out that the plane cubic curve C1 : x3 + 5x2 y + 5x2 z + 2xy 2 + xyz + xz 2 + y 3 − 5y 2 z + 2yz 2 + 6z 3 = 0

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deﬁnes an element ξ (up to inverse) of order three in X(E1 /Q). The contravariants, denoted P0 , Q0 , are given by P0 = −478x3 + 2525x2 y + 916x2 z − 1127xy 2 + 29xyz −160xz 2 + 753y 3 − 1228y 2z + 260yz 2 + 301z 3, Q0 = −122314x3 + 618551x2y + 191092x2z − 271157xy 2 − 7825xyz −28120xz 2 + 184011y 3 − 264916y 2z + 55892yz 2 + 73663z 3. Now the curve C2 : 55033P0 − 235Q0 = 0 has a rational point [x : y : z] = [10 : 8 : 7] and its jacobian is the elliptic curve 681c1, given by the minimal Weierstrass equation E2 : y 2 + y = x3 − x2 + 2. To construct the corresponding genus two curve C such that ξ becomes visible in its jacobian we could take the curve in C1 × C2 ⊂ P2 × P2 with coordinates ([x : y : z], [u : v : w]) given by the equation xu + yv + zw = 0, and take its image under C1 × C2 → E1 × E2 , since this is a twist of [3] × [3] : E1 × E2 → E1 × E2 anyway. We will follow Section 5 more closely. Obviously, E1 is given by F = 0 if we deﬁne F := y 2 z + xyz − (x3 + x2 z − 1154xz 2 − 15345z 3). The contravariants of the ternary cubic F are given by P = −2308x3 + 3462x2 y − 5x2 z − 275056xy 2 + 5xyz +6xz 2 + 136951y 3 + 13853y 2z − 3yz 2, Q = −725020x3 + 1087530x2y + 27721x2z − 65861608xy 2 − 27721xyz −30xz 2 + 32749549y 3 + 3217559y 2z + 15yz 2 + 24z 3 . Write j(s, t) for the j-invariant of the curve given by sP +tQ = 0. The j-invariant of E2 equals −4096/2043 and the equation j(s, t) = −4096/2043 has exactly one solution in P1 (Q), namely [s : t] = [55033 : −235] (compare with the deﬁnition of C2 ). This gives us a new model for E2 , namely E2 : 55033P − 235Q = 0. We consider the surface E1 × E2 embedded in P2 × P2 as F (x, y, z) = 0,

55033P (u, v, w) − 235Q(u, v, w) = 0.

The curve D on this surface is given by xu + yv + zw = 0.

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The image of D under multiplication by 3 on E1 × E2 is the genus two curve C. Using the deﬁning properties of C from Section 5 (such as the invariance under multiplication by −1), we get that as a curve on E1 × E2 it must be of the form axu + byv + czw + dxw + ezu = 0 for some a, b, c, d, e ∈ Q. We simply generate 4 points on C (over a number ﬁeld), compute the image under multiplication by 3 of these points and solve for a, b, c, d, e. If the dimension of the solution space is greater than 1, we must of course add points (or take 4 better ones) so that the solution space becomes 1-dimensional. This gives us our equation for C. By a linear change of the u, v, w coordinates we can change the model for E2 back to the original minimal Weierstrass model. Thus, the model for E1 × E2 embedded in P2 × P2 is E1 : y 2 z + xyz = x3 + x2 z − 1154xz 2 − 15345z 3, E2 : v 2 w + vw2 = u3 − u2 w + 2w3 and C is the curve on this surface given by 4xu − 155zu + xv + 2yv − 40xw + yw + 1314zw = 0. Hyperelliptic models for C are Y 2 + (X + 1)Y = 3X 5 + 5X 4 + X 3 − 8X 2 − 5X + 2 or Y 2 = (3X − 1)(X + 1)(4X 3 + 4X 2 − 9). Next, consider the elliptic curve 2006e1, given by the minimal Weierstrass equation E1 : y 2 + xy = x3 + x2 − 58293654x − 171333232940. It turns out that the plane cubic curve C1 : 20x3 +44x2 y +21x2z −77xy 2 +71xyz +44xz 2 +31y 3 +3y 2 z +150yz 2 +z 3 = 0 deﬁnes an element ξ (up to inverse) of order three in X(E1 /Q). In the sixth example in [4, Table 1] the elliptic curve E2 which ‘explains’ X(E1 /Q) is 2006d1. However, for this choice of E2 , there only exists an isometry between E1 [3] and E2 [3] and not an anti-isometry. The corresponding abelian surface (E1 × E2 )/Δ visualizing ξ will not be the jacobian of a genus 2 curve. If instead we take for E2 the elliptic curve 6018c1, then we do have an anti-isometry between E1 [3] and E2 [3]. Following the same route as in the ﬁrst example, we ﬁnd that ξ is visible in the jacobian of the genus 2 curve C with hyperelliptic models Y 2 + (X 2 + X)Y = − 9675X 6 − 94041X 5 − 914X 4 + 1301674X 3 − 352310X 2 − 2071181X − 945269

or

Y = 43(2X + 13)(18X − 81X + 89)(25X 3 + 193X 2 + 224X + 76). 2

2

Visualizing Elements of Sha[3] in Genus 2 Jacobians

7

123

Applications to 3-Descent

In this section we survey some of the ways in which explicit visibility might aid computations of Mordell-Weil groups and related quantities of elliptic curves. We recall that given an abelian variety A over a number ﬁeld k, the group X(A/k) ⊂ H 1 (k, A) consists of the cocycle classes that are everywhere locally trivial. It measures the diﬀerence between the Mordell-Weil group A(k) and the Selmer 1 group S(A/k) ⊂ lim ←−n H (k, A[n]) which is an everywhere local approximation to A(k), in the sense that the following sequence is exact. 0 → A(k) → S(A/k) → X(A/k) → 0 An n-descent usually means an explicit computational process to compute S (n) (A/k) = S(A/k)/nS(A/k) ⊂ H 1 (k, A[n]). It provides a bound on rkA(k) and conversely, if A(k) is known, then we can use 0 → A(k)/nA(k) → S (n) (A/k) → X(A/k)[n] → 0 to compute #X(A/k)[n] and thus obtain information on #X(A/k). In principle, one can use visibility to reﬁne this information. We will argue using an example. Stein and Watkins [12] found the following elliptic curve E : y 2 + xy = x3 − x2 + 94x + 9. Using a 2-descent and some point searching (with for instance Magma [1]) it is straightforward to verify that E(Q) Z × Z and that #X(E/Q)[2] = 1. Using a 3-descent (see [2, 3, 11], implemented in Magma), with unproved S-unit data we ﬁnd that C1 : x3 + 2x2 z + 2xy 2 + xyz − xz 2 − y 3 + 3y 2 z − 6yz 2 + z 3 = 0, C2 : x3 − 2xy 2 + 3xyz + 2y 3 + y 2 z + yz 2 + 3z 3 = 0 are 3-coverings of E that have points everywhere locally and we can verify by looking at preimages of representatives of E(Q)/3E(Q) that C1 , C2 have no rational points. The same process allows us to ﬁnd more than 18 such spaces, verifying unconditionally that #X(E/Q)[3] ≥ 9. The conditional 3-descent computation suggests that C1 , C2 represent cocycles generating S (3) (E/Q)/E(Q), so one expects that #X(E/Q)[3] = 9 and indeed BSD predicts that #X(E/Q) = 9. Visibility could help with proving that #X(E/Q)[3∞ ] = 9. The construction in this paper yields an abelian surface A = Jac(C), together with a map φ∗ : X(E/Q) × X(E /Q) → X(A/Q) where we know that ker(φ∗ ) is contained in the 3-torsion, because multiplicationby-three factors through φ. If we can make sure that ker(φ∗ ) contains the classes

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represented by C1 , C2 (this implies that E (Q) is of rank at least 2), it may well be that #X(A/Q)[3] = 1. If we can compute S (3) (A/Q), we can check this and the result would follow. Thus, visibility allows us to substitute a 9-descent on an elliptic curve with a 3-descent on the Jacobian of a genus 2 curve. Both are theoretically computable, but in neither case does it seem practical at this point. Since A has a 3-isogeny to E × E , the 3-torsion algebra (generically of degree 80), splits in two algebras of degrees 72 and 8 respectively. However, doing class group computations for degree 72 algebras over Q still seems well out of range. It is conceivable that some appropriate galois-stable set S of divisors on C exists with #S < 72. The group Sp4 (F3 ) has an index 27 subgroup, for instance, predicting a transitive action on 27 objects somewhere. If for some ﬁxed divisor D0 we have that A[3] = [D − D0 ] : D ∈ S, it may be possible to adapt ideas about fake Selmer groups [9] for application to A and only require class group information for algebras of degree #S. At this point it is unclear if this approach has any advantages to a direct 9descent on E and whether either method can be made practical for the example given in this section. Acknowledgments. The authors would like to thank the referees, who provided various helpful comments which found their way into this article.

References [1] Bosma, W., Cannon, J., Playoust, C.: The Magma computer algebra system is described in the Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997) [2] Cremona, J.E., Fisher, T.A., O’Neil, C., Simon, D., Stoll, M.: Explicit n-descent on elliptic curves. I. Algebra. J. Reine Angew. Math. 615, 121–155 (2008) [3] Cremona, J.E., Fisher, T.A., Stoll, M.: Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, arXiv: 0908.1741 (2009), http://arxiv.org/abs/0908.1741 [4] Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich-Tate group. Experiment. Math. 9(1), 13–28 (2000) [5] Mazur, B.: Visualizing elements of order three in the Shafarevich-Tate group. Asian J. Math. 3(1), 221–232 (1999); Sir Michael Atiyah: a great mathematician of the twentieth century [6] O’Neil, C.: The period-index obstruction for elliptic curves. J. Number Theory 95(2), 329–339 (2002) [7] Fisher, T.: The Hessian of a genus one curve, arXiv: math/0610403 (2006), http://arxiv.org/abs/math/0610403 [8] Frey, G., Kani, E.: Curves of genus 2 covering elliptic curves and an arithmetical application. In: Arithmetic Algebraic Geometry (Texel, 1989), pp. 153–176 (1991) [9] Poonen, B., Schaefer, E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997)

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[10] Salmon, G.: A treatise on the higher plane curves, 3rd edn. Hodges, Foster, and Figgis, Grafton Street, Dublin (1879) [11] Schaefer, E.F., Stoll, M.: How to do a p-descent on an elliptic curve. Trans. Amer. Math. Soc. 356(3), 1209–1231 (2004) [12] Stein, W.A., Watkins, M.: A database of elliptic curves—first report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002), http://wstein.org/Tables/ecdb/

On Weil Polynomials of K3 Surfaces Andreas-Stephan Elsenhans1, and J¨ org Jahnel2 1

2

Universit¨ at Bayreuth, Mathematisches Institut, Universit¨ atsstraße 30, D-95447 Bayreuth, Germany [email protected] Fachbereich 6, Mathematik, Universit¨ at Siegen, Walter-Flex-Straße 3, D-57072 Siegen, Germany [email protected]

Abstract. For K3 surfaces, we derive some conditions the characteristic polynomial of the Frobenius on the ´etale cohomology must satisfy. These conditions may be used to speed up the computation of Picard numbers and the decision of the sign in the functional equation∗∗ . Our investigations are based on the Artin-Tate formula.

1

Introduction

√ An algebraic integer such that all its conjugates have absolute value r is called an r-Weil number. Correspondingly, a possibly √ reducible monic polynomial Φ ∈ Z[T ] such that all roots have absolute value r is called an r-Weil polynomial. Let q be a prime power and r = q k . Then, for every smooth projective variety V over Fq , the eigenvalues of the Frobenius endomorphism Frob on the ´etale cohomology H´ekt (VFq, Ql ) are r-Weil numbers [3, Lemme 1.7]. Conversely, every q k -Weil number is an eigenvalue of Frob on H´ekt (VFq, Ql ) for a suitable smooth projective variety V over Fq . Actually, this fact is a direct consequence of the results of T. Honda [9]. In this note, we will study the Weil numbers of K3 surfaces. As the second Betti number of a K3 surface is b2 (V ) = 22 and q is always a root of the characteristic polynomial, the possible Weil numbers are of degree at most 20. We will show that not all q 2 -Weil polynomials Φ ∈ Z[T ] satisfying deg Φ = 22 and Φ(q) = 0 occur as characteristic polynomials of Frob on the ´etale cohomology of K3 surfaces. Concerning K3 surfaces of ﬁxed degree, even more restrictions result. Our investigations are based on the Artin-Tate formula which we will recall in section 3.

The first author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) through a funded research project. The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Computing at the G¨ ottingen Mathematisches Institut. Both authors are grateful to Prof. Y. Tschinkel for the permission to use these machines as well as to the system administrators for their support.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 126–141, 2010. c Springer-Verlag Berlin Heidelberg 2010

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An application. The characteristic polynomial of Frob may be computed by counting points over extensions of the ground ﬁeld. Indeed, for V a K3 surface over Fq , the Lefschetz trace formula [13, Ch. VI, §12] yields tr(Frobe ) = #V (Fqe ) − q 2e − 1. When we denote the eigenvalues of Frob by r1 , . . . , r22 , we have e tr(Frobe ) = r1e + · · · + r22 =: σe (r1 , . . . , r22 ). Newton’s identity [20] sk (r1 , . . . , r22 ) =

k−1 1 (−1)k+j+1 σk−j (r1 , . . . , r22 )sj (r1 , . . . , r22 ) k j=0

shows that the knowledge of σe (r1 , . . . , r22 ), for e = 1, . . . , k, is suﬃcient in order to determine the coeﬃcient (−1)k sk of T 22−k of the characteristic polynomial Φ of Frob. Further, there is the functional equation q deg Φ Φ(T ) = ±T deg Φ Φ(q 2/T )

(1)

which, as deg Φ = 22, relates the coeﬃcient of T k with that of T 22−k . Nevertheless, this method is time-consuming. The size of the ﬁelds to be considered grows exponentially. One would like to avoid point counting over large ﬁelds and, nevertheless, determine Φ suﬃciently well in order to decide things such as the sign in (1). Algorithms of this type were presented in [6]. For example, Algorithm 22 of [6] veriﬁes that the geometric Picard rank is 2, having counted points over Fp , . . . , Fp9 for p a prime number. The main result of the present article leads to a more substantial approach to this problem. In fact, we will show that certain hypothetical characteristic polynomials are impossible, in general. This leads to an improvement of [6, Algorithm 22]. Sections 7 and 8 will be devoted to examples showing how this improvement works in practice. Remark 1. A continuation of this application, which we have in mind, is the computation of the geometric Picard rank for K3 surfaces over Q. Here, the general strategy is to use reduction modulo p. One applies the inequality rk Pic(VQ ) ≤ rk Pic(VFp ) which is true for every smooth variety V over Q and every prime p of good reduction. Then, the number of eigenvalues of Frob which are roots of unity is an upper bound for the Picard number. More details are given in [6] and [7].

2

The Galois Group of a Weil Polynomial

For a randomly chosen irreducible polynomial over Q, one expects the Galois group to be the full symmetric group. In this sense, the irreducible factors of a Weil polynomial are not very random. When we consider the operation of Frob on a cohomology group of even degree, cyclotomic factors do arise. They correspond to the algebraic part of the cohomology, i.e., to the image of the Picard group and its analogues in higher codimension. The corresponding Galois group is always abelian.

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Concerning the remaining factors, still, there are restrictions on the Galois group. Note that, for each root of an irreducible r-Weil polynomial not of degree 1, the complex conjugate is a root, too. This means, the roots come in pairs. The product of each pair is equal to r. The Galois group therefore acts on the pairs. For a suitable integer n, it is a subgroup of the semi-direct product (Z/2Z)n Sn ⊂ S2n . Here, each factor (Z/2Z) acts on one pair by complex conjugation. The complex conjugation itself belongs to the center of the group. An experimental result. One could ask for further restrictions on the Galois group. For that, we computed the characteristic polynomial of Frob for a few thousand randomly chosen K3 surfaces. In each case, the factorization of that polynomial had precisely one irreducible factor which was not cyclotomic. This coincides with Zarhin’s results [18] for ordinary K3 surfaces. Furthermore, in the vast majority of the examples, the Galois group of the last factor was actually equal to the semi-direct product (Z/2Z)n Sn ⊂ S2n . For example, this was true for 875 out of 1 000 K3 surfaces of degree 2 over F3 and 923 out of 1 000 K3 surfaces of degree 2 over F7 . The resolvent algebra. Let Φ ∈ Q[T ] be a polynomial such that its set of roots is of the particular form {r1 , r1 , . . . , rn , rn } such that r1 r1 = . . . = rn rn =: r ∈ Q. Then, the sums r1 + r1 , . . . , rn + rn are the roots of a polynomial R ∈ Q[T ] of half the degree. We will call R the resolvent polynomial and A := Q[T ]/R the resolvent algebra of Φ. Remarks 2. a) When Φ √ is an r-Weil polynomial of even degree, the assumption is satisﬁed if and only if r is a root of even multiplicity (or no root) of Φ. In this √ case, (− r) has even multiplicity, too. In fact, this means exactly that Φ fulﬁlls the functional equation (1) with the plus sign. b) On the other hand, when one wants to verify that a given polynomial satisfying the functional equation is, in fact, a Weil polynomial, the resolvent is helpful. √ Observe that the roots of the initial polynomial are all of absolute value √ √r if and only if the roots of the resolvent are all real and in the interval [−2 r, 2 r]. That property may easily be checked using Sturm’s chain theorem. This is a fast and exact replacement of [6, Algorithm 23].

3

The Artin-Tate Formula

Let us recall the Artin-Tate conjecture in the special case of a K3 surface. Conjecture 3 (Artin-Tate). Let V be a K3 surface over a ﬁnite ﬁeld Fq . Denote by ρ the rank and by Δ the discriminant of the Picard group of V, deﬁned over Fq . Then, lim Φ(T )ρ T →q (T −q) . |Δ| = 21−ρ q #Br(V )

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Here, Φ denotes the characteristic polynomial of Frob on H´e2t (VFq, Ql ). Finally, Br(V ) is the Brauer group of V . Remarks 4. i) The characteristic polynomial Φ is independent of the choice of the auxiliary prime l as long as l = p for q = pe [3, Th´eor`eme 1.6]. ii) For a general non-singular, projective surface, the exponent of q in the numerator is b2 (V ) − h02 (V ) − ρ. Here, h02 (V ) denotes the Hodge number. iii) The Artin-Tate conjecture is proven for most K3 surfaces. Most notably, the Tate conjecture implies the Artin-Tate conjecture [11, Theorem 6.1]. iv) The Tate conjecture claims that all zeroes of Φ of the form qζ for ζ a root of unity belong to the algebraic part of H´e2t (VFq, Ql ). I.e., it asserts that the transcendental part never generates a zero of this form. The evidence for this is overwhelming as far as K3 surfaces are concerned. The Tate conjecture is proven for elliptic K3 surfaces [1] and ordinary K3 surfaces [15]. In characteristic diﬀerent from 2 and 3, even more particular cases were successfully treated [16]. v) It is expected that Br(V ) is always a ﬁnite group. This is actually equivalent to the Tate conjecture. In this case, #Br(V ) is automatically a perfect square. We may therefore compute the square class of Δ making use of the ArtinTate conjecture. An unconditional version of the Artin-Tate formula Notation 5. i) For n a positive integer, we will denote by μn the sheaf of n-th roots of unity with respect to the fppf topology. When l is a prime number, d d we put Hfppf (VFq, Tl μ) := lim Hfppf (VFq, μle ). ←e− ii) For l a prime number and M an abelian group, the notation Ml-pow shall be used for the l-power torsion subgroup of M . Similarly, we will write Ml-div ⊆ Ml-pow for the subgroup of inﬁnitely l-divisible elements. iii) We will denote by M Frob and MFrob the invariants, respectively coinvariants, under the operation of Frob on the abelian group M . The coinvariants may have torsion even when M is torsion-free. Write MFrob for the torsion-free quotient. Proposition 6. Let V be a K3 surface over a ﬁnite ﬁeld Fq and l be any prime. Write Φ for the characteristic polynomial of Frob on the ´etale cohomology of VFq and ρ for the multiplicity of q as a zero of Φ. i) Then, the Brauer group Br(V ) is a torsion group. The quotient Br0 (V, l) := Br(V )l-pow / Br(V )l-div is a ﬁnite group of square order. 2 ii) Further, Hfppf (VFq, Tl μ)Frob is a free Zl -module of rank ρ. iii) Denote by Δl the discriminant of the bilinear form 2 2 (VFq, Tl μ)Frob × Hfppf (VFq, Tl μ)Frob −→ Zl Hfppf

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deﬁned by Poincar´e duality. Then, νl (Δl ) = νl

Φ(T ) lim ρ T →q (T −q) q21−ρ #Br0 (V,l)

.

Proof. i) Finiteness of Br0 (V, l) follows immediately from [8, (8.9)]. Further, there is a non-degenerate alternating pairing Br0 (V, l) × Br0 (V, l) → Ql /Zl constructed in [19, Lemma 3.4.1]. This ensures that the group order is a perfect square. ii) and iii) We denote the zeroes of Φ by r1 , . . . , r22 . 2 (VFq, Tl μ) = H´e2t (VFq, Zl (1)) is the same as First case. l = p. Here, H := Hfppf l-adic ´etale cohomology. It is a free Zl -module of rank 22. In the present case, the operation of Frob on H is known to be semi-simple [4, Corollary 1.10]. The eigenvalues are r1 /q, . . . , r22 /q. Assertion ii) follows immediately from this. Further, we have νl (Δl ) = νl (#coker(H Frob → Hom(H Frob , Zl ))), the map being induced by Poincar´e duality. Identifying Hom(H, Zl ) with H, the module Hom(H Frob , Zl ) goes over into HFrob . Here, as shown in [19, Proposition 1.4.2], (HFrob )tors ∼ ho= Br0 (V, l). Further, the order of the cokernel of the canonical momorphism H Frob → HFrob is equal to the l-primary part of (1 − rj /q). rj =q Altogether, this implies the claim. Second case. l = p. Here, some modiﬁcations are necessary which are described in [11]. More concretely, the short exact sequence 2 n 0 → Pic(VFq)⊗Z Zp → Hfppf (VFq, Tp μ) → lim ←− Br(VFq)p → 0 2 immediately shows that H := Hfppf (VFq, Tp μ) is a torsion-free Zp -module. Otherwise, its structure is rather diﬀerent from the previous case. The rank of H is, in general, less than 22. Eigenvalues of Frob are only those rj /q which are units in Qp [11, 1.4]. But this is enough to show ii). Generally, there are unipotent connected quasi-algebraic groups U d and ´etale group schemes Dnd for d = 2, 3 and n 0 which provide short exact sequences d 0 → U d (Fq ) → Hfppf (VFq, μpn) → Dnd (Fq ) → 0. For varying n, the vector groups 3 U (Fq ) are connected by identities. Further, Dn3 = 0. Hence, if dim U 3 = s then 3 #Hfppf (VFq, Tp μ)Frob = q s the operation of Frob being semi-simple. Actually, one has s = 0 except when V is supersingular. Poincar´e duality is available [12, Theorem 5.2 and Corollary 2.7.c)] only at the level of torsion coeﬃcients. Thereby, U 2 (Fq ) and U 3 (Fq ) are dual to each other. 2 1 2 One has ← lim − U (Fq )2= 0 and R lim ←∼ − U (Fq2) = 0 as the connecting homomorphisms are zero. Hence, Hfppf (VFq, Tp μ) = lim Dn (Fq ). Further, it turns out that the ho←− momorphism HFrob → Hom(H Frob , Zp ) does not need to be bijective. It has a cokernel exactly of order q s (cf. [11, Lemma 5.2]). Summarizing, we ﬁnd that Δp has the same p-adic valuation as qs · (1 − rj /q) . νp (rj /q)=0 rj =q

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For iii), it remains to show p-adic units, the product of the following. Up tos−1 the remaining factors, i.e. (1 − rj /q), equals q . This is worked out in [11, νp (rj /q)=0 sec. 7]. 2 Remark 7. The Tate conjecture implies Hfppf (VFq, Tl μ)Frob ∼ = Pic(VFq )⊗ZZl . Further, it is equivalent to Br(V )l-div = 0. Thus, Proposition 6 goes over into the Artin-Tate formula in its usual form. However, the Tate conjecture is unknown in general, even for K3 surfaces. For this reason, we prefer to apply the version of the Artin-Tate formula which holds unconditionally.

4

The Rank-1 Condition

Let V be a K3 surface of degree d over a ﬁnite ﬁeld Fq . Assume that q is a simple zero of the characteristic polynomial of Frob. Then, the Tate conjecture is true for V and the arithmetic Picard rank is equal to 1. The discriminant of Pic(V ) is equal to d. A comparison with the analytic discriminant computed via the ArtinTate formula leads to a non-trivial condition for hypothetical Weil polynomials. Remarks 8. a) This is a condition for rank-1 surfaces of a given degree d. It is not a condition for K3 surfaces, in general. b) The degree of a K3 surface may be any even integer greater than zero. On the other hand, when the arithmetic Picard rank is 1, the number (−q) is necessarily among the Frobenius eigenvalues. Hence, the Artin-Tate formula can generate only even numbers. c) The Artin-Tate conjecture implies the inequality #Br(V )|Δ| ≤ 222−ρ q. Thus, the left hand side is O(q). Observe the following striking consequence. Over the ﬁeld Fq , there is no K3 surface of a square-free degree d > 221 q and arithmetic Picard rank 1. Remark 9. The rank-1 condition may be extended to other situations where a subgroup of the Picard group is known. For this, one has to compare the predicted ranks and discriminants with the known ones.

5

The Field Extension Condition

Notation 10. For q a positive integer, let Φ be a q 2 -Weil polynomial. Then, we will write q c − rjc (c) EΦ := q (c−1)(21−ρ) . q − rj rj =q

Here, rj runs over all the zeroes of Φ. Further, ρ is the multiplicity of the zero q. Observation 11 (Field extension for the characteristic polynomial). Let V be any smooth, projective variety over Fq and j (T − rj ) the characteristic polynomial on H´e2t (VFq, Ql ). Then, the corresponding polynomial for VFqd of Frob d is j (T − rj ).

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Theorem 12. Let V be a K3 surface over Fq . Further, let c be a positive in(c) teger. Then, for Φ the characteristic polynomial of Frob, the expression EΦ is a perfect square in Q. (c) Proof. If there is an rj = q such that rjc = q c then EΦ = 0. Otherwise, for every 2 Frobqc 2 prime l, Hfppf(VFq, Tl μ) is a sublattice of ﬁnite index in Hfppf (VFq, Tl μ)Frobq . In particular, the discriminants diﬀer by a factor being a perfect square. Dividing the Artin-Tate formulas for VFqc and VFq through each other yields that (c) (c) νl (EΦ ) is even for every l. Finally, it is easy to see that EΦ > 0. (c)

Remark 13. Assume the Tate conjecture. Then, EΦ is non-zero if and only if rk Pic(VFq) = rk Pic(VFqc). (c)

Definition 14. We will call the condition on EΦ to be a perfect square, the ﬁeld extension condition for the ﬁeld extension Fqc /Fq . (c)

Explicit computation of the expression EΦ . Our goal is now to describe (c) the square class of EΦ more explicitly. It will turn out that, for an arbitrary Weil (c) polynomial, EΦ may be a non-square. In other words, Theorem 12 provides a non-trivial condition. Remark 15. A priori, there are inﬁnitely many conditions, one for each value of c. The main result of this section is that there is in fact only one condition. Further, this condition may be checked easily. Lemma 16. Let f ∈ Q[T ] be a q 2 -Weil polynomial. Suppose f (q) = 0 and f (−q) = 0. Then, for r1 , . . . , r2l the zeroes of f , 2l q c − rjc (Q∗ )2 ∪ {0} for c odd, ∈ f (−q)(Q∗ )2 ∪ {0} for c even. q − rj j=1 Further, the left hand side is actually in f (−q)(Q∗ )2 for c = 2. Proof. First observe that, for c = 2, the numerators q 2 − rj2 are all non-zero according to the assumption. Hence, the additional assertion is clear once we showed the main one. For that, let us start with the contribution of one pair of complex conjugate roots. Put rj = q(u + iv). Then, the corresponding factor is (q c − rjc )(q c − r cj ) (q c − q c (u + iv)c )(q c − q c (u − iv)c ) = (q − rj )(q − r j ) (q − q(u + iv))(q − q(u − iv) = q 2(c−1)

c−1

(1 − ζck (u + iv))(1 − ζck (u − iv)) .

k=1

Using (u + iv)(u − iv) = 1, we get q 2(c−1)

c−1 k=1

(1 − 2ζck u + ζc2k ) .

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Next, for k = c/2, let us multiply the factors for k and c − k. This yields (1 − 2ζck u + ζc2k )(1 − 2ζcc−k u + ζc2c−2k ) = 2 + 4u2 − 8u Re(ζck ) + 2 Re(ζc2k ) . As Re(ζc2k ) = 2 Re(ζck )2 − 1, the latter term is the same as 4u2 − 8u Re(ζck ) + 4 Re(ζck )2 = (2u − 2 Re(ζck ))2 . Multiplying over all k such that 1 ≤ k < c/2, we ﬁnd a square in Q(u). Consequently, up to the factor for k = c/2, if present, the contribution of the pair {rj , rj } is a square in the resolvent algebra A of f . Multiplying over all l pairs means to form a norm for the extension A/Q. As the norm of a square is a square, the result is a perfect square in Q. For c odd, this completes the argument. For c even, the factors for k = c/2 are still missing. These are the ones for ζck = −1. We ﬁnd the product l

(1 + rj /q)(1 + r j /q) = q −2l f (−q) .

j=1

The assertion follows. Proposition 17. Let Φ be a q -Weil polynomial of even degree. Then, (Q∗ )2 ∪ {0} for c odd, (c) EΦ ∈ qΦ(−q)(Q∗ )2 ∪ {0} for c even. 2

For c = 2, we actually have EΦ ∈ qΦ(−q)(Q∗ )2 . Proof. First case: c is odd. Then, the denominator q (c−1)(21−ρ) is a perfect square. The zeroes (−q) con(c) tribute factors q c−1 which are squares, too. Finally, the contribution to EΦ of the zeroes not being real is a perfect square according to Lemma 16. Second case: c is even. (c) If (−q) is a zero of Φ then EΦ = 0. This coincides with the claim as Φ(−q) = 0. Otherwise, write Φ(T ) = (T − q)ρ f (T ) where f (q) = 0 and f (−q) = 0. By assumption, ρ is even. Hence, q (c−1)(21−ρ) is in the square class of q. Further, the zeroes of Φ diﬀering from q are exactly the zeroes of f . Their contribution is in f (−q)(Q∗ )2 for c = 2 and in f (−q)(Q∗ )2 ∪ {0}, in general. As ρ is even, f (−q)(Q∗ )2 is the same class as Φ(−q)(Q∗ )2 . The assertion follows. (c)

Corollary 18. Let f ∈ Z[T ] be a q 2 -Weil polynomial. i) Then, all ﬁeld extension conditions for Fqc /Fq are satisﬁed if only if the condition for the quadratic extension Fq2 /Fq does hold. ii) For extensions of odd degree, the ﬁeld extension condition is always satisﬁed. iii) If Fq and Fq2 lead to diﬀerent Picard ranks then all the ﬁeld extension conditions are satisﬁed. Remark 19. One might want to study the ﬁeld extension conditions for Fqac /Fqa , i.e., for an extended ground ﬁeld. Our calculations show that this does not lead to new conditions.

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Simplification of the field extension test. Denote by φn the n-th cyclotomic polynomial. Correspondingly, there is the monic polynomial ψn given by ψn (T ) := q ϕ(n) φn (T /q). This is a q 2 -Weil polynomial. Lemma 20. Let n > 1 be an integer. Then, ⎧ ⎨ (Q∗ )2 if n is not a power of 2 , ψn (−q) ∈ 2(Q∗ )2 for n = 2m , m ≥ 2 , ⎩ {0} for n = 2 . Proof. It is well known (see, e.g., [14, sec. 3]) that φn (−1) = 1 unless n is e−1 a power of 2. Further, the formula φ2e (t) = t2 + 1 shows φ2 (−1) = 0 and φ2e (−1) = 2 for e > 1. Observe, ﬁnally, that ϕ(n) is always even for n > 2. Remark 21. The result used here is a very special case of the value of a cyclotomic polynomial at a root of unity. Theorem 22. Let Φ ∈ Z[T ] be a q 2 -Weil polynomial of even degree. Factorize Φ as Φ(T ) = (T − q)r (T + q)s ψn1 (T ) · . . . · ψnk (T )Φ1 (T ) such that Φ1 has no root being a root of unity multiplied by q. Denote by M the number of the powers of 2 among the n1 , . . . , nk . Then, i) if c is odd then EΦ ∈ (Q∗ )2 ∪ {0}. (c)

(c)

ii) If c is even and s > 0 then EΦ = 0 for every c.

iii) Finally, if c is even and s = 0 then EΦ ∈ 2MqΦ1 (−q)(Q∗ )2 ∪ {0}. Furthermore, for c = 2, one actually has (c)

EΦ ∈ 2MqΦ1 (−q)(Q∗ )2 . (2)

Proof. i) and ii) are immediate consequences from Proposition 17. For iii), observe the assumption implies that r is even. In particular, (−2q)r is a perfect square. The assertion now follows from Proposition 17 together with Corollary 20. Remark 23. Suppose Φ ∈ Z[T ] is a q 2 -Weil polynomial of degree 22. In order to show that Φ may not be the characteristic polynomial of the Frobenius for a K3 surface over Fq , it suﬃces to verify that s = 0 and 2MqΦ1 (−q) is a non-square. Example 24. As an example, we look at K3 surfaces of Picard rank 18 such that the Picard group is deﬁned over an extension of odd degree. Then, (−q) is not an eigenvalue of the Frobenius. The transcendental part of the characteristic polynomial is given by (T 4 + aT 3 + bT 2 + aq 2 T + q 4 ). Hence, the ﬁeld extension condition usually requires that (2q 2 − 2aq + b)q is a perfect square. If, however, the cyclotomic factors contain an odd number of type ψ2n then 2(2q 2 − 2aq + b)q is required to be a square.

On Weil Polynomials of K3 Surfaces

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135

The Special Case of a Degree-2 Surface – Twisting

When a K3 surface has a non-trivial automorphism, one can hope to get more conditions by inspecting the corresponding twist. This is the case for degree-2 surfaces. The Twist. Let the K3 surface V be given by the equation w2 = f6 (x, y, z) . Then, for n a non-square in Fq , consider the twist V of V given by nw2 = f6 (x, y, z) . Fact 25. Assume that q, r2 , . . . , r22 are the eigenvalues of Frob for V . Then, the eigenvalues for V are q, −r2 , . . . , −r22 . Proof. For e even, VFqe and V Fqe are isomorphic. When e is odd, we have #V (Fqe ) + #V (Fqe ) = 2·#P2 (Fqe ) = 2q 2e + 2q e + 2 . It is easy to check that the Lefschetz trace formula, applied to the eigenvalues q, −r2 , . . . , −r22 , implies exactly this relation. Proposition 26. Let V be a K3 surface of degree 2 over Fq . Denote by Φ the the corresponding polynomial characteristic polynomial of Frob for V and by Φ for the twist V . does not have a zero at (−q). i) Then, Φ has a simple zero at q if and only if Φ I.e., the rank-1 condition can be applied to the one precisely when the ﬁeld extension condition is non-empty for the other one. ii) The two conditions are equivalent to each other. Proof. i) immediately follows from Fact 25. ii) By assumption, we can write Φ(T ) = (T − q)(T + q)2n−1 f (T ). Here both, f (q) and f (−q) are non-zero. Fact 25 shows, the corresponding polyno ) = (T − q)2n f (−T ). Using these two formulas, one mial for the twist is Φ(T can make the conditions explicit. The rank-1 condition for Φ simply means (2q)2n−1 f (q) = 2 in Q∗/(Q∗ )2 which is equivalent to saying that qf (q) is a per fect square. This is precisely the ﬁeld extension condition for Φ.

7

Examples

Let us show in detail the data for a few examples. Our goal is to illustrate how the Artin-Tate conditions work in practice. Example 27 (A K3 surface of degree 2 over F7 ). Consider the surface V over F7 , given by w2 = y 6 + 3z 6 + 5xz 5 + 5x2 y 4 + x2 z 4 + 3x3 y 3 + x3 z 3 + 5x4 y 2 + x4 z 2 + 5x5 y + 2x6 . Over F7 , . . . , F79 , there are exactly 66, 2 378, 118 113, 5 768 710, 282 535 041,

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13 841 275 877, 678 223 852 225, 33 232 944 372 654, and 1 628 413 551 007 224 points. We claim that rk Pic(VF7 ) = 2. Assuming the characteristic polynomial of the Frobenius has more than two zeroes of the form 7 times a root of unity, [6, Algorithm 22] leaves us with three candidates Φ1 , Φ2 , Φ3 . Φi (t) = t22 − 16 t21 + 140 t20 − 1 029 t19 + 5 831 t18 − 36 015 t17 + 268 912 t16 − 1 882 384 t15 + 11 529 602 t14 − 46 118 408 t13 + ai t12 + bi t11 + ci t10 + (−1)ji [−110 730 297 608 t9 + 1 356 446 145 698 t8 − 10 851 569 165 584 t7 + 75 960 984 159 088 t6 − 498 493 958 544 015 t5 + 3 954 718 737 782 519 t4 − 34 196 685 556 119 429 t3 + 227 977 903 707 462 860 t2 − 1 276 676 260 761 792 016 t + 3 909 821 048 582 988 049]

for j1 = 0,

(a1 , b1 , c1 ) = (161 414 428, −1 129 900 996, 7 909 306 972) ,

j2 = 1,

(a2 , b2 , c2 ) = ( 80 707 214,

0, −3 954 653 486) ,

j3 = 1,

(a3 , b3 , c3 ) = (121 060 821,

0, −5 931 980 229) .

Each of the three polynomials leads to an upper bound of 4 for the rank of the geometric Picard group. All three have roots of absolute value 7, only. Applying the Artin-Tate formula, we ﬁnd the following. Table 1. Hypothetical ranks and discriminants polynomial Φ1 Φ2 Φ3

ﬁeld F7 F49 F7 F49 F7 F49

arithmetic Picard rank 2 2 1 2 1 2

#Br(V )|Δ| 58 4524 4 1996 6 2997

The polynomial Φ1 is excluded by the ﬁeld extension condition as the two values in the rightmost column deﬁne diﬀerent square classes. On the other hand, the rank-1 condition excludes Φ2 and Φ3 since we have a degree-2 example. Thus, relative to the Tate conjecture, geometric Picard rank 2 is proven. Example 28 (continuation). On the same surface, point counting over F710 leads to a number of 79 792 267 067 823 523. For the characteristic polynomial of the Frobenius, we ﬁnd the two candidates Φ4 , Φ5 , Φi (t) = t22 − 16 t21 + 140 t20 − 1 029 t19 + 5 831 t18 − 36 015 t17 + 268 912 t16 − 1 882 384 t15 + 11 529 602 t14 − 46 118 408 t13 + 40 353 607 t12 + ai t11 + (−1)ji [ −1 977 326 743 t10 + 110 730 297 608 t9 − 1 356 446 145 698 t8 + 10 851 569 165 584 t7 − 75 960 984 159 088 t6 + 498 493 958 544 015 t5 − 3 954 718 737 782 51 9t4 + 34 196 685 556 119 429 t3 − 227 977 903 707 462 860 t2 + 1 276 676 260 761 792 016 t − 3 909 821 048 582 988 049]

On Weil Polynomials of K3 Surfaces

137

for j4 = 0, a4 = 0, j5 = 1, and a5 = 564 950 498. Φ4 corresponds to the minus sign in the functional equation, Φ5 to the case of the plus sign. Both candidates, according to the Tate conjecture, imply geometric Picard rank 2. To decide which sign is the right one, one would ﬁrst check the absolute values of the roots. Unfortunately, both polynomials only have roots of absolute value 7. The Artin-Tate formula provides the picture given in the table below. Table 2. Hypothetical ranks and discriminants polynomial Φ4 Φ5

ﬁeld F7 F49 F7 F49

arithmetic Picard rank 1 2 2 2

#Br(V )|Δ| 2 997 55 4125

Thus, Φ5 is excluded by the ﬁeld extension condition. The minus sign in the functional equation is correct. Example 29 (A K3 surface of degree 8 over F3 ). Consider the complete intersection V of the three quadrics in P5F3 , given by q1 , q2 , and q3 , q1 := −xy + xz + xu + xv + xw − y 2 − yz − yv + yw + z 2 + zu + zw − u2 − uw + v 2 + w2 , q2 := −x2 + xy + xz − xv + xw − y 2 + yz − yu − yv + yw − zu − zw + uw − v 2 + vw , q3 := xu − yz . V is smooth and, therefore, a K3 surface. As q3 is of rank 4, V carries an elliptic ﬁbration. There are precisely 14, 98, 794, 6 710, 59 129, 532 460, 4 784 990, 43 049 510, and 387 374 024 points over F3 , . . . , F39 . From these data, let us check whether one can prove rk Pic(VF3 ) = 2. Assume that the characteristic polynomial of the Frobenius has more than two zeroes of the form 3 times a root of unity. Then, [6, Algorithm 22] leaves us with ﬁve polynomials Ψ1 , . . . , Ψ5 , Ψi (t) = t22 − 4 t21 + 27 t18 + 81 t17 − 243 t16 + 6 561 t13 + a1 t12 + b1 t11 + c1 t10 + (−1)ji [531 441 t9 − 14 348 907 t6 + 43 046 721 t5 + 129 140 163 t4 − 13 947 137 604 t + 31 381 059 609]

for

j1 = 0,

(ai , bi , ci ) = (−59 049, 236 196, −531 441) ,

j2 = 0,

(a2 , b2 , c2 ) = (

j3 = 0,

(a3 , b3 , c3 ) = ( 19 683, −236 196, 177 147) ,

j4 = 1,

(a4 , b4 , c4 ) = (−59 049,

0, 531 441) ,

j5 = 1,

(a5 , b5 , c5 ) = (−39 366,

0, 354 294) .

0, −118 098,

0) ,

Applying the Artin-Tate formula to these polynomials, we obtain the following data.

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A.-S. Elsenhans and J. Jahnel Table 3. Hypothetical ranks and discriminants polynomial Ψ1 Ψ2 Ψ3 Ψ4 Ψ5

ﬁeld F3 F9 F3 F9 F3 F9 F3 F9 F3 F9

arithmetic Picard rank 2 4 2 2 2 2 3 4 1 2

#Br(V )|Δ| 24 1116 27 81 28 112 144 1152 2 65

Observe that an elliptic surface of Picard rank 2 automatically has a discriminant of the form (−n2 ) for n an integer. We may therefore exclude everything except for Ψ4 . Note that Ψ2 is, in addition, incompatible with the ﬁeld extension condition. Thus, using the numbers of points over the ﬁelds up to F39 , we only obtain that, either the geometric Picard rank is equal to 2, or Ψ4 is the characteristic polynomial of the Frobenius in which case it is 4. Example 30 (continuation). The number of points over F310 is 34 871 648 631. This additional information reproduces Ψ1 and Ψ4 as possible characteristic polynomials of Frob. Consequently, the minus sign holds in the functional equation and the geometric Picard rank of V is equal to 4.

8

Statistics

We tested the Artin-Tate conditions on samples of K3 surfaces of degrees 2, 4, 6, and 8. The possibilities of computing are limited by the fact that point counting over large ﬁnite ﬁelds is slow. In degree 2, decoupling [6, Algorithm 17] (see also [5]) leads to a substantial speed-up. In higher degrees, one may focus on elliptic K3 surfaces and exploit the fact that point counting on the elliptic ﬁbers is fast. The numbers and particularities of the examples treated are listed in Table 4. Table 4. Numbers of examples computed d d d d

= = = =

2 4 6 8

p=2 1000 rand 1000 rand 1000 rand 1000 rand

p=3 1000 rand 1000 ell 1000 ell 1000 ell

p=5 1000 dec

p=7 1000 dec

dec = decoupled, ell = elliptic, rand = random

The remaining parameters of the surfaces were chosen by a random number generator. We stored the equations and the numbers of points over Fp , . . . , Fp10 in a ﬁle.

On Weil Polynomials of K3 Surfaces

139

Results I. Point counting until Fp9 . First, we tried to show that the geometric Picard-rank was equal to 2 only using the numbers of rational points over Fp , . . . , Fp9 . I.e., we applied [6, Algorithm 22]. This algorithm produces a list of hypothetical Weil polynomials for each surface. If one is able to exclude all of them then, relative to the Tate conjecture, rank 2 is proven. To exclude a particular polynomial, we ﬁrst checked whether the roots are of absolute value p. When a surface was known to be elliptic over Fp , we checked in addition that the predicted Picard rank over Fp was at least equal to 2. Then, we applied the Artin-Tate conditions to the polynomials. We checked the ﬁeld extension condition and the rank-1 condition. For surfaces known to be elliptic over Fp , we observed the fact that arithmetic Picard rank 2 forces the discriminant to be minus a perfect square. The results are summarized in Table 5. Table 5. Distribution of the remaining hypothetical characteristic polynomials d = 2, p = 2 d = 2, p = 3 d = 2, p = 5 d = 2, p = 7 d = 4, p = 2 d = 4, p = 3 d = 6, p = 2 d = 6, p = 3 d = 8, p = 2 d = 8, p = 3

Number of polynomials without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions without with A-T conditions

0 84 149 116 214 85 158 92 214 40 81 22 53 39 83 16 50 25 29 12 20

1 479 598 480 573 581 651 534 611 532 638 669 785 549 645 713 797 657 723 720 803

2 312 218 285 193 209 169 232 154 303 249 242 161 312 257 217 148 268 239 236 175

3 89 28 88 20 96 20 98 21 87 27 57 1 70 14 47 5 38 5 27 2

4 21 7 24 0 25 2 37 0 29 5 9 0 22 1 7 0 8 4 4 0

5 12 0 4 0 4 0 7 0 8 0 1 0 6 0 0 0 4 0 1 0

6 3 0 3 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0

Results II. Point counting until Fp10 . Using data up to Fp10 , one obtains two hypothetical Weil polynomials for each of the surfaces. The two polynomials correspond to the possible signs in the functional equation (1). One has to exclude one of them. For this, we ﬁrst checked the absolute values of the roots. For surfaces known to be elliptic over Fp , we then tested whether the predicted arithmetic Picard rank is at least 2. Then, we applied the Artin-Tate conditions. We checked the ﬁeld extensions and the rank-1 condition. For elliptic surfaces, supposed to be of arithmetic Picard rank 2, we tested, in addition, whether the predicted discriminant was minus a square. Table 6 shows the number of surfaces with known signs. In the case that the sign is not known, we computed the numbers of points predicted over further extensions of Fp . Comparing these numbers for both hypothetical polynomials indicates whether further point counting would lead to a decision of the sign. We count how often which ﬁelds had to be considered in order to decide the sign.

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A.-S. Elsenhans and J. Jahnel Table 6. Sign decision in the functional equation p d Known signs without A-T Known signs using A-T Remaining unknown signs Data up to Fp11 insuﬃcient Data up to Fp12 insuﬃcient Data up to Fp13 insuﬃcient Data up to Fp14 insuﬃcient Data up to Fp15 insuﬃcient Data up to Fp16 insuﬃcient Data up to Fp17 insuﬃcient Data up to Fp18 insuﬃcient Data up to Fp19 insuﬃcient Data up to Fp20 insuﬃcient

2 2 768 863 137 84 41 22 13 7 4 4 4 2 0

3 2 843 940 60 23 11 5 2 0 0 0 0 0 0

5 2 864 940 60 15 2 1 0 0 0 0 0 0 0

7 2 869 961 39 12 1 0 0 0 0 0 0 0 0

2 4 761 863 137 69 39 24 12 8 3 2 0 0 0

3 4 876 943 57 19 3 2 0 0 0 0 0 0 0

2 6 790 868 132 77 42 20 13 7 2 2 1 1 0

3 6 888 933 67 25 11 2 1 0 0 0 0 0 0

2 8 822 867 133 72 47 24 8 5 4 0 0 0 0

3 8 897 944 56 21 7 2 0 0 0 0 0 0 0

Using these data, we repeated our attempt to prove that the geometric Picard rank is equal to 2. More precisely, we checked whether only two roots of the characteristic polynomial are of the form p times a root of unity. The numbers of surfaces for which we succeeded are listed in Table 7. Table 7. Numbers of rank-2 cases using Fp10 -data p = 2, d = 2 p = 3, d = 2 p = 5, d = 2 p = 7, d = 2 p = 2, d = 4 p = 3, d = 4 p = 2, d = 6 p = 3, d = 6 p = 2, d = 8 p = 3, d = 8

without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T without with A-T

conditions conditions conditions conditions conditions conditions conditions conditions conditions conditions

rank 2 proven 271 278 397 409 353 360 460 464 132 138 79 79 145 152 74 74 65 65 23 23

rank 2 possible 330 301 460 428 425 382 511 476 197 163 114 81 183 163 101 81 93 74 47 25

Conclusion. The Artin-Tate conditions usually halve the number of cases with unknown signs. Furthermore, they double the number of cases where geometric Picard rank 2 may be proven only using data up to Fp9 . Comparing Table 5 with Table 7, we see, however, that still only about one half of the cases with Picard rank 2 may be detected when counting until Fp9 . Remark 31. Let us ﬁnally mention that the Artin-Tate conditions came to us as a big surprise. It is astonishing that the Artin-Tate formula may be incompatible with itself under ﬁeld extensions. Thus, it seems not entirely unlikely that there are even more constraints and one can still do better.

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References 1. Artin, M., Swinnerton-Dyer, S.P.: The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math. 20, 249–266 (1973) 2. Beauville, A.: Surfaces alg´ebriques complexes, Ast´erisque 54, Soci´et´e Math´ematique de France, Paris (1978) 3. Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974) 4. Deligne, P.: Rel`evement des surfaces K3 en caract´eristique nulle. In: Prepared for publication by Luc Illusie, Algebraic surfaces (Orsay 1976–78). LNM, vol. 868, pp. 58–79. Springer, Berlin (1981) 5. Elsenhans, A.-S., Jahnel, J.: The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds. In: Algorithmic Number Theory (ANTS 7), pp. 317–332. Springer, Berlin (2006) 6. Elsenhans, A.S., Jahnel, J.: K3 surfaces of Picard rank one and degree two. In: Algorithmic Number Theory (ANTS 8), pp. 212–225. Springer, Berlin (2008) 7. Elsenhans, A.S., Jahnel, J.: On the computation of the Picard group for K3 surfaces (2009) (preprint) 8. Grothendieck, A.: Le groupe de Brauer, III: Exemples et compl´ements. In: Grothendieck, A. (ed.) Dix expos´es sur la Cohomologie des sch´emas, pp. 88–188. North-Holland, Amsterdam (1968) 9. Honda, T.: Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan 20, 83–95 (1968) 10. van Luijk, R.: K3 surfaces with Picard number one and infinitely many rational points. Algebra & Number Theory 1, 1–15 (2007) 11. Milne, J.S.: On a conjecture of Artin and Tate. Ann. of Math. 102, 517–533 (1975) ´ 12. Milne, J.S.: Duality in the flat cohomology of a surface. Ann. Sci. Ecole Norm. Sup., 4e s´erie 9, 171–201 (1976) ´ 13. Milne, J.S.: Etale Cohomology. Princeton University Press, Princeton (1980) 14. Motose, K.: On values of cyclotomic polynomials. VIII. Bull. Fac. Sci. Technol. Hirosaki Univ. 9, 15–27 (2006) 15. Nygaard, N.O.: The Tate conjecture for ordinary K3 surfaces over finite fields. Invent. Math. 74, 213–237 (1983) 16. Nygaard, N.O., Ogus, A.: Tate’s conjecture for K3 surfaces of finite height. Ann. of Math. 122, 461–507 (1985) 17. Tate, J.: Conjectures on algebraic cycles in l-adic cohomology. In: Motives, Proc. Sympos. Pure Math., vol. 55(1), pp. 71–83. Amer. Math. Soc., Providence (1994) 18. Zarhin, Y.I.: Transcendental cycles on ordinary K3 surfaces over finite fields. Duke Math. J. 72, 65–83 (1993) 19. Zarhin, Y.I.: The Brauer group of an abelian variety over a finite field. Izv. Akad. Nauk SSSR Ser. Mat. 46, 211–243 (1982) (Russian) 20. Zeilberger, D.: A combinatorial proof of Newtons’s identities. Discrete Math. 49, 319 (1984)

Class Invariants by the CRT Method Andreas Enge1 and Andrew V. Sutherland2 2

1 INRIA Bordeaux–Sud-Ouest, France Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract. We adapt the CRT approach for computing Hilbert class polynomials to handle a wide range of class invariants. For suitable discriminants D, this improves its performance by a large constant factor, more than 200 in the most favourable circumstances. This has enabled record-breaking constructions of elliptic curves via the CM method, including examples with |D| > 1015 .

1

Introduction

Every ordinary elliptic curve E over a ﬁnite ﬁeld Fq has complex multiplication by an imaginary quadratic order O, by which we mean that the endomorphism ring End(E) is isomorphic to O. The Deuring lifting theorem implies that E ˆ is the reduction of an elliptic curve E/C that also has complex multiplication by O. Let K denote the fraction ﬁeld of O. The j-invariant of Eˆ is an algebraic integer whose minimal polynomial over K is the Hilbert class polynomial HD , where D is the discriminant of O. Notably, the polynomial HD actually lies in Z[X], and its splitting ﬁeld is the ring class field KO for the order O. Conversely, an elliptic curve E/Fq with complex multiplication by O exists whenever q satisﬁes the norm equation 4q = t2 − v 2 D, with t, v ∈ Z and t ≡ 0 modulo the characteristic of Fq . In this case HD splits completely over Fq , and its roots are precisely the j-invariants of the elliptic curves E/Fq that have complex multiplication by O. Such a curve has q + 1 ± t points, where t is determined, up to a sign, by the norm equation. With a judicious selection of D and q one may obtain a curve with prescribed order. This is known as the CM method. The main challenge for the CM method is to obtain the polynomial HD , which has degree equal to the class number h(D), and total size O(|D|1+ ). There are three approaches to computing HD , all of which, under reasonable assumptions, can achieve a running time of O(|D|1+ ). These include the complex analytic method [12], a p-adic algorithm [9, 7], and an approach based on the Chinese Remainder Theorem (CRT) [2]. The ﬁrst is the most widely used, and it is quite eﬃcient; the range of discriminants to which it may be applied is limited not by its running time, but by the space required. The polynomial HD is already likely to exceed available memory when |D| > 109 , hence one seeks to apply the CM method to alternative class polynomials that have smaller coeﬃcients than HD . This makes computations with |D| > 1010 feasible. Recently, a modiﬁed version of the CRT approach was proposed that greatly reduces the space required for the CM method [30]. Under the Generalised Riemann Hypothesis (GRH), this algorithm is able to compute HD mod P using G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 142–156, 2010. c Springer-Verlag Berlin Heidelberg 2010

Class Invariants by the CRT Method

143

O(|D|1/2+ log P ) space and O(|D|1+ ) time. (Here and in the following, all complexity estimates refer to bit operations.) The reduced space complexity allows it to handle much larger discriminants, including examples with |D| > 1013 . An apparent limitation of the CRT approach is that it depends on some speciﬁc features of the j-function. As noted in [2], this potentially precludes it from computing class polynomials other than HD . The purpose of the present article is to show how these obstructions may be overcome, allowing us to apply the CRT method to many functions other than j, including two inﬁnite families. Subject to suitable constraints on D, we may then compute a class polynomial with smaller coeﬃcients than HD (by a factor of up to 72), and, in certain cases, with smaller degree (by a factor of 2). Remarkably, the actual running time with the CRT method is typically better than the size diﬀerence would suggest. Fewer CRT moduli are needed, and we may choose a subset for which the computation is substantially faster than on average. We start §2 with a brief overview of the CRT method, and then describe a new technique to improve its performance, which also turns out to be crucial for certain class invariants. After discussing families of invariants in §3, we consider CRT-based approaches applicable to the diﬀerent families and give a general algorithm in §4. Computational results and performance data appear in §5.

2 2.1

Hilbert Class Polynomials via the CRT The Algorithm of Belding, Br¨ oker, Enge, Lauter and Sutherland

The basic idea of the CRT-based algorithm for Hilbert class polynomials is to compute HD modulo many small primes p, and then lift its coeﬃcients by Chinese remaindering to integers, or to their reductions modulo a large (typically prime) integer P , via the explicit CRT [4, Thm. 3.1]. The latter approach sufﬁces for most applications, and while it does not substantially reduce the running time (the same number of small primes is required), it can be accomplished using only O(|D|1/2+ log P ) space with the method of [30, §6]. For future reference, we summarise the algorithm to compute HD mod p for a prime p that splits completely in the ring class ﬁeld KO . Let h = h(D). Algorithm 1 (Computing HD mod p) 1. Find the j-invariant j1 of an elliptic curve E/Fp with End(E) ∼ = O. 2. Enumerate the other roots j2 , . . . , jh of HD mod p. 3. Compute HD (X) mod p = (X − j1 ) · · · (X − jh ). The ﬁrst step is achieved by varying j1 (systematically or randomly) over the elements of Fp until it corresponds to a suitable curve; details and many practical improvements are given in [2, 30]. The third step is a standard building block of computer algebra. Our interest lies in Step 2.

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Enumerating the Roots of HD mod p

2.2

The key idea in [2] leading to a quasi-linear complexity is to apply the Galois action of Cl(O) Gal(KO /K). The group Cl(O) acts on the roots of HD , and when p splits completely in KO there is a corresponding action on the set EllO (Fp ) = {j1 , . . . , jh } containing the roots of HD mod p. For an ideal class [a] in Cl(O) and a j-invariant ji ∈ EllO (Fp ), let us write [a]ji for the image of ji under the Galois action of [a]. We then have EllO (Fp ) = {[a]j1 : [a] ∈ Cl(O)}. As in [30, §5], we use a polycyclic presentation deﬁned by a sequence of ideals l1 , . . . , lm with prime norms 1 , . . . , m whose classes generate Cl(O). The relative order rk is the least positive integer for which [lrkk ] ∈ [l1 ], . . . , [lk−1 ]. We may then uniquely write [a] = [le11 ] · · · [lemm ], with 0 ≤ ek < rk . To maximise performance, we use a presentation in which 1 < · · · < m , with each k as small as possible subject to rk > 1. Note that the relative order rk divides the order nk of [lk ] in Cl(O), but for k > 1 we can (and often do) have rk < nk . For each ji ∈ EllO (Fp ) and each O-ideal l of prime norm , the j-invariant [l]ji corresponds to an -isogenous curve, which we may obtain as a root of Φ (ji , X), where Φ ∈ Z[J, J ] is the classical modularpolynomial [31, §69]. The polynomial Φ has the pair of functions j(z), j(z) as roots, and parameterises isogenies of degree . Fixing an isomorphism End(E) ∼ = O, we let π ∈ O denote the Frobenius endomorphism. When the order Z[π] is maximal at , the univariate polynomial Φ (ji , X) ∈ Fp [X] has exactly two roots [l]ji and [¯l]ji when splits in O, and a single root [l]ji if is ramiﬁed [25, Prop. 23]. To simplify matters, we assume here that Z[π] is maximal at each k , but this is not necessary, see [30, §4]. We may enumerate EllO (Fp ) = {[a]j1 : [a] ∈ [l1 ], . . . , [lm ]} via [30, Alg. 1.3]: Algorithm 2 (Enumerating EllO (Fp ) — Step 2 of Algorithm 1) 1. Let j2 be an arbitrary root of Φm (j1 , X) in Fp . 2. For i from 3 to rm , let ji be the root of Φm (ji−1 , X)/(X − ji−2 ) in Fp . 3. If m > 1, then for i from 1 to rm : Recursively enumerate the set {[a]ji : [a] ∈ [l1 ], . . . , [lm−1 ]}. In general there are two distinct choices for j2 , but either will do. Once j2 is chosen, j3 , . . . , jrm are determined. The sequence (j1 , . . . , jrm ) corresponds to a path of m -isogenies; we call this path an m -thread. The choice of j2 in Step 1 may change the order in which EllO (Fp ) is enumerated. Three of the sixteen possibilities when m = 2, r1 = 4, and r2 = 3 are shown below; we assume [l32 ] = [l1 ], and label each vertex [le2 ]j1 by the exponent e. 0 l2 1 l2 2

l1 l1 l1

3 4 5

l1 l1 l1

6 7 8

l1 l1 l1

9

0 l2

10

1 l2

11

2

¯l1 ¯l1 l1

9 10 5

¯l1 ¯l1 l1

6 7 8

¯l1 ¯l1 l1

3

0 ¯l2

4

11 ¯l2

11

10

l1 ¯l1 l1

3 8 1

l1 ¯l1 l1

6 5 4

l1 ¯l1 l1

9 2 7

Bold edges indicate where a choice was made. Regardless of these choices, Algorithm 2 correctly enumerates EllO (Fp ) in every case [30, Prop. 5].

Class Invariants by the CRT Method

2.3

145

Finding Roots with Greatest Common Divisors (gcds)

The potentially haphazard manner in which Algorithm 2 enumerates EllO (Fp ) is not a problem when computing HD , but it can complicate matters when we wish to compute other class polynomials. We could distinguish the actions of l and ¯l using an Elkies kernel polynomial [10], as suggested in [7, §5], however this slows down the algorithm signiﬁcantly. An alternative approach using polynomial gcds turns out to be much more eﬃcient, and actually speeds up Algorithm 2, making it already a useful improvement when computing HD . We need not distinguish the actions of l and ¯l at this stage, but we wish to ensure that our enumeration of EllO (Fp ) makes a consistent choice of direction each time it starts an -thread. The ﬁrst -thread may be oriented arbitrarily, but for each subsequent -thread (j1 , j2 , . . . , jr ), we apply Lemma 1 below. This allows us to “square the corner” by choosing j2 as the unique common root of Φ (X, j1 ) and Φ (X, j2 ), where (j1 , . . . , jr ) is a previously computed -thread and j1 is -isogenous to j1 . The edge (j1 , j1 ) lies in an -thread that has already been computed, for some > . j1 l j1

l j 2 l

l j2

l j 3

l

· · · l jr

j1 l j1

l j 2

l j 3

l

l

l

j2

l

j3

l

· · · l jr

l

l · · · l jr

Having computed j2 , we could compute j3 , . . . , jr as before, but it is usually better to continue using gcds, as depicted above. Asymptotically, both rootﬁnding and gcd computations are dominated by the O(2 M(log p)) time it takes to instantiate Φ (X, ji ) mod p, but in practice is small, and we eﬀectively gain a factor of O(log p) by using gcds when ≈ . This can substantially reduce the running time of Algorithm 2, as may be seen in Table 1 of §5. With the gcd approach described total number of root-ﬁnding m above,the m operations can be reduced from k=1 rk to k=1 rk . When m is large, this is a big improvement, but it is no help when m = 1, as necessarily occurs when h(D) is prime. However, even in this case we can apply gcds by looking for an auxiliary ideal l1 , with prime norm 1 , for which [l1 ] = [le1 ]. When r1 is large, such an l1 is easy to ﬁnd, and we may choose the best combination of 1 and e available. This idea generalises to k -threads, where we seek [lk ] ∈ [l1 ] . . . , [lk ]\[l1 ] . . . , [lk−1 ]. Lemma 1. Let j1 , j2 ∈ EllO (Fp ), and let 1 , 2 = p be distinct primes with 421 22 < |D|. Then gcd Φ1 (j1 , X), Φ2 (j2 , X) has degree at most 1. Proof. It follows from [25, Prop. 23] that Φ1 (X, j1 ) and Φ2 (X, j2 ) have at most two common roots in the algebraic closure Fp , which in fact lie in EllO (Fp ). If there are exactly two, then both 1 = l1 l1 and 2 = l2 l2 split in O, and one of l21 l22 or l21¯l22 is principal with a non-rational generator. We thus have a norm equation 421 22 = a2 − b2 D with a, b ∈ Z and b = 0, and the lemma follows.

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Class Invariants

Due to the large size of HD , much eﬀort has been spent seeking smaller generators of KO . For a modular function f and O = Z[τ ], with τ in the upper half plane, we call f (τ ) a class invariant if f (τ ) ∈ KO . The class polynomial for f is HD [f ](X) = (X − [a]f (τ )) . [a]∈Cl(O)

The contemporary tool for determining class invariants is Shimura’s reciprocity law; see [28, Th. 4] for a fairly general result. Class invariants arising from many diﬀerent modular functions have been described in the literature; we brieﬂy summarise some of the most useful ones. Let η be Dedekind’s function, and let ζn = exp(2πi/n). Weber considered z+1 √ η(2z) η z2 −1 η 2 , f1 (z) = , f2 (z) = 2 , f = ζ48 η(z) η(z) η(z) √ 3 powers of which yield class invariants when D j, which 2 = −1, and also γ2 = is a class invariant whenever 3 D. The Weber functions can be generalised [15, 16, 21, 20, 23], and we have the simple and double η-quotients z z z η η N p1 η p2 wN (z) = ; wp1 ,p2 = with N = p1 p2 , z η(z) η(z) η p1 p2

where p1 and p2 are primes. Subject to constraints on D, including that no prime dividing N is inert in O, suitable yield class invariants, powers of these functions see [15, 16]. For s = 24/ gcd 24, (p1 − 1)(p2 − 1) , the canonical power wsp1 ,p2 0 is invariant under the Fricke involution W |N : z → −N z for Γ (N ), equivalently, the Atkin-Lehner involution of level N , by [17, Thm. 2]. The theory of [28] applies to any functions for Γ0 (N ), in particular to those of primelevel N invariant under the Fricke involution, which yield class invariants D when N = −1. Atkin developed a method to compute such functions AN , which are conjectured to have a pole of minimal order at the unique cusp [10, 26]. These are used in the SEA algorithm, and can be found in Magma or Pari/GP. The functions above all yield algebraic integers, so HD [f ] ∈ OK [X]. Except for weN or when gcd(N, D) = 1, in which cases additional restrictions may apply, one actually has HD [f ] ∈ Z[X], cf. [16, Cor. 3.1]. The (logarithmic) height of HD [f ] = ai X i is log max |ai |, which determines the precision needed to compute the ai . We let cD (f ) denote the ratio of the heights of HD [j] and HD [f ]. With c(f ) = lim|D|→∞ cD (f ), we have: c(γ2 ) = 3; c(f) = 72 (when D 2 = 1); c(weN ) =

24(N + 1) ; e(N − 1)

c(wsp1 ,p2 ) =

12ψ(p1 p2 ) ; s(p1 − 1)(p2 − 1)

c(AN ) =

N +1 , 2|vN |

where e divides the exponent s deﬁned above, vN is the order of the pole of AN at the cusp, and ψ(p1 p2 ) is (p1 + 1)(p2 + 1) when p1 = p2 , and p1 (p1 + 1) when

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p1 = p2 . Morain observed in [27] that c(A71 ) = 36, which is so far the best value known when D 2 = −1. We conjecture that in fact for all primes N > 11 with +1 , and that for N ≡ −1 mod 60 we N ≡ 11 mod 60 we have c(AN ) = 30 NN−11 have c(AN ) = 30. This implies that given an arbitrary discriminant D, we can always choose N so that AN yields class invariants with cD (AN ) ≥ 30 + o(1). When the prime divisors of N are all ramiﬁed in K, both wp1 ,p2 and AN yield class polynomials that are squares in Z[X], see [11, §1.6] and [18]. Taking the square root of such a class polynomial reduces both its degree and its height by a factor of 2. For a composite fundamental discriminant D (the most common case), this applies to HD [AN ] for any prime N | D. In the best case, D is divisible by 71, and we obtain a class polynomial that is 144 times smaller than HD . 3.1

Modular Polynomials

Each function f (z) considered above is related to j(z) by a modular polynomial Ψf ∈ Z[F, J] satisfying Ψf (f (z), j(z)) = 0. For primes not dividing the level N , we let Φ,f denote the minimal polynomial satisfying Φ ,f (f (z), f (z)) = 0; it is a factor of ResJ ResJ (Φ (J, J ), Ψf (F, J)), Ψf (F , J ) , and as such, an element of Z[F, F ]. Thus Φ,f generalises the classical modular polynomial Φ = Φ,j . The polynomial Φ,f has degree d(+1) in F and F , where d divides degJ Ψf , see [6, §6.8], and 2d divides degJ Ψf when f is invariant under the Fricke involution. In general, d is maximal, and d = 1 is achievable only in the relatively few cases where X0 (N ), respectively X0+ (N ), is of genus 0 and, moreover, f is a hauptmodul, that is, it generates the function ﬁeld of the curve. Happily, this includes many cases of practical interest. The polynomial Ψf characterises the analytic function f in an algebraic way; when d = 1, the polynomials Φ and Φ,f algebraically characterise -isogenies between elliptic curves given by their j-invariants, or by class invariants derived from f , respectively. These are key ingredients for the CRT method.

4

CRT Algorithms for Class Invariants

To adapt Algorithm 1 to class invariants arising from a modular function f (z) other than j(z), we only need to consider Algorithm 2. Our objective is to enumerate the roots of HD [f ] mod p for suitable primes p, which we are free to choose. This may be done in one of two ways. The most direct approach computes an “f -invariant” f1 , corresponding to j1 , then enumerates f2 , . . . , fh using the modular polynomials Φ,f . Alternatively, we may enumerate j1 , . . . , jh as before, and from these derive f1 , . . . , fh . The latter approach is not as eﬃcient, but it applies to a wider range of functions, including two inﬁnite families. Several problems arise. First, an elliptic curve E/Fp with CM by O unambiguously deﬁnes a j-invariant j1 = j(E), but not the corresponding f1 . The f1 we seek is a root of ψf (X) = Ψf (X, j1 ) mod p, but ψf may have other roots, which may or may not be class invariants. The same problem occurs for the

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p-adic lifting algorithm and can be solved generically [6, §6]; we describe some more eﬃcient solutions, which are in part speciﬁc to certain types of functions. When ψf has multiple roots that are class invariants, these may be roots of distinct class polynomials. We are generally happy to compute any one of these, but it is imperative that we compute the reduction of “the same” class polynomial HD [f ] modulo each prime p. The lemma below helps to address these issues for at least two inﬁnite families of functions: the double η-quotients wp1 ,p2 and the Atkin functions AN . 0 Lemma 2. Let f be a modular function Γ (N ), invariant under the Fricke −1for involution W |N , such that f (z) and f z have rational q-expansions. Let the imaginary quadratic order O have conductor coprime to N and contain an √ √ B02 −D 0+ D ideal n = N, B0 +2 D . Let A0 = 4N and τ0 = −B2A , and assume that 0 gcd(A0 , N ) = 1. Then f (τ0 ) is a class invariant, and if f (τ ) is any of its conjugates under the action of Gal(KO /K) we have and Ψf f (τ ), [n]j(τ ) = 0. Ψf f (τ ), j(τ ) = 0 Proof. By deﬁnition, Ψf f (z), j(z) = 0. Applying the Fricke involution yields = Ψf f (z), j Nz . The 0 = Ψf ((W |N f )(z), (W |N j)(z)) = Ψf f (z), j −N z value f (τ0 ) is a class invariant by [28, Th. 4]. By the same result, we may assume −B+√D that τ is the basis quotient of an ideal a = A, with gcd(A, N ) = 1 2 √ τ −B+ D . It is the basis quotient of an = AN, and B ≡ B0 mod 2N. Then N 2 τ , and replacing z above by τ completes the proof. follows that [n]j(τ ) = j N

If we arrange the roots of HD into a graph of n-isogeny cycles corresponding to the action of n, the lemma yields a dual graph deﬁned on the roots of HD [f ], in which vertices f (τ ) correspond to edges j(τ ), [n]j(τ ) . In computational terms, f (τ ) is a root of gcd Ψf X, j(τ ) , Ψf X, [n]j(τ ) . Generically, we expect this gcd to have no other roots modulo primes p that split completely in KO . For a ﬁnite number of such primes, there may be additional roots. We have observed this for p dividing the conductor of the order generated by f (τ ) in the maximal order of KO . Such primes may either be excluded from our CRT computations, or addressed by one of the techniques described in §4.3. 4.1

Direct Enumeration

When the polynomials Φ,f have degree + 1 we can apply Algorithm 2 with essentially no modiﬁcation; the only new consideration is that must not divide the level N , but we can exclude such when choosing a polycyclic presentation for Cl(O). When the degree is greater than + 1 the situation is more complex, moreover the most eﬃcient algorithms for computing modular polynomials do not apply [8, 13], making it diﬃcult to obtain Φ,f unless is very small. Thus in practice we do not use Φ,f in this case; instead we apply the methods of §4.3 or §4.4. For the remainder of this subsection and the next we assume that we do have polynomials Φ,f of degree + 1 with which to enumerate f1 , . . . , fh , and

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consider how to determine a starting point f1 , given the j-invariant j1 = j(E) of an elliptic curve E/Fp with CM by O. When ψf (X) = Ψf (X, j1 ) mod p has only one root, our choice of f1 is immediately determined. This is usually not the case, but we may be able to ensure it by restricting our choice of p. As an example, for f = γ2 with 3 D, if we require that p ≡ 2 mod 3, then f1 is the unique cube root of j1 in Fp . If we additionally have D ≡ 1 mod 8 and p ≡ 3 mod 4, then the equation γ2 = (f24 − 16)/f8 uniquely determines the square of the Weber f function, by [8, Lem. 7.3]. To treat f itself we need an additional trick described in §4.2. The next simplest case occurs when only one of the roots of ψf is a class invariant. This necessarily happens when f is invariant under the Fricke involution and all the primes dividing N are ramiﬁed in O. In the context of Lemma 2, each root of HD [f ] then corresponds to an isolated edge j(τ ), [n]j(τ ) in the n-isogeny graph on the roots of HD , and we compute f1 as the unique root of ¯, and each f (τ ) occurs twice gcd Ψf (X, j1 ), Ψf (X, [n]j1 ) . In this situation n = n as a root of HD [f ]. By using a polycyclic presentation for Cl(O)/[n] rather than Cl(O), we enumerate each double root of HD [f ] mod p just once. Even when ψf has multiple roots that are class invariants, it may happen that they are all roots of the same class polynomial. This applies to the Atkin functions f = AN . When N is a split prime, there are two N -isogenous pairs (j1 , [n]j1 ) and ([¯ n]j1 , j1 ) in EllO (Fp ), and under Lemma 2 these correspond to roots f1 and [¯ n]f1 of ψf . Both are roots of HD [f ], and we may choose either. The situation is slightly more complicated for the double η-quotients wp1 ,p2 , with N = p1 p2 composite. If p1 = p1 ¯ p1 and p2 = p2 ¯p2 both split and p1 = p2 , then there are four distinct N -isogenies corresponding to four roots of ψf . Two of these roots are related by the action of [n] = [p1 p2 ]; they belong to the same class polynomial, which we choose as HD [f ] mod p. The other two are related by [p1 ¯ p2 ] and are roots of a diﬀerent class polynomial. We make an arbitrary choice for f1 , explicitly compute [n]f1 , and then check whether it occurs among the other three roots; if not, we correct the initial choice. The techniques of §4.3 may be used to eﬃciently determine the action of [n]. Listed below are some of the modular functions f for which the roots of HD [f ] mod p may be directly enumerated, with suﬃcient constraints on D and p. In each case p splits completely in KO and D < −4N 2 has conductor u. (1) γ2 , with 3 D and p ≡ 2 mod 3; (2) f2 , with D ≡ 1 mod 8, 3 D, and p ≡ 11 mod 12; (3) wsN , for N ∈ {3, 5, 7, 13} and s = 24/ gcd(24, N − 1), with N | D and N u; (4) w25 , with 3 D, 5 | D, and 5 u; (5) AN , for N ∈ {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}, with and N u.

D N

= −1

(6) wsp1 ,p2 , for (p1 , p2 ) ∈ {(2, 3), (2, 5), (2, 7), (2, 13), (3, 5), (3, 7), (3, 13), (5, 7)} and s = 24/ gcd 24, (p1 − 1)(p2 − 1) , with pD1 , pD2 = −1 and p1 , p2 u. (7) w63,3 with D 3 = 1 and 3 u.

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The Trace Trick

In §4.1 we were able to treat the square of the Weber f function but not f itself. To remedy this, we generalise a method suggested to us by Reinier Br¨oker. We consider the situation where there are two modular functions f and f that are roots of Ψf (X, j(z)), both of which yield class invariants for O, and we wish to apply the direct enumeration approach. We assume that p is chosen so that ψf (X) = Ψf (X, j1 ) mod p has exactly two roots, and depending on which root we take as f1 , we may compute the reduction of either HD [f ](X) or HD [f ](X) modulo p. In the case of Weber f, we have f = −f , and HD [f ] diﬀers from HD [f ] only in the sign of every other coeﬃcient. Consider a ﬁxed coeﬃcient ai of HD [f ](X) = ai X i ; most of the time, the trace t = −ah−1 = f1 + · · · + fh will do (if f = −f , we need to use ai with i ≡ h mod 2). The two roots f1 and f1 lead to two possibilities t and t modulo p. However, the elementary symmetric functions T1 = t + t and T2 = tt are unambiguous modulo p. Computing these modulo many primes p yields T1 and T2 as integers (via the CRT), from which t and t are obtained as roots of the quadratic equation X 2 − T1 X + T2 . If these are diﬀerent, we arbitrarily pick one of them, which, going back, determines the set of conjugates {f1 , . . . , fh } or {f1 , . . . , fh } to take modulo each of the primes p t − t . In the unlikely event that they are the same (the suspicion t = t being conﬁrmed after, say, looking at the second prime), we need to switch to a diﬀerent coeﬃcient ai . If f and f diﬀer by a simple transformation (such as f = −f ), the second set of conjugates and the value t are obtained essentially for free. As a special case, when h is odd and the class invariants are units (as with Weber f), we can simply ﬁx t = a0 = 1, and need not compute T1 = 0 and T2 = −1. The key point is that the number of primes p we use to determine t is much less than the number of primes we use to compute HD [f ]. Asymptotically, the logarithmic height of the trace is smaller than the height bound we use for HD [f ] by a factor quasi-linear in log |D|, under the GRH. In practical terms, determining t typically requires less than one tenth of the primes used to compute HD [f ], and these computations can be combined. The approach described above generalises immediately to more than two roots, but this case does not occur for the functions we examine. Unfortunately it can be used only in conjunction with the direct enumeration approach of §4.1; otherwise we would have to consistently distinguish not only between f1 and f1 , but also between fi and fi for i = 2, . . . , h. 4.3

Enumeration via the Fricke Involution

For functions f to which Lemma 2 applies, we can readily obtain the roots of HD [f ] mod p without using the polynomials Φ,f . We instead enumerate the roots of HD mod p (using the polynomials Φ ), and arrange them into a graph G of n-isogeny cycles, where n is the ideal of norm N appearing in Lemma 2. We then obtain roots of HD [f ] mod p by computing gcd Ψf (X, ji ), Ψf (X, [n]ji ) for each edge (ji , [n]ji ) in G.

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The graph G is composed of h/n cycles of length n, where n is the order of [n] in Cl(O). We assume that the O-ideals of norm N are all non-principal and inequivalent (by requiring |D| > 4N 2 if needed). When every prime dividing N is ramiﬁed in O we have n = 2; as noted in §4.1, every root of HD [f ] then occurs with multiplicity 2, and we may compute the square-root of HD [f ] by taking each root just once. Otherwise we have n > 2. Let [l1 ], . . . , [lm ] be a polycyclic presentation for Cl(O) with relative orders √ r1 , . . . , rm , as in §2.2. For k from 1 to m let us ﬁx lk = k , −Bk2+ D with Bk ≥ 0. To each vector e = (e1 , . . . , em ) with 0 ≤ ek < rk , we associate a unique root je enumerated by Algorithm 2, corresponding to the path taken from j1 to je , where ek counts steps taken along an k -thread. For o = (0, . . . , 0) we have jo = j1 , and in general je = [lσ1 1 e1 · · · lσmm em ]jo , with σk = ±1. Using the method of §2.3 to consistently orient the k -threads ensures that each σk depends only on the orientation of the ﬁrst k -thread. To compute the graph G we must determine the signs σk . For those [lk ] of order 2, we let σk = 1. We additionally ﬁx σk = 1 for the least k = k0 (if any) for which [lk ] has order greater than 2, since we need not distinguish the actions of n ¯. It suﬃces to show how to determine σk , given that we know σ1 , . . . , σk−1 . and n We may assume [lk0 ] and [lk ] both have order greater than 2, with k0 < k ≤ m. Let l be an auxiliary ideal of prime norm such that [l] = [ab] = [le11 · · · lekk ], with 0 ≤ ei < ri , where b = lekk , and [a] and [b] have order greater than 2. Our ˇ assumptions guarantee that such an l exists, by the Cebotarev density theorem, and under the GRH, is relatively small [1]. The fact that [a] and [b] have order ¯ is distinct from [l] and its inverse. It follows that greater than 2 ensures that [ab] σk = 1 if and only if Φ (jo , je ) = 0, where e = (e1 , . . . , ek , 0, . . . , 0). Having determined the σk , we compute the unique vector v = (v1 , . . . , vm ) for which [n] = [lσ1 1 v1 · · · lσmm vm ]. We then have [n]jo = jv , yielding the edge (jo , jv ) of G. In general, we obtain the vector corresponding to [n]je by computing e + v xk−1 and using relations [lrkk ] = [lx1 1 · · · lk−1 ] to reduce the result, cf. [30, §5]. This method may be used with any function f satisfying Lemma 2, and in particular it applies to two inﬁnite families of functions: D (8) AN , for N > 2 prime, with N = −1 and N u. s (9) wp1 ,p2 , for p1 , p2 primes not both 2, with pD1 , pD2 = −1 and p1 , p2 u. As above, u denotes the conductor of D < −4N 2 . As noted earlier, for certain primes p we may have diﬃculty computing the edges of G when gcd Ψf (X, ji ), Ψf (X, [n]ji ) has more than one root in Fp . While we need not use such primes, it is often easy to determine the correct root. Here we give two heuristic techniques for doing so. The ﬁrst applies when N is prime, as with the Atkin functions. In this case problems can arise when HD [f ] has repeated roots modulo p. By Kummer’s criterion, this can happen only when p divides the discriminant of HD [f ], and even then, a repeated root x1 is only actually a problem when it corresponds to two alternating edges in G, say (j1 , j2 ) and (j3 , j4 ), with the edge (j2 , j3 ) between them.

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In this scenario we will get two roots x1 and x2 of gcd Ψf (X, j2 ), Ψf (X, j3 ) . But if we already know that x1 corresponds to (j1 , j2 ), we can unambiguously choose x2 . In each of the N -isogeny cycles of G, it is enough to ﬁnd a single edge that yields a unique root. If no such edge exists, then every edge must yield the same two roots x1 and x2 , and we count each with multiplicity n/2. The second technique applies when the roots of HD [f ] are units, as with the double η-quotients [16, Thm. 3.3]. The product of the roots is then ±1. Assuming that the number of edges in G for which multiple roots arise is small (it is usually zero, and rarely more than one or two), we simply test all the possible choices of roots and see which yield ±1. If only one combination works, then the correct choices are determined. This is not guaranteed to happen, but in practice it almost always does. 4.4

A General Algorithm

We now brieﬂy consider the case of an arbitrary modular function f of level N , and sketch a general algorithm to compute HD [f ] with the CRT method. Let us assume that f (τ ) is a class invariant, and let D be the discriminant and u the conductor of the order O = [1, τ ]. The roots of Ψf (X, j(τ )) ∈ KO [X] lie in the ray class ﬁeld of conductor uN over K, and some number n of these, including f (τ ), actually lie in the ring class ﬁeld KO . We may determine n using the method described in [6, §6.4], which computes the action of (O/N O)∗ /O∗ on the roots of Ψf (X, j(τ )). We note that the complexity of this task is essentially ﬁxed as a function of |D|. Having determined n, we use Algorithm 2 to enumerate the roots j1 , . . . , jh of HD mod p as usual, but if for any ji we ﬁnd that Ψf (X, ji ) mod p does not have (1) (n) exactly n roots fi , . . . , fi , we exclude the prime p from our computations. The number of such p is ﬁnite and may be bounded in terms of the discriminants of the polynomials Ψf (X, α) as α ranges over the roots of HD [f ]. We then h n (r) compute the polynomial H(X) = i=1 r=1 X − fi of degree nh in Fp [X]. After doing this for suﬃciently many primes p, we can lift the coeﬃcients by Chinese remaindering to the integers. The resulting H is a product of n distinct class polynomials, all of which may be obtained by factoring H in Z[X]. Under suitable heuristic assumptions (including the GRH), the total time to compute HD [f ] is quasi-linear in |D|, including the time to factor H. This approach is practically eﬃcient only when n is small, but then it can be quite useful. A notable example is the modular function g for which Ψg (X, J) = (X 12 − 6X 6 − 27)3 − JX 18 . This function was originally proposed by Atkin, and is closely related to certain class invariants of Ramanujan [3, Thm. 4.1]. The function g yields class invariants when D ≡ 13 mod 24. In terms of our generic algorithm, we have n = 2, and for p ≡ 2 mod 3 we get exactly two roots of Ψg (X, ji ) mod p, which diﬀer only in sign. Thus H(X) = HD [g 2 ](X 2 ) = HD [g](X)HD [g](−X), and from this we easily obtain HD [g 2 ], and also HD [g] if desired.

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5

153

Computational Results

This section provides performance data for the techniques developed above. We used AMD Phenom II 945 CPUs clocked at 3.0 GHz for our tests; the software was implemented using the gmp [22] and zn poly [24] libraries, and compiled with gcc [19]. To compute the class polynomial HD [f ], we require a bound on the size of its coeﬃcients. Unfortunately, provably accurate bounds for functions f other than j are generally unavailable. As a heuristic, we take the bound B on the coeﬃcients of HD given by [30, Lem. 8], divide log2 B by the asymptotic height factor c(f ), and add a “safety margin” of 256 bits. We note that with the CM method, the correctness of the ﬁnal result can be eﬃciently and unconditionally conﬁrmed [5], so we are generally happy to work with a heuristic bound. 5.1

Class Polynomial Computations Using the CRT Method

Our ﬁrst set of tests measures the improvement relative to previous computations with the CRT method. We used discriminants related to the construction of a large set of pairing-friendly elliptic curves, see [30, §8] for details. We reconstructed many of these curves, ﬁrst using the Hilbert class polynomial HD , and then using an alternative class polynomial HD [f ]. In each case we used the explicit CRT to compute HD or HD [f ] modulo a large prime q (170 to 256 bits). Table 1 gives results for four discriminants with |D| ≈ 1010 , three of which appear in [30, Table 2]. Each column lists times for three class polynomial computations. First, we give the total time Ttot to compute HD mod q, including the time Tenum spent enumerating EllD (Fp ), for all the small primes p, using Algorithm 2 as it appears in §2.2. We then list the times Tenum and Ttot obtained when Algorithm 2 is modiﬁed to use gcd computations whenever it is advantageous to do so, as explained in §2.3. The gcd approach typically speeds up Algorithm 2 by a factor of 2 or more. For the third computation we selected a function f that yields class invariants for D, and computed HD [f ] mod q. This polynomial can be used in place of HD in the CM method (one extracts a root x0 of HD [f ] mod q, and then extracts a root of Ψf (x0 , J) mod q). For each function f we give a “size factor”, which approximates the ratio of the total size of HD to HD [f ] (over Z). In the ﬁrst three examples this is just the height factor c(f ), but in Example 4 it is 4c(f ) because the prime 59 is ramiﬁed and we actually work with the square root of HD [A59 ], as noted in §4.1, reducing both the height and degree by a factor of 2. We then list the speedup Ttot /Ttot [f ] attributable to computing HD [f ] rather than HD . Remarkably, in each case this speedup is about twice what one would expect from the height factor. This is explained by a particular feature of the CRT method: The cost of computing HD mod p for small primes p varies significantly, and, as explained in [30, §3], one can accelerate the CRT method with a careful choice of primes. When fewer small primes are needed, we choose those for which Step 1 of Algorithm 1 can be performed most quickly. The last line in Table 1 lists the total speedup Ttot /Ttot [f ] achieved.

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A. Enge and A.V. Sutherland Table 1. Example class polynomial computations (times in CPU seconds) Example 1

Example 2

Example 3

Example 4

|D| h(D) log2 B r (r11 , . . . , kk )

13569850003 20203 2272564 (720203 )

11039933587 11280 1359134 (171128 , 1910 )

12901800539 54706 5469776 (327038 , 52 )

12042704347 9788 1207412 (292447 , 312 , 432 )

Tenum (roots) Ttot

6440 19900

10200 23700

10800 52200

21700 42400

Tenum (gcds) Ttot

2510 15900

2140 15500

3440 44700

4780 25300

A71 36 213

A47 24 305

A71 36 629

A59 120* 191

75 93

51 78

71 83

132 222

Function f Size factor [f ] Ttot Speedup (Ttot /Ttot [f ]) [f ]) Speedup (Ttot /Ttot

5.2

Comparison to the Complex Analytic Method

Our second set of tests compares the CRT approach to the complex analytic method. For each of the ﬁve discriminants listed in Table 2 we computed class polynomials HD [f ] for the double η-quotient w3,13 and the Weber f function, using both the CRT approach described here, and the implementation [14] of the complex analytic method as described in [12]. With the CRT we computed HD [f ] both over Z and modulo a 256-bit prime q; for the complex analytic method these times are essentially the same. Table 2. CRT vs. complex analytic (times in CPU seconds) complex analytic

CRT mod q

CRT

|D|

h(D)

w3,13

f

w3,13

f

w3,13

f

6961631 23512271 98016239 357116231 2093236031

5000 10000 20000 40000 100000

15 106 819 6210 91000

5.4 33 262 1900 27900

2.2 10 52 248 2200

1.0 4.1 22 101 870

2.1 9.8 47 213 1800

1.0 4.0 22 94 770

We also tested a “worst case” scenario for the CRT approach: the discriminant D = −85702502803, for which the smallest non-inert prime is 1 = 109. Choosing the function most suitable to each method, the complex analytic method computes HD [w109,127 ] in 8310 seconds, while the CRT method computes HD [A131 ]

Class Invariants by the CRT Method

155

in 7150 seconds. The CRT approach beneﬁts from the attractive height factor of the Atkin functions, c(A131 ) = 33 versus c(w109,127 ) ≈ 12.4, and the use of gcds in Algorithm 2. Without these improvements, the time to compute HD with the CRT method is 1460000 seconds. The techniques presented here yield more than a 200-fold speedup in this example. 5.3

A Record-Breaking CM Construction

To test the scalability of the CRT approach, we constructed an elliptic curve using |D| = 1000000013079299 > 1015 , with h(D) = 10034174 > 107 . This yielded a curve y 2 = x3 − 3x + c of prime order n over the prime ﬁeld Fq , where c = 12229445650235697471539531853482081746072487194452039355467804333684298579047; q = 28948022309329048855892746252171981646113288548904805961094058424256743169033; n = 28948022309329048855892746252171981646453570915825744424557433031688511408013.

This curve was obtained by computing the square root of HD [A71 ] modulo q, a polynomial of degree h(D)/2 = 5017087. The height bound of 21533832 bits was achieved with 438709 small primes p, the largest of which was 53 bits in size. The class polynomial computation took slightly less than a week using 32 cores, approximately 200 days of CPU time. Extracting a root over Fq took 25 hours of CPU time using NTL [29]. We estimate that the size of HD [A71 ] is over 13 terabytes, and that the size of the Hilbert class polynomial HD is nearly 2 petabytes. The size of HD [A71 ] mod q, however, is under 200 megabytes, and less than 800 megabytes of memory (per core) were needed to compute it.

References [1] Bach, E.: Explicit bounds for primality testing and related problems. Mathematics of Computation 55(191), 355–380 (1990) [2] Belding, J., Br¨ oker, R., Enge, A., Lauter, K.: Computing Hilbert class polynomials. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 282–295. Springer, Heidelberg (2008) [3] Berndt, B.C., Chan, H.H.: Ramanujan and the modular j-invariant. Canadian Mathematical Bulletin 42(4), 427–440 (1999) [4] Bernstein, D.J.: Modular exponentiation via the explicit Chinese Remainder Theorem. Mathematics of Computation 76, 443–454 (2007) [5] Bisson, G., Sutherland, A.V.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory (2009) (to appear), http://arxiv.org/abs/0902.4670 [6] Br¨ oker, R.: Constructing elliptic curves of prescribed order. Universiteit Leiden, Proefschrift (2006) [7] Br¨ oker, R.: A p-adic algorithm to compute the Hilbert class polynomial. Mathematics of Computation 77, 2417–2435 (2008) [8] Br¨ oker, R., Lauter, K., Sutherland, A.V.: Modular polynomials via isogeny volcanoes (2009) (preprint), http://arxiv.org/abs/1001.0402

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[9] Couveignes, J.-M., Henocq, T.: Action of modular correspondences around CM points. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 234– 243. Springer, Heidelberg (2002) [10] Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory, pp. 21–76. AMS, Providence (1998) [11] Enge, A.: Courbes alg´ebriques et cryptologie. In: Habilitation ` a diriger des recherches, vol. 7. Universit´e Denis Diderot, Paris (2007) [12] Enge, A.: The complexity of class polynomial computation via floating point approximations. Mathematics of Computation 78(266), 1089–1107 (2009) [13] Enge, A.: Computing modular polynomials in quasi-linear time. Mathematics of Computation 78(267), 1809–1824 (2009) [14] Enge, A.: cm, 0.2 edition (2010), http://cm.multiprecision.org/ [15] Enge, A., Morain, F.: Generalised Weber functions. I. Technical Report 385608, HAL-INRIA (2009), http://hal.inria.fr/inria-00385608 [16] Enge, A., Schertz, R.: Constructing elliptic curves over finite fields using double eta-quotients. Journal de Th´eorie des Nombres de Bordeaux 16, 555–568 (2004) [17] Enge, A., Schertz, R.: Modular curves of composite level. Acta Arithmetica 118(2), 129–141 (2005) [18] Enge, A., Schertz, R.: Singular values of multiple eta-quotients for ramified primes (in preparation 2010) [19] Free Software Foundation. GNU Compiler Collection, 4.2.4 edition (2008), http://gcc.gnu.org/ [20] Gee, A.: Class fields by Shimura reciprocity. Universiteit Leiden, Proefschrift (2001) [21] Gee, A., Stevenhagen, P.: Generating class fields using Shimura reciprocity. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 441–453. Springer, Heidelberg (1998) [22] Granlund, T., et al.: gmp, 4.3.1 edition (2009). http://gmplib.org/. [23] Hajir, F., Villegas, F.R.: Explicit elliptic units, I. Duke Mathematical Journal 90(3), 495–521 (1997) [24] Harvey, D.: zn poly: a library for polynomial arithmetic, 0.9 edn. (2008), http://cims.nyu.edu/~ harvey/zn_poly [25] Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California at Berkeley (1996) [26] Morain, F.: Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques. Journal de Th´eorie des Nombres de Bordeaux 7(1), 111–138 (1995) [27] Morain, F.: Advances in the CM method for elliptic curves. In: Slides of Fields Cryptography Retrospective Meeting, May 11-15 (2009), http://www.lix.polytechnique.fr/~ morain/Exposes/fields09.pdf [28] Schertz, R.: Weber’s class invariants revisited. Journal de Th´eorie des Nombres de Bordeaux 14(1), 325–343 (2002) [29] Shoup, V.: NTL: A library for doing number theory, 5.5 edn. (2008), http://www.shoup.net/ntl/ [30] Sutherland, A.V.: Computing Hilbert class polynomials with the Chinese Remainder Theorem. Mathematics of Computation (to appear 2010), http://arxiv.org/abs/0903.2785 [31] Weber, H.: Lehrbuch der Algebra, 3rd edn., vol. III. Chelsea, New York (1961)

Short Bases of Lattices over Number Fields Claus Fieker1 and Damien Stehl´e1,2 1

Magma Computer Algebra Group, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia 2 CNRS and Macquarie University [email protected], [email protected]

Abstract. Lattices over number ﬁelds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, “short” (pseudo)-basis is important in many applications. In this article, we provide the ﬁrst algorithm that computes such a “short” (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to ﬁnd a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a ﬁeld invariant. As it runs in polynomial time, this provides an eﬀective variant of Minkowski’s second theorem for lattices over number ﬁelds.

1

Introduction

Let K be a number ﬁeld and OK be its maximal order. An OK -module M is a ﬁnitely generated set of elements which is closed under addition and multiplication by elements in OK . Frequently, we have M ⊆ K m for some m. In the case of K being Q, we have OK = Z, thus OK -modules are just the classical Z-lattices. Since Z is a principal ideal domain, every (torsion free) module is free, thus there exists a basis b1 , . . . , bn ∈ M for some n ≤ m such that M = ⊕i≤n Zbi . Any two bases (bi )i and (ci )i have the same cardinality and are linked by some unimodular matrix T ∈ GL(n, Z). The choice of a good basis is crucial for almost all computational problems attached to M . Generally one tries to ﬁnd a basis whose vectors have short Euclidean norms, using, for example, the LLL algorithm [15]. Replacing Z by the maximal order OK makes the classiﬁcation more complicated since OK may no longer be a principal ideal domain. However, since OK is still a Dedekind domain, the modules M ⊆ K m have a well known structure ([7, Cor. 1.2.25], [23, Th. 81:3]): there exist linearly independent elements b1 , . . . , bn ∈ K m and (non-zero fractional) ideals b1 , . . . , bn such that M = ⊕i≤n bi bi , i.e., every b ∈ M has a unique representation as b = i≤n xi bi with xi ∈ bi for all i ≤ n. Such a representation is commonly called a pseudo-basis. It should be noted that bi may not belong to M , and in fact bi ∈ M if and only if 1 ∈ bi . Similarly to the case of Z-lattices, diﬀerent pseudo-bases share the same cardinality, and it is known how to move from a pseudo-basis to another. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 157–173, 2010. c Springer-Verlag Berlin Heidelberg 2010

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As for Z-lattices, the choice of the pseudo-basis is of utmost importance. However, a key diﬀerence is that no analogue of LLL is known, as repeatedly noted in [7]. There have been attempts [10,22,11] but the algorithms are either limited to certain ﬁelds or give no guaranteed bounds on the output size. While every OK -module is also a Z-lattice and can thus be analyzed with all the tools available over Z, for many applications the additional structure as an OK -module is important. This structure is typically lost when applying techniques over Z. Originally, OK -modules mainly came from the study of ﬁnite extensions of K but now they occur in a wider range of problems from group theory (matrix groups and representations [9]) to applications in geometry (automorphism algebras of Abelian varieties). OK -modules also occur in lattice-based cryptography [17,19,24,25,26], and in that context the module rank n is usually polylogarithmic in the degree of the number ﬁeld. Cryptography based on OK modules is increasingly popular, as on one side they lead to compact representations and to fast operations, and on the other side they enjoy a worst-case to average-case reduction for variants of the shortest vector problem, which allows the cryptographic security to be based on worst-case hardness assumptions. As diverse as the applications are the requirements: only one (or more) short module element(s) may be needed, or a short (pseudo)-basis may be required, some applications rely on canonical representations, while any representation may suﬃce for others. We note that canonical representations tend to have components that are much larger than short representations as obtained by lattice reduction or our techniques. To ﬁnd one short element it suﬃces to consider the underlying Z-module (of dimension nd with d = [K : Q]). For Z-lattices contained in Qm , a canonical representation is the Hermite Normal Form (HNF). It has been generalized (BPC-HNF) to OK -modules contained in K m by Bosma and Pohst [4] and Cohen [7, Chap. 1.4] (see also [12]). Our results. In the present work, we describe an algorithm that computes a pseudo-basis made of short vectors. Given an arbitrary pseudo-basis [(ai )i , (ai )i ] of a module M ⊆ K m , it returns a pseudo-basis [(bi )i , (bi )i ] such that: 2

∀i ≤ n : bi ∈ M, N (bi ) ∈ [2−O(d ) , 1] and bi ≤ 2O(dn) λi (M ), where the O(·)’s depend only on the ﬁeld K and the choice of a given LLLreduced integral basis, the euclidean norm · is a module extension of the T2 -norm over K, and the λi (M )’s correspond to the module minima. We refer to Corollary 1 for a precise statement. Overall, this provides a module equivalent to LLL-reduced bases of Z-lattices in the sense that the vectors cannot be arbitrarily longer than the minima. Since it runs in polynomial time, it can also be interpreted as an eﬀective approximate variant of the adaptation to OK -modules of Minkowski’s second theorem (given in Theorem 2). We also study the representation of one-dimensional OK -modules, i.e., modules that are isomorphic to ideals of OK . We show how to modify Belabas’ 2-element representation algorithm [2, Alg. 6.15] so that the output is provably small. Combining the latter and our module pseudo-reduction algorithm leads to compact representations of OK -modules.

Short Bases of Lattices over Number Fields

159

The most natural approach to obtain reduced pseudo-bases consists in trying to generalize LLL, but as mentioned earlier all previous attempts have only partially succeeded. In contrast, we start by viewing the OK -module as a highdimensional Z-lattice. We ﬁnd short module elements by applying LLL to a basis of the latter lattice and interpreting the output as module elements. At this point, we have a pseudo-basis (the input) and a full-rank set of short module vectors (produced by LLL). If we had a Z-lattice instead of an OK -module, we would then use a technique common in the lattice-based cryptography community (see, e.g., [20, Le. 7.1]), consisting in using the HNF to convert a full rank set of short lattice vectors to a short basis. We adapt this technique to number ﬁelds, using the BPC-HNF and introducing a size-reduction algorithm for pseudo-bases. Let us compare (pseudo-)LLL-reduced and BPC-HNF pseudo-bases. A theoretical advantage of the LLL approach is that it is not restricted to K m but also works in a continuous extension (similarly to LLL-reduction being welldeﬁned for real lattices). It should also be signiﬁcantly more eﬃcient to work with pseudo-bases made of short vectors because smaller integers and polynomials of smaller degrees are involved. On the other side, (pseudo-)LLL-reduced pseudo-bases are far from being unique, and seem more expensive to obtain. Road-map. In Section 2, we give some reminders and elementary results on lattices, number ﬁelds and modules. In Section 3, we modify Belabas’ 2-element representation algorithm for ideals of OK , as described above. We then give our module reduction algorithm in Section 4. Finally, in Section 5 we describe our implementation and give some examples. Implementation. The algorithms have been implemented in the Magma computer algebra system [3,18] and are available on request. They will be part of upcoming releases.

2

Preliminaries

We assume the reader is familiar with the geometry of numbers and algebraic number theory. We refer to [16,20], [5,21] and [7, Chap. 1] for introductions to the computational aspects of lattices, elementary algebraic number theory and to modules over Dedekind domains, respectively. 2.1

Lattices

In this work, we will call any ﬁnitely generated free Z-module L a lattice. A usual lattice corresponds to the case where L is a discrete additive subgroup of Rn for some n. Any lattice can be written L = ⊕i≤d Zbi . If the bi ’s are Z-free, they are called a basis of L. A given lattice may have inﬁnitely many bases but their cardinality d is constant and called rank. Any two bases are related by a unimodular transformation, i.e., one is obtained from the other by multiplying by a matrix in Zd×d of determinant ±1. If L ⊆ Qn is of rank d, then there exists a basis B = (bi )i ∈ Qn×d of L such that μj = min{i : Bi,j = 0} (strictly) increases with j, and for all j > k we

160

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have Bμj ,j > Bμj ,k ≥ 0. If d = n, this means that B is a row-wise diagonally strictly dominant lower triangular matrix and that its entries are non-negative. This basis is unique and called the Hermite Normal Form (HNF) of L. It can be computed in polynomial time from any basis [13]. In order to quantify the smallness of an element of a lattice L, we associate to L a positive deﬁnite bilinear form q : LR × LR → R. We use it to map a basis (bi )i to its Gram matrix Gq (b1 , . . . , bd ) := (q(bi , bj ))i,j . We denote q(b, b) by bq , and may omit the subscript if it is clear from the context. The determinant of L, deﬁned as detq (L) = det(Gq (b1 , . . . , bd ))1/2 , does not depend on the particular choice of the basis of L. Note that if L ⊆ Rn and q is the euclidean inner product, then det(L) is the d-dimensional volume of the parallelepiped { i yi bi : yi ∈ [0, 1]}. We deﬁne the lattice minima as follows: ∀i ≤ d, λi,q (L) = min{r : ∃c1 , . . . , ci ∈ L free, maxk≤i ck q ≤ r}. √ d Minkowski’s second theorem states that i≤d λi,q (L) ≤ d detq (L). Frequently one tries to represent a lattice L by a basis that approximates the minima. In this article, we assume that we have an algorithm LatRed that takes as input an arbitrary basis of L and returns a reduced basis satisfying bi ≤ γλi (L), for all i ≤ d. For example, if we use the LLL algorithm [15], then we can take γ = 2d/2 . We proceed as follows: compute the Gram matrix G of the input basis; use the Gram matrix LLL algorithm (see, e.g., [5, p. 88]), to ﬁnd U unimodular such that U t GU is reduced; apply U to the input lattice basis. If the arithmetic over L is eﬃcient, and if q can be eﬃciently computed or approximated with high accuracy, then this provides an eﬃcient algorithm. Apart from being well-deﬁned for more general lattices (not only for lattices on a rational vector space), a signiﬁcant advantage of the LLL-reduction over the HNF is that it provides small lattice elements. However, it seems more expensive to obtain and the uniqueness of the representation is lost. Taking√ the HKZ-reduction instead of the LLLreduction allows one to take γ = 1/2 d + 3 (see [14]), but the complexity of the best algorithm for computing it [1] is exponential in d. ∗ ∗ ∗ Let (bi )i≤d be a lattice j, we deﬁne μi,j = q(b i , bj )/q(bj , bj ), basis. For any i > ∗ ∗ where bi = argminbi + ji |xj |, which gives the bound. To complete the proof, note that the reducedness of the ri ’s gives minj rj∗ ≥ √ 2−d/2 minj rj , and that rj ≥ d for all j. 2.3

OK -Modules

Let b1 , . . . , bn ∈ KRm with n = rankK (bi )i , and b1 , . . . , bn be fractional ideals of OK . The OK -module M [(bi )i , (bi )i ] spanned by the pseudo-basis [(bi )i , (bi )i ] is bi bi . The bi ’s are called the coeﬃcient ideals. As each bi is a Z-lattice, so (j) (j) is M . More precisely, if bi = j≤d Zβi , then M = i,j Zβi bi . Two pseudobases [(bi )i , (bi )i ] and [(ci )i , (ci )i ] represent the same OK -module M if and only if there exists a non-singular U ∈ K n×n with ([23, §81 C]): 1. (c1 , . . . , cn ) = (b1 , . . . , bn )U ; 2. For all i, j, we have Ui,j ∈ bi c−1 j ; −1 3. For all i, j, we have Ui,j ∈ ci b−1 . j , where U = U Cohen [6] generalized the HNF to modules in K m . The algorithm of [4] may also be interpreted as such a generalization. We refer to [12, Chap. 4] for a detailed exposure and comparison.

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163

Theorem 1. Let M ⊆ K m be an OK -module of rank n. There exists a pseudobasis [(bi )i , (bi )i ] of M such that μj = min{i : Bi,j = 0} (strictly) increases with j, for all j we have Bμj ,j = 1 and for all j > k the entry Bμj ,k ∈ K is size-reduced modulo the HNF of bj b−1 k . This unique pseudo-basis is called the HNF of M . It can be computed in polynomial time from any pseudo-basis of M . Similarly to the HNF for lattices, the above HNF can only handle OK -modules M ⊆ K m (as opposed to KRm ) and does not necessarily contain small elements of M . We now deﬁne the concept of small-ness for elements of KRm . For any two vectors b = (b1 , . . . , bm )t , b = (b1 , . . . , bm )t ∈ KRm , we deﬁne T2⊗m (b, b ) = T2⊗m (b, b) by b. Notice that for any (r, b) ∈ i≤m T2 (bi , bi ), and we denote KR × KRm , we have rb ≤ r · b. With this deﬁnition at hand, we can deﬁne the minima of M : ∀i ≤ n, λi (M ) = min{r : ∃c1 , . . . , ci ∈ M, rankK (ck )k = i and max ck ≤ r}. Let [(bi )i , (bi )i ] be a pseudo-basis of an OK -module M ⊆ KRm . Assume that bi = (j) j≤d Zβi . We deﬁne det(M ) as the square root of the determinant of the nd×nd (j )

symmetric positive deﬁnite matrix T2⊗m (βi bi , βi bi )i,j;i ,j . This is a module invariant. When M is a non-zero fractional ideal of OK , this matches detT2 (M ). It should be noted that det(M ) is not immediately related to the (Steinitz) class of M nor to the maximal exterior power of M . The following is a direct consequence of Minkowski’s second theorem over Z-lattices. Theorem 2. Let M ⊆ KRm be an OK -module of rank n. Then i≤n λi (M ) ≤ √ n dn det(M )1/d . (j)

Proof. The module M can be seen as a lattice L of dimension nd, with det(M ) = √ dn det(L). Minkowski’s second theorem asserts that i≤nd λi (L) ≤ dn det(L). Let c1 , . . . , cnd ∈ M be free over the integers such that ci = λi (L) holds for all i. For all i ≤ n, let φ(i) = min(j : rankK (c1 , . . . , cj ) = i). As OK has rank d as a Z-module, we have φ(i) ≤ (i − 1)d + 1. We conclude with the following sequence of inequalities: √ n 1 1 λi (M ) ≤ cφ(i) ≤ λ(i−1)d+1 (L) ≤ λi (L) d ≤ dn det(M ) d . i≤n

i≤n

i≤n

i≤dn

We now extend the concept of GSO. Let [(bi ) i , (bi )i ] be a pseudo-basis of an OK -module M . We deﬁne b∗i = argminbi + j _ := InvariantForm(N); // compute the form > SetVerbose("RLLL", 1); > O := Nice(N); > #Sprint(ActionGenerators(M)); 1359862 > #Sprint(ActionGenerators(N)); 327378 > #Sprint(ActionGenerators(O)); 4577 The function Nice implements the procedure outlined above. Note that the actual result can vary substantially as several parts use randomized algorithms. The Sprint statements are only used as a very crude indication of the output size, they simply give the number of characters neccessary to write the generating matrices for G.

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4. Bosma, W., Pohst, M.: Computations with ﬁnitely generated modules over Dedekind domains. In: Proc. ISSAC 1991, pp. 151–156. ACM, New York (1991) 5. Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1995) 6. Cohen, H.: Hermite and Smith normal form algorithms over Dedekind domains. Math. Comp. 65, 1681–1699 (1996) 7. Cohen, H.: Advanced topics in Computational Number Theory. Springer, Heidelberg (2000) 8. Evertse, J.-H.: Reduced bases of lattices over number ﬁelds. Indag. Mathem. N.S. 2(3), 153–168 (1992) 9. Fieker, C.: Minimizing representations over number ﬁelds II: Computations in the Brauer group. J. Algebra 3(322), 752–765 (2009) 10. Fieker, C., Pohst, M.E.: Lattices over number ﬁelds. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 147–157. Springer, Heidelberg (1996) 11. Gan, Y.H., Ling, C., Mow, W.H.: Complex lattice reduction algorithm for lowcomplexity full-diversity MIMO detection. IEEE Trans. Signal Processing 57, 2701– 2710 (2009) 12. Hoppe, A.: Normal forms over Dedekind domains, eﬃcient implementation in the computer algebra system KANT. PhD thesis, Technical University of Berlin (1998) 13. Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499–507 (1979) 14. Lagarias, J.C., Lenstra Jr., H.W., Schnorr, C.P.: Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10, 333–348 (1990) 15. Lenstra, A.K., Lenstra Jr., H.W., Lov´ asz, L.: Factoring polynomials with rational coeﬃcients. Math. Ann. 261, 515–534 (1982) 16. Lov´ asz, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. CBMSNSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1986) 17. Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006) 18. Magma. The Magma computational algebra system for algebra, number theory and geometry, http://magma.maths.usyd.edu.au/magma/ 19. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and eﬃcient oneway functions. Comput. Complexity 16(4), 365–411 (2007) 20. Micciancio, D., Goldwasser, S.: Complexity of lattice problems: a cryptographic perspective. Kluwer Academic Press, Dordrecht (2002) 21. Mollin, R.A.: Algebraic Number Theory. Chapman and Hall/CRC Press (1999) 22. Napias, H.: A generalization of the LLL-algorithm over Euclidean rings or orders. J. th´eorie des nombres de Bordeaux 2, 387–396 (1996) 23. O’Meara, O.T.: Introduction to Quadratic Forms. In: Grundlehren der Mathematischen Wissenschaften, vol. 117. Springer, Heidelberg (1963) 24. Peikert, C., Rosen, A.: Eﬃcient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006) 25. Peikert, C., Rosen, A.: Lattices that admit logarithmic worst-case to average-case connection factors. In: Proc. STOC 2007, pp. 478–487. ACM, New York (2007) 26. Stehl´e, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Eﬃcient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009)

On the Complexity of the Montes Ideal Factorization Algorithm David Ford and Olga Veres Concordia University, 1455 de Maisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1J1 [email protected], [email protected] Abstract. Let p be a rational prime and let Φ(X) be a monic irreducible polynomial in Z[X], with nΦ = deg Φ and δΦ = vp (disc Φ). In [13] Montes describes an algorithm for the decomposition of the ideal p OK in the algebraic number field K generated by a root of Φ. A simplified version of the Montes algorithm, merely testing Φ(X) for irreducibility over Qp , is given in [19], together with a full Maple implementation and a demonstration that in the worst case, when Φ(X) is irreducible over Qp , the expected 2+ number of bit operations for termination is O(n3+ Φ δΦ ). We now give a 2+ 2+ refined analysis that yields an improved estimate of O(n3+ Φ δΦ +nΦ δΦ ) bit operations. Since the worst case of the simplified algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the complete Montes algorithm.

1

Introduction

In an algebraic number ﬁeld K with ring of integers OK , factorization of the ideal pOK , for p prime, can be determined via polynomial factorization over the ﬁeld of p-adic numbers Qp [12]. If K = Q(α) for a given α ∈ OK such that the index OK : Z[α] is not divisible by p then the factorization of the ideal pOK can be determined by polynomial factorization modulo p [5,6,7]. In practice, eﬃcient techniques for polynomial factorization modulo p [1,2,4] combined with Hensel lifting [12,20] solve the problem of factoring pOK in a straightforward and eﬀective manner when p does not divide the index. The complications arising when p divides the index OK : Z[α] have been the subject of considerable study. Current ideas are derived from the “Round Four” algorithm of Zassenhaus [20], which has evolved into two main variations, the “one-element” method [8] and the “two-element” method [16]. Versions of the one-element method are used by Maple and PARI. The two-element method is used, e.g., by Magma. The algorithm of Montes [13] is in a separate category. Given a monic irreducible polynomial Φ(X) in Z[X], the Montes algorithm determines the number of irreducible factors of Φ(X) in Zp [X] and their respective degrees. The algorithm exploits classical results of Ore [15,14] on Newton G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 174–185, 2010. c Springer-Verlag Berlin Heidelberg 2010

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polygons and provides an alternative to the methods based on ideas of Zassenhaus. A familiar application of Newton polygons gives the p-adic valuations of roots of a polynomial in Zp [X]. If Φ(X) ∈ Zp [X] has two roots with diﬀerent p-adic values then Hensel-lifting techniques can be applied to construct a non-trivial p-adic factorization of Φ to any desired degree of precision. This process constitutes “level 0” of the Montes algorithm. For each factor of Φ revealed at level 0, the algorithm proceeds to higher levels, either to discover a reﬁned factorization or to establish irreducibility. At level r, with ϕr (X) an irreducible monic polynomial in Zp [X] and Vr a valuation of Qp [X], the algorithm constructs the ϕr -adic expansion of a given polynomial and then computes • a ﬁnite ﬁeld Fqr , • the Newton polygon Nr (Φ) of Φ with respect to the valuation Vr , • a slope −dr /er , with dr and er coprime positive integers, of an edge of Nr (Φ), (r)

• the “associated polynomial” ΨS,Φ (Y ) ∈ Fqr [Y ] for each segment S of Nr (Φ), (r)

• a monic irreducible factor ψr of ΨS,Φ with ξr a root of ψr and fr = deg ψr , • a valuation Vr+1 of Qp [X], • an irreducible monic polynomial ϕr+1 (X) ∈ Zp [X]. The number of edges of Nr (Φ) and the number of distinct irreducible factors of (r) ΨS,Φ give information for the factorization of Φ; if either is greater than one then Φ is reducible. Our goal being to give an estimate of the complexity of the worst case of the Montes algorithm, we have restricted the algorithm merely to decide the question of irreducibility of a given polynomial. When Φ is irreducible over Qp the Newton polygon at each level is a single segment. It is apparent that this is the most costly case, i.e., the case that reaches the highest level, for the full algorithm. So our restricted algorithm operates under the assumption that Nr (Φ) has just one edge at each level r; the failure of this condition terminates the restricted algorithm. In [19, Chapter 3] a complete Maple implementation of the restricted Montes algorithm is given, together with a demonstration that in the worst case, when Φ is irreducible over Qp , the expected number of bit operations for termination 2+ is O(n3+ Φ δΦ ), with nΦ = deg Φ and δΦ = vp (disc Φ). In the present paper we 2+ 2+ give a reﬁned analysis that yields an improved estimate of O(n3+ Φ δΦ + n Φ δΦ ) bit operations. Since the worst case of the simpliﬁed algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the full Montes algorithm.

2

Definitions and Notation

Definition 1. Let ϕ0 (X) = X and let V0 denote the standard p-adic valuation of Qp . For K(X) ∈ Qp [X] and r ≥ 1, the level-r Newton polygon of K, denoted

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Nr (K), is the Newton polygon of K with respect to the valuation Vr of Qp [X], which can be defined recursively as Vr (K) = min er−1 Vr−1 Ar−1,k + kVr ϕr−1 0 ≤ k ≤ n n with K(X) = k=0 Ar−1,k (X) ϕr−1 (X)k the ϕr−1 -adic expansion of K(X). Remark 1. Nr (K) is the lower convex hull of the set { (k, Vr (Ar,k ϕkr )) | 0 ≤ k ≤ n, Ar,k (X) = 0 } , and if deg K < deg ϕr then Nr (K) = {(0, Vr (K))} and Vr+1 (K) = er Vr (K). Definition 2. For r ≥ 1 and K(X) a nonzero polynomial in Zp [X] we define Sr,K to be the segment of Nr (K) having slope −dr /er . Definition 3. For positive integers r and ν we define αr,ν = ν d−1 r mod er , βr,ν = (ν − αr,ν dr )/er , Tr,ν = { (αr,ν + λer , βr,ν − λdr ) | 0 ≤ λ ≤ βr,ν /dr } . Remark 2. If L is the line through the point (0, ν/er ) with slope −dr /er then Tr,ν is the longest segment of L with endpoints having nonnegative integer coordinates. Definition 4. For r ≥ 0 we define μr = 0 ,

νr = 0 ,

if r = 0 ,

μr = dr−1 + er−1 ν r−1 ,

ν r = er−1 fr−1 μr ,

if r ≥ 1 .

Remark 3. For r ≥ 1 it is easily seen that μr = Vr (ϕr−1 ) and ν r = Vr (ϕr ). Definition 5 (Associated Polynomial ). Let r ≥ 0, let α and β be nonnegative integers, and let S be an arbitrary segment of slope −dr /er with left endpoint (α, β). Let m0 = 0 and for r ≥ 1 and k ≥ 0 define mr = (1/dr ) mod er ,

1 Ωr = er−1 fr−1 mr−1 fr−1 μr ξr−1 Ωr−1

if r = 1 , if r > 1 ,

(β − kdr ) − (α + ker ) ν r , Θ(S, r, k) = mr−1 er−1 Θ(S,r,k)

ΓS,r,k = Ωrα+ker ξr−1

∈ Fqr .

Let K(X) ∈ Zp [X] have ϕr -adic expansion K(X) = A0 (X) + A1 (X) ϕr (X) + · · · + An (X) ϕr (X)n

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with dr j + er Vr (Aj ϕjr ) ≥ dr α + er β for j = 0, . . . , n and let r ) ∈S . J = k 0 ≤ k ≤ (n − α)/er , α + ker , Vr (Aα+ker ϕα+ke r We define the level-r associated polynomial of K with respect to S to be (r) ΨS,K (Y ) = k∈J ηk Y k with ηk ∈ Fqr defined as ⎧ A ⎪ ⎪ ⎨ α+ke0 ηk = B k (ξ0 ) , ⎪ ⎪ ⎩ Γ −1 Ψ (r−1)

with Bk (X) = Aα+ke1 (X) p

S,r,k Tr−1,νk,Aα+ker (ξr−1 ) ,

if r = 0 , β−kd1

,

if r = 1 ,

with νk = Vr (Aα+ker ) ,

if r ≥ 2 .

We further define the natural level-r associated polynomial of K to be (r) (r) ΨK (Y ) = ΨSr,K ,K (Y ) . (r) Remark 4. The polynomial ΨK (Y ) has nonzero constant term.

3

Outline of the Restricted Montes Algorithm

A complete Maple implementation of the restricted Montes algorithm, with proofs and explanatory comments interspersed, is given in [19]. Here we give an outline showing the three major phases of the algorithm. The algorithm begins in phase M0 (level 0), then alternates between phase M1 and phase M2 (level r, for r = 1, 2, . . . ) until reaching a terminating condition. • input: Φ(X) ∈ Z[X] monic and irreducible, p ∈ Z prime

TRUE if Φ(X) is irreducible over Qp [X], • output: FALSE if Φ(X) is reducible over Qp [X]. M0 :

1. Factorize Φ modulo p: a

a0,κ

0,1 · · · ψ0,κ00 Φ ≡ ψ0,1

(mod p) .

2. If κ0 > 1 then return FALSE. If κ0 = 1 and a0,1 = 1 then return TRUE. 3. Deﬁne ϕ0 (X) = X, n0 = 1, d0 = 0, e0 = 1, ψ0 = ψ0,1 , f0 = deg ψ0 , ξ0 a root of ψ0 . 4. Set r ← 1. M1 :

5. If r = 1 let ϕ1 (X) be a monic polynomial in Z[X] such that ϕ1 = ψ0 . If r > 1 construct Hr−1 according to Algorithm 1 in Sect. 6 below and let er−1 fr−1 ϕr = ϕr−1 + Hr−1 .

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6. Deﬁne nr = er−1 fr−1 nr−1 = deg ϕr . 7. If r > 1 and er−1 fr−1 = 1 then replace ϕr−1 ← ϕr and r ← r − 1. M2 :

8. If ϕr = Φ then return TRUE. If ϕr | Φ and ϕr = Φ then return FALSE. 9. Let Sr,1 , . . . , Sr,λr be the segments of Nr (Φ) and let ζr,k + 1 be the number of points on Sr,k with integer coordinates, for k = 1, . . . , λr . 10. If λr > 1 then return FALSE. If λr = 1 and ζr,1 = 1 then return TRUE. 11. Let −dr /er be the slope of Sr,1 , with dr and er relatively prime and (r) er > 0, and construct ΨΦ (Y ) ∈ Fqr [ Y ]. 12. Factorize ar,1 ar,κr (r) ΨΦ = cr ψr,1 · · · ψr,κ r over Fqr , with cr ∈ Fqr a nonzero constant. 13. If κr > 1 then return FALSE. If κr = 1 and ar,1 = 1 then return TRUE. 14. Deﬁne ψr = ψr,1 , fr = deg ψr , ξr a root of ψr . 15. Replace r ← r + 1. Go to M1 .

4

Complexity of Fundamental Operations

Notation. We use alpha F and alpha Q to denote the number of operations p in Fp and Q respectively required for the execution of the procedure alpha. We use the notation f (n) ∈ O(nk+ ) as an alternative to the “soft-O” notation f (n) ∈ O∼(nk ) ≡ f (n) ∈ O(nk (ln n)c ) for some positive constant c (see [9]). For n ≥ 3 and q a prime power we deﬁne the following. L(n) = ln n ln ln n M(n) = n L(n)

F(n, q) = n M(n) ln(qn) K(q) = M(ln q) ln ln q

We are concerned with the reducibility of the monic polynomial Φ(X) ∈ Zp [X] ∗ for some prime p. We let δΦ denote vp (disc Φ) and we let pδΦ denote the p-adic ∗ reduced discriminant of Φ [8, Appendix A]. It is clear that δΦ ≤ δΦ . Magnitude of p. To simplify the subsequent discussion we impose the condition that p ∈ O(1), by which we mean that p is a small prime, not exceeding the magnitude of a single machine word.

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∗

Arithmetic in Zp . If F (X) ∈ Z[X] with F (X) ≡ Φ(X) (mod p2δΦ +1 Zp [X]) then Φ(X) is reducible in Zp [X] if and only if F (X) is reducible in Zp [X]. Thus in our computations p-adic integers are represented as rational approximations ∗ with 2δΦ + 1 p-adic digits of precision, i.e., as rational integers reduced modulo ∗ 2δΦ +1 p . Sch¨ onhage and Strassen have shown that the time required to perform an arithmetic operation on two rational integers of length m is O(M(m)); see [9, Ch.8, §8.3]. It follows that if we represent p-adic integers in this fashion then the cost of an arithmetic operation is O(ΔΦ ), with ∗ ln p) . ΔΦ = M(δΦ

Arithmetic in Fq . By [9, Ch.14, §14.7], a single operation in Fq can be per∗ formed in O(K(q)) word operations. If q = pf the assumption that ln p ∈ O(1) gives ln q = f ∗ ln p ∈ O(f ∗ ) and thus the cost of an operation in Fq is O(K(q)) = O M(ln q) ln ln q ⊆ O f ∗ (ln f ∗ )2 ln ln f ∗ ⊆ O f ∗ (1+) . For α ∈ Fq and any integer n the cost of computing αn is O(ln q K(q)) ⊆ O(f ∗ f ∗ (1+) ) = O(f ∗ (2+) ) since we may assume 0 ≤ n ≤ q − 1. By [18, Theorem 10], the asymptotic cost for constructing an irreducible polynomial of degree n over the ﬁnite ﬁeld Fq is O (n2 ln n + n ln q) L(n) . Polynomial Arithmetic. The number of operations required to evaluate a polynomial of degree n at a given point using Horner’s rule is O(n). By [17] and [3], the number of operations needed to multiply two polynomials of degree at most n is O(M(n)). It follows that the number of operations needed to compute the mth power of a polynomial of degree n is O nm ln2 (nm) ⊆ O (nm)1+ . By [9, Ch 14, §14.4 and §14.5], the expected number of operations in Fq needed to factorize a polynomial of degree n over Fq is O(F(n, q)) ⊆ O(n2+ ln q) . Let ϕ(X) be a monic polynomial in Zp [X] of degree nϕ , let f (X) be a polynomial in Zp [X] of degree n, and let kϕ = n/nϕ . Let E(f, kϕ ) denote the number of operations in Zp needed to compute the ϕ-adic expansion kϕ f (X) = i=1 ai (X) ϕi (X) . From [9, Ch 5, §5.11], we have E(f, kϕ ) ∈ O(kϕ (kϕ + 1)n2ϕ ) = O(n2ϕ kϕ2 ) = O(n2 ) .

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Complexity of the Algorithm

Finite Fields. For r ≥ 0 the ﬁnite ﬁeld Fqr+1 is implemented as Fp [ρr ], with • ρr of a root of ψr∗ , • ψr∗ (Y ) an arbitrary irreducible monic polynomial in Fp [Y ] of degree fr∗ , • fr∗ = f0 · · · fr .

∗

Thus Fqr+1 = Fqr [ξr ] = Fp [ξ0 , . . . , ξr ] = Fp [ρr ] and qr+1 = qrfr = pfr . Computing the Newton Polygon. It follows from [19, Theorem 15] that the recursive computation of Vr (Φ) requires O(n2+ Φ ΔΦ ) operations in Q and that this dominates the cost of constructing Nr (Φ). e

r−1 Computing ϕr . The construction of ϕr = ϕr−1 er−1 fr−1 Sect. 6 below. The cost of computing ϕr−1 is

fr−1

+ Hr−1 is explained in

er−1 fr−1 ϕr−1 = 0, Fp er−1 fr−1 ϕr−1 ∈ O (nr−1 er−1 fr−1 )1+ ΔΦ = O n1+ r ΔΦ . Q

A slight modiﬁcation of the proof of [19, Theorem 17] shows that the cost of constructing Hr−1 = Hr−1,ν r ,γr−1 is

∗ (3+) Hr−1 Fp ∈ O rfr−1 fr−2 ⊆ O(rn3+ ), r 1+ Hr−1 Q ∈ O rnr ΔΦ .

Thus the cost of computing ϕr is dominated by the cost of computing Hr−1 . Computing the Associated Polynomial. It follows from [19, Theorem 16] that if r ≥ 2 then (r) 2+ ΨΦ Fp ∈ O(nΦ n1+ r ) ⊆ O(nΦ ) ,

(r) ΨΦ

Q

2+ ∈ O nΦ n1+ r ΔΦ ⊆ O nΦ ΔΦ .

Total Complexity. The cost of phase M0 is dominated by the cost of factorizing Φ over Fp . Hence M0 Fp ∈ O(F(nΦ , p)) ⊆ O(n2+ Φ ), M0 Q ∈ O(1) .

On the Complexity of the Montes Ideal Factorization Algorithm

181

The cost of phase M1 is dominated by the cost of constructing ϕr . Hence ), M1 (r) Fp ∈ O(rn3+ r M1 (r) Q ∈ O rn1+ ΔΦ . r

The cost in Q-operations of phase M2 is dominated by the construction of the (r) Newton polygon Nr (Φ) and of the associated polynomial ΨΦ , each of which 2+ require O(nΦ ΔΦ ) operations in Q. Since Fqr+1 = Fp [ρr ], the necessity of expressing ξr and ρr−1 in terms of ρr arises. This is achieved in each case by ∗ factoring ψr−1 over Fp [ρr ], which requires O(fr∗ 3+ ) ⊆ O(n3+ Φ ) operations in Fp . These are the dominant ﬁnite-ﬁeld operations in M2 , hence M2 (r) Fp ∈ O(n3+ Φ ), M2 (r) Q ∈ O(n2+ Φ ΔΦ ) .

We now estimate the number of operations required for the chain of computations M0 (Φ) → M1 (1) → M2 (1) → M1 (2) → M2 (2) → · · · → M1 (m) → M2 (m) with the algorithm terminating at level m. We note that at level r we have n0 < n1 < · · · < nr with n0 | n1 | · · · | nr . Hence 2r ≤ nr and thus r ∈ O(ln nr ). It follows that m ∈ O(ln nΦ ) and we have m M0 (F ) Fp + r=1 M1 (r) Fp + M2 (r) Fp m m = M0 (F ) Fp + r=1 M1 (r) Fp + r=1 M2 (r) Fp ∈ O n2+ + m2 n3+ + mn3+ Φ Φ Φ ⊆ O n3+ , Φ m M0 (F ) Q + r=1 M1 (r) Q + M2 (r) Q m m = M0 (F ) Q + r=1 M1 (r) Q + r=1 M2 (r) Q 2+ ∈ O nΦ + m2 n1+ Φ ΔΦ + m nΦ ΔΦ ⊆ O n2+ Φ ΔΦ .

From [16, Proposition 4.1] it follows that the case er−1 fr−1 = 1 can occur at most e∗ 2 r−2 vp (disc Φ) ≤ 2 vp (disc Φ) nΦ times. Hence the sequence M1 (r) → M2 (r − 1) → M1 (r)

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can occur at most 2vp (disc Φ) times in the course of the computation. From the results above we have 3+ M1 (r) Fp + M2 (r − 1) Fp ∈ O(rn3+ + n3+ r Φ ) ⊆ O(nΦ ) , 2+ M1 (r) Q + M2 (r − 1) Q ∈ O(rn1+ + n2+ r Φ ΔΦ ) ⊆ O(nΦ ΔΦ ) . ∗ Since δΦ ≤ δΦ and ln p ∈ O(1) we have 1+ ∗ ln p) ∈ O(δΦ ). ΔΦ = M(δΦ

It now follows that the expected number of operations required for the restricted Montes algorithm to terminate is 3+ 2+ 2+ . + n2+ O 2δΦ (n3+ Φ Φ ΔΦ ) ⊆ O nΦ δΦ + nΦ δΦ 2+ Remark 5. This is a slight improvement on the estimate O(n3+ Φ δΦ ) from [19]. By way of comparison, Pauli [16] gives an estimate of 1+ 2+ O n3+ + n2+ Φ δΦ Φ δΦ

bit operations for factorization of a univariate polynomial over Qp via the “twoelement” method.

6

The Construction of ϕr

Algorithm 1 (Montes). Given ds , es , fs , etc., for 1 ≤ s ≤ r and given • an integer t in the range 1 ≤ t ≤ r, • an integer ν ≥ ν t+1 , • a nonzero polynomial δ(Y ) ∈ Fqt [Y ] of degree less than ft , to construct a polynomial Ht,ν,δ (X) ∈ Zp [X] such that • deg Ht,ν,δ < nt+1 , • Vt+1 (Ht,ν,δ ) = ν, (t)

• ΨTt,ν , Ht,ν,δ(Y ) = δ(Y ). Construction. Let ζ0 , . . . , ζft −1 in Fqt be such that ft −1 δ(Y ) = i=0 ζi Y i . Since δ(Y ) = 0 the set Jδ = { i | 0 ≤ i ≤ ft − 1, ζi = 0 } is not empty. For i ∈ Jδ we construct Ki (X) as follows. • We take δi (Y ) to be the unique polynomial in Fqt−1 [Y ] of degree less than ft−1 such that δi (ξt−1 ) = ΓTt,ν ,t,i ζi .

On the Complexity of the Montes Ideal Factorization Algorithm

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• If t = 1 we take Pi (X) to be a polynomial in Zp [X] of degree less than f0 such that P i (Y ) = δi (Y ) and we set Ki (X) = pβ1,ν −id1 Pi (X) . • If t ≥ 2 we let νi = (βt,ν − idt ) − (αt,ν + iet )ν t and we set Ki (X) = Ht−1, νi , δi(X) . Having constructed Ki (X) for i ∈ Jδ , we set Ht,ν,δ (X) = i∈Jδ Ki (X) ϕt (X)αt,ν +iet .

Remark 6. It follows from [13, Proposition 3.2] that Algorithm 1 correctly constructs the polynomial Ht,ν,δ with the indicated properties. The construction of δi (Y ) in Algorithm 1 being rather complicated, we provide some implementation details. f ∗ ×fr ×f ∗

r−1 Computing Υr . If r > 0 we construct Υr ∈ Fp r such that ∗ fr −1 ρkr−1 ξrj = h=0 (Υr )h,j,k ρhr

f ∗ ×fr∗

∗ for j = 0, . . . , fr − 1, k = 0, . . . , fr−1 − 1. In practice we construct Υr ∈ Fp r ∗ ∈ Fpfr such that and M

(Υr )1+h,1+j+kfr = (Υr )h,j,k ,

1+j+kfr = Mj,k , M

∗ for h = 0, . . . , fr∗ − 1, j = 0, . . . , fr − 1, k = 0, . . . , fr−1 − 1.

Deriving δi from Υt−1 . Given i ∈ Jδ and t ≥ 2, let f∗

t−1 ∗ ΓTt,ν ,t,i ζi = κi,0 + κi,1 ρt−1 + · · · + κi,ft−1 −1 ρt−1

∗ ft−2

For j = 0, . . . , ft−1 − 1, k = 0, . . . , ∗ ft−1 −1 ft−2 −1 j=0

k=0

−1

∈ Fp [ρt−1 ] = Fqt .

− 1, let Mj,k ∈ Fp satisfy

(Υt−1 )h,j,k Mj,k = κi,h

∗ for h = 0, . . . , ft−1 − 1, and let ∗ ft−1 −1 ft−2 −1 Mj,k ρkt−2 Y j . δi (Y ) = j=0 k=0

Then δi (Y ) ∈ Fp [ρt−2 ][Y ] = Fqt−1 [Y ] and ∗ ft−1 −1 ft−2 −1 j Mj,k ρkt−2 ξt−1 δi (ξt−1 ) = j=0 k=0 = = =

∗ ft−1 −1 ft−2 −1

j=0

k=0

Mj,k

∗ ft−1 −1

h=0

∗ ∗ ft−1 −1 ft−1 −1 ft−2 −1

j=0

h=0

∗ ft−1 −1

h=0

k=0

(Υt−1 )h,j,k ρht−1

(Υt−1 )h,j,k Mj,k ρht−1

κi,h ρht−1

= ΓTt,ν ,t,i ζi . The essential properties of ϕr are as follows (see [19, Proposition 9]).

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Proposition 1 (Montes). Let ds , es , fs , ϕs , ψs , etc., be given for 1 ≤ s ≤ r−1 and let −e

γr−1 (Y ) = Ωr−1r−1

fr−1

(ψr−1 (Y ) − Y fr−1 ),

ϕr (X) = ϕr−1 (X)er−1 fr−1 + Hr−1,ν r ,γr−1 (X) . Then ϕr (X) is a monic polynomial in Zp [X] with the following properties. • deg ϕr = nr . • Nr−1 (ϕr ) consists of the single segment Sr−1,ϕr . • Vr (ϕr ) = ν r . (r−1) −e f • Ψϕr (Y ) = Ω r−1 r−1 ψr−1 (Y ). r−1

• ϕr is irreducible over Zp .

7

Supplementary Remarks

The Maple code from [19], including an example, can be found at this URL. http://www.mathstat.concordia.ca/faculty/ford/Student/Veres/mmtest.mpl

Two recent monographs by Gu`ardia, Montes, and Nart give a thorough revision of the theory underlying the Montes algorithm [10] and a detailed description of the algorithm [11]. Algorithm 1 and Proposition 1 in Sect. 6 above appear in [10]. A simpler choice for Ωr (see Deﬁnition 5) is also given, but with no eﬀect on the complexity of the algorithm.

References 1. Berlekamp, E.R.: Factoring Polynomials over Finite Fields. Bell Systems Technical Journal 46, 1853–1859 (1967) 2. Berlekamp, E.R.: Factoring Polynomials over Large Finite Fields. Math. Comp. 24, 713–735 (1970) 3. Cantor, D.G., Kaltofen, E.: On Fast Multiplication of Polynomials over Arbitrary Algebras. Acta Informatica 28(7), 693–701 (1991) 4. Cantor, D.G., Zassenhaus, H.: A New Algorithm for Factoring Polynomials Over Finite Fields. Math. Comp. 36, 587–592 (1981) 5. Dedekind, R.: Supplement X to Vorlesungen u ¨ber Zahlentheorie von P.G. Lejeune Dirichlet (2nd ed.). Vieweg, Braunschweig (1871); Also Werke 3, 223–261 (1932) (in part) 6. Dedekind, R.: Sur la th´eorie des nombres entiers alg´ebriques. Gauthier-Villars (1877); Also Bull. des Sci. Math. Astron. 11(1), 278–288 (1876); 1(2), 17–41, 69–92, 144–164, 207–248 (1877) and Werke 3, 263–296 (1932) (in part) ¨ 7. Dedekind, R.: Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der h¨ oheren Kongruenzen. Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen 23, 1–23 (1878)

On the Complexity of the Montes Ideal Factorization Algorithm

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8. Ford, D., Pauli, S., Roblot, X.-F.: A Fast Algorithm for Polynomial Factorization over Qp . Journal de Th´eorie des Nombres de Bordeaux 14, 151–169 (2002) 9. von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999) 10. Gu` ardia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory (2008), arXiv:0807.2620v2[math.NT] 11. Gu` ardia, J., Montes, J., Nart, E.: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields (2008), arXiv:0807.4065v3[math.NT] 12. Hensel, K.: Theorie der algebraischen Zahlen. Teubner, Leipzig (1908) 13. Montes, J.: Pol´ıgonos de Newton de orden superior y aplicaciones aritm´eticas. PhD thesis, Universitat de Barcelona (1999) 14. Montes, J., Nart, E.: On a theorem of Ore. Journal of Algebra 146, 318–334 (1992) 15. Ore, Ø.: Newtonsche Polygone in der Theorie der algebraischen K¨ orper. Math. Ann. 99 (1928) 16. Pauli, S.: Factoring Polynomials over Local Fields. Journal of Symbolic Computation 32(5), 533–547 (2001) 17. Sch¨ onhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971) 18. Shoup, V.: Fast Construction of Irreducible Polynomials over Finite Fields. Journal of Symbolic Computation 17, 371–394 (1994) 19. Veres, O.: On the Complexity of Polynomial Factorization over p-adic Fields. PhD Dissertation, Concordia University (2009), http://www.mathstat.concordia.ca/faculty/ford/Student/Veres/vthp.pdf 20. Zassenhaus, H.: On Hensel factorization II. In: Symposia Mathematica XV, Instituto Di Alta Matematica, pp. 499–513. Academic Press, New York (1975)

Congruent Number Theta Coeﬃcients to 1012 William B. Hart1, , Gonzalo Tornar´ıa2, and Mark Watkins3, 1 2 3

Mathematics Institute, Warwick University, Coventry, United Kingdom Centro de Matem´ atica, Universidad de la Rep´ ublica, Montevideo, Uruguay Department of Mathematics and Statistics, University of Sydney, Australia

Abstract. We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 1012 . The computation involves multiplying long theta series as per Tunnell (1983). The method, which we describe in some detail, uses a multimodular disk based technique for multiplying polynomials out-ofcore which minimises expensive disk access by keeping data truncated.

1

History

The congruent number problem ﬁrst makes its appearance in the literature of the classical Islamic period, e.g. in al-Karaji’s text the al-Fakhri. Dickson [11] states that an anonymous Arab manuscript written before 972 A.D. contains reference to the problem. The problem was initially studied in terms of squares of rational numbers: a natural number n is congruent iﬀ there exist rational numbers x, y, z, w such that x2 + ny 2 = z 2 and x2 − ny 2 = w2 . In other words n is congruent iﬀ there exist three rational squares in arithmetic progression with common diﬀerence n. It suﬃces to consider squarefree n. Bachet, in translating Diophantus’ Arithmetica, wrote an appendix of problems on right triangles. Problem 20 was “to ﬁnd a right-angled triangle such that its area is equal to a given number”. This equivalent problem refers to right triangles with rational sides whose area n is a natural number. The problem was studied by Fermat and Fibonacci the latter of which referred to a common diﬀerence of squares in arithmetic progression as a congruum. Euler referred to such numbers as congruere meaning to “come together”. Many authors have contributed to the study of the properties of and computation of congruent numbers, including Alter, Curtz and Kubota [1] who conjectured that if n is congruent to 5, 6 or 7 modulo 8 then n is a congruent number. This was shown to be true, subject to the weak Birch and Swinnerton-Dyer conjecture by Stephens [35] in 1975.

Supported by EPSRC grant number EP/G004870/1. All authors were supported at workshops administered by AIMath under NSF Grant number DMS-0757627.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 186–200, 2010. c Springer-Verlag Berlin Heidelberg 2010

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The earliest computations of congruent numbers are due to the classical Islamic mathematicians, the congruent numbers 5, 6, 14, 15, 21, 30, 34, 65, 70, 110, 154, 190, 210, 221, 231, 246, 290, 390, 429, 546 and ten other substantially larger congruent numbers being known to them. Fibonacci, Genocchi and G´erardin added 7, 22, 41, 69, 77 and forty-three other values below 1000. Fermat showed that 1 is not congruent in 1659, something which had been stated but not proved by Fibonacci in 1225. By scaling this is equivalent to the fact that no square number can be congruent. Bastien [5] observed that numbers which are prime and 3 modulo 8, products of two such primes, twice a prime which is 5 modulo 8, twice a product of two such primes or twice a prime which is 9 modulo 16 are not congruent. Numerous congruent numbers were demonstrated by Alter, Curtz and Kubota [1] and by Jean Lagrange in his thesis [23]. See Guy [17] for further details on the history of the computation of congruent numbers. More recently Monsky [28] showed that, for example, two times the product of primes p ≡ 1 (mod 8) and q ≡ 7 (mod 8) with (p/q) = −1 is a congruent number. For a history of results along these lines see Feng [13]. Also see [27]. By 1980 there were numerous values below 1000 not yet decided either way. By 1986 Kramarz [26] had handled all cases up to 2000, and Noe’s list up to 10000 is included in Sloane’s database. Matsuno had reached 300000 in 2005. Subject to a conjecture of Birch and Swinnerton-Dyer (see Tunnell’s Criterion below), Rogers [32] had computed all congruent numbers up to 107 by the year 2000 and Mike Rubinstein (personal communication) had computed all congruent numbers up to 109 a few years prior to the current work. We had raised that limit to 2 × 1010 by 2008 and with this paper the current plateau is now 1012 . By counting representations of n or n/2 by ternary quadratic forms, previous 3 computations had the asymptotic running time O(N 2 ) for computing coeﬃcients up to a limit N . In this paper we describe a multimodular Fast Fourier Transform technique with quasilinear runtime. We demonstrate that the method is practical as it permits computations whose data is considerably larger than main memory.

2

Relating Congruent Numbers to Elliptic Curves

If three rational squares in arithmetic progression have common diﬀerence n, their product is a square: v 2 = (u2 − n)u2 (u2 + n) = (u2 )3 − n2 (u2 ). This shows immediately that if n is congruent then it corresponds to a point (u2 , v) on the elliptic curve En : y 2 = x3 − n2 x. Along similar lines, in 1877 Lucas showed that n is congruent iﬀ y 2 = x4 − n2 has a positive rational solution. The group of points on the curve En is isomorphic to (Z/2Z × Z/2Z) × Zr where r is the rank. The three non-trivial 2-torsion points do not yield congruent numbers and so n is congruent iﬀ En has positive rank.

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There has been considerable interest in verifying that the curves En for which n is thought to be congruent do in fact have positive rank. See for example the tables of Elkies [12]. As the sign of the functional equation of L(En /Q, s) is +1 for n ≡ 1, 2, 3 (mod 8) and −1 for n ≡ 5, 6, 7 (mod 8) [7] then by the Parity Conjecture (a special case of the Birch and Swinnerton-Dyer Conjecture) we expect that the rank of En is even in the +1 case and odd in the −1 case. This is an interesting test of the Birch and Swinnerton-Dyer Conjecture. 2.1

Tunnell’s Criterion

In 1983 Jerrold Tunnell gave the following criterion: Theorem 1 (Tunnell). Let n be an odd squarefree positive integer. Set a(n) = #{(x, y, z) ∈ Z3 | x2 + 2y 2 + 8z 2 = n} − 2 #{(x, y, z) ∈ Z3 | x2 + 2y 2 + 32z 2 = n}, b(n) = #{(x, y, z) ∈ Z3 | x2 + 4y 2 + 8z 2 = n} − 2 #{(x, y, z) ∈ Z3 | x2 + 4y 2 + 32z 2 = n}. If n is congruent then a(n) = 0. If 2n is congruent then b(n) = 0. Moreover, if the weak BSD conjecture is true for the curve y 2 = x3 − n2 x then the converses also hold: a(n) = 0 implies n is congruent and b(n) = 0 implies 2n is congruent. We explain brieﬂy the connection between the curves En and Tunnell’s criterion. The curve En is a quadratic twist of the curve E : y 2 = x3 − x. Associated to E is a weight 2 newform F (z) = η(4z)2 η(8z)2 ∈ S2new (Γ0 (32)) such that L(E, s) = L(F, s), where L(E, s) is the Hasse-Weil L-series of the elliptic curve E and L(F, s) is the Mellin transform of the modular form F . If we write L(E, s) = bm m−s then L(En , s) = LF (χD , s) = χD (m)bm m−s , where D = n if n ≡ 1 (mod 4) and D = 4n if n ≡ 2, 3 (mod 4). The importance of this fact is that the conjecture of Birch and SwinnertonDyer (applied to En ) then gives a condition on when n can be congruent: Conjecture 1 (Birch and Swinnerton-Dyer). If E is an elliptic curve deﬁned over Q then L(E, 1) = 0 iﬀ E has positive rank. The following theorem of Shimura gives a link between modular forms of half integer weight k/2 and forms of integer weight k − 1. The correspondence is called a Shimura lift. We are interested in this theorem in the case k = 3. m Theorem 2 (Shimura). Let f (z) = ∞ ∈ Sk/2 (4N, χ) be a modum=1 a(m)q lar form of weight k/2 for Γ0 (4N ) (actually Δ0 (4N )) with χ a Dirichlet character modulo 4N and suppose that Tp2 (f ) = ωp f for all primes p, where Tp2 are the

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m Hecke operators. Deﬁne F (z) = ∞ where the values A(m) are given m=1 A(m)q by ∞ A(m)m−s = (1 − ω p ps + χ(p)2 pk−2−2s )−1 . p

m=1

Then for some integer N0 divisible by the conductor of χ2 we have that F (z) ∈ Mk−1 (N0 , χ2 ), i.e. F (z) is an integer weight modular form of weight k − 1. As mentioned above, we are interested in whether or not the L-series L(En , s) vanishes at s = 1. Tunnell made use of a result of Waldspurger to access information about the value of these L-series at s = 1. The basic idea behind Waldspurger’s Theorem and related results is that if F (z) is the Shimura lift of f (z) as per the previous theorem, then the value of L(Fn , s) at s = (k − 1)/2 for squarefree n, is proportional to the n-th Fourier coeﬃcient of f (z). In particular if suitable forms f (z) can be identiﬁed then it is possible to determine when L(Fn , s) vanishes at the centre of the critical strip, s = (k − 1)/2. The following result (which is a reformulation of the theorem of Waldspurger, see [30]) formulates this more precisely. Theorem 3 (Waldspurger). If F (z) = ∞ a(m)q m ∈ S new (Γ (M )) and k−1

m=1

0

δ = ±1 is the sign of the functional equation of L(F, s) then there is a Dirichlet character χ modulo 4N , a positive integer M |N , a nonzero complex number ΩF and a nonzero Hecke eigenform ∞

f (z) =

bF (m)q m ∈ Sk/2 (Γ0 (4N ), χ)

m=1

such that there are fundamental disciminants n, coprime to 4N and with the same sign as δ that lie in arithmetic progressions and for which bF (n0 )2 = εn ·

k/2

L(Fn , (k − 1)/2)n0 ΩF

,

where εn is algebraic and n0 = |n| if n is odd, otherwise n0 = |n|/4. For all other n with the same sign as δ the Fourier coeﬃcients bF (n0 ) vanish. By careful examination of the conditions of Waldspurger’s Theorem, Tunnell was able to construct modular forms which allowed for identiﬁcation of the values of n for which L(En , s) vanishes at s = 1. Even better yet, he was able to write these weight 3/2 modular forms as the product of explicit theta series. ∞ 2 Following Tunnel we let g = (θ1 − θ4 )(θ8 − 2θ32 ), where θt = m=−∞ q tm . Then ∞ g θ2 = a(m)q m ∈ S 32 (Γ0 (128)), g θ4 =

m=1 ∞ m=1

b(m)q m ∈ S 32 (Γ0 (128), χ),

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where χ(r) = 8r . Note this agrees with the formulas for a(n) and b(n) given above for odd n. Tunnell proved that these were Hecke eigenforms whose Shimura lift was F (z). He then showed that if n is an odd positive squarefree integer then Ω L(En , 1) = a(n)2 · √ , 4 n

Ω and L(E2n , 1) = b(n)2 · √ , 2 2n

for a certain real period Ω. For further information on Tunnell’s approach, see Tunnell’s original paper [39] and the books by Ono [30] and Koblitz [25]. The above result of Tunnell allows us to determine congruent numbers, subject to the BSD conjecture, simply by checking whether the Fourier coeﬃcients a(n) and b(n) are zero. Thus the entire problem of determining congruent numbers is reduced to computing the theta series g and θt and performing power series multiplications. We actually use slight modiﬁcations of these θ-functions, which allow us to exploit additional information on arithmetic progressions. 2.2

Our Θ-Functions

Rather than use the modular forms of Tunnell given above, we note (as suggested to us by N. D. Elkies) that we can split the problem(s) up by a factor of two. The series g θ2 and g θ4 can each be split into a sum of two similar products, each of which is supported on (approximately) half as many coeﬃcients. Indeed, we have the following product expressions: θ8 (θ1 − θ4 ) × (θ8 − 2θ32 ) = a(n) q n , n≡1

(θ2 − θ8 )(θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡3

θ16 (θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡1

(θ4 − θ16 )(θ1 − θ4 ) × (θ8 − 2θ32 ) = n≡5

(mod 8)

a(n) q n ,

(mod 8)

b(n) q n ,

(mod 8)

b(n) q n .

(mod 8)

As each factor above is a (shifted) power series in q 8 , our complexity reduces by a factor of 8. Indeed, the second factor above is θ8 − 2θ32 = C(q 8 ) where C = θ1 − 2θ4 is a sparse power series which can be quickly computed. For the ﬁrst factor, we can easily compute theta series A1 , A3 , B1 and B5 such that θ8 (θ1 − θ4 ) = q A1 (q 8 ),

(θ2 − θ8 )(θ1 − θ4 ) = q 3 A3 (q 8 ),

θ16 (θ1 − θ4 ) = q B1 (q 8 ),

(θ4 − θ16 )(θ1 − θ4 ) = q 5 B5 (q 8 ).

These series can be computed directly by counting lattice points in 2 dimensions, taking approximately linear time. So we only need one convolution for each of

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the four cases: two convolutions of 1.25 × 1011 coeﬃcients (for the a(n),) and two convolutions of 6.25 × 1010 coeﬃcients (for the b(n).) The computation of the Θ-series can be done eﬃciently in intervals, √ √ taking essentially N time to compute the coeﬃcients between N and N + N . For N up to 1.25×1011 this ensures each interval includes less than 500, 000 coeﬃcients, ﬁtting comfortably in a typical L2 cache. This cache locality is essential for the computation.

3

“Out-of-Core” Fast Fourier Transform Methods

The complex FFT algorithm was essentially known to Gauss in 1805 (see [19]) but developed in its current form by Cooley and Tukey in 1965 [9]. In 1971 Sch¨ onhage and Strassen presented two algorithms for multiplication of large integers based on the FFT [33]. One of these methods, where the ﬁeld of complex numbers is replaced by a ﬁnite ring Z/pZ containing a principal root of unity of order 2K , has become known as the Sch¨onhage-Strassen method. It can multiply two n bit numbers in asymptotic time O(n logn log logn). Power series multiplication can be eﬀected by truncating a full polynomial multiplication of two n term polynomials to length n and by encoding the polynomial multiplication as an integer multiplication using Kronecker Segmentation. The latter technique is that of evaluating the polynomials at a power of 2 chosen suﬃciently large that the product coeﬃcients can be identiﬁed from their binary representation in the output of the large integer multiplication. In the literature, FFT computations whose data exceeds the size of available memory are referred to as out-of-core FFT methods. The literature is replete with many references to methods for defunct vector architectures, or for distributed memory systems, including those with tree, mesh or hypercube architectures (see [2], [8], [24], [36] and [38] for examples), where the emphasis is often on minimising interprocess communication. In our case, we used a shared memory system where available memory was a limiting factor for the computation, forcing an “out-of-core” computation. The principal issue with standard FFT algorithms in a hierarchical memory system (e.g. where disk is one level of the hierarchy) is that at least K complete passes over the data are required for a convolution of length 2K . However disk access is typically a couple of orders of magnitude slower than memory access, making such algorithms prohibitively slow. The ﬁrst FFT technique to deal with a memory hierarchy is that of Gentleman and Sande [20]. The method has become known as Bailey’s Four Step method (in the context of complex FFT’s), see [3]. The idea is to break the data into a two dimensional array and perform small FFT’s in the horizontal and then in the vertical directions, with certain “twiddle factors” applied between the two stages. A ﬁnal transpose stage then follows. This basic strategy is also sometimes referred to as the Matrix Fourier Algorithm. Bailey’s method can be extended to a six (or ﬁve) step three dimensional method and beyond. See the above cited paper of Bailey’s for older references,

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or [31] for a more recent reference. For applications to integer multiplication, see for example [21]. Some other algorithms for out-of-core FFT’s include the algorithm of Cormen [10] based on the in-core method of Swarztrauber, the method of Takahashi [37] for the Parallel Disk Model (PDM) of Vitter and Shriver and the parallel FFT method of Vitter and Shriver [40] for a two level memory system. Another technique commonly used for out-of-core FFT computations is the method of performing Number Theoretic Transforms (NTTs) with Chinese Remainder Theorem reconstitution. A Number Theoretic Transform is an FFT in the ring R = Z/pZ for a specially chosen small prime p sometimes called an “FFT prime”. Usually p is chosen to ﬁt into a single machine word, i.e. 32 or 64 bits. For this to work, R must have suﬃciently many roots of unity to support the convolution. FFT primes p can be chosen to be of the form p = m2K + 1 for some small value m. Let x be a primitive root modulo p, i.e. a value x such that xp−1 ≡ 1 (mod p), but such that xa is not 1 (mod p) for any value of a dividing p − 1. Then xm is a 2K -th root of unity, supporting convolutions of length 2K . In order to perform an out-of-core polynomial multiplication h(x) = fA (x) × gB (x) using NTTs the coeﬃcients of the two polynomials are ﬁrst reduced modulo a number of FFT primes. Then the Chinese Remainder Algorithm can be used to reconstitute the full product from the results of the NTTs. The NTT transform method is a standard one for computing large numbers of digits of π. See for example the paper of Bailey, [4] where two FFT primes were used, in that case to avoid the necessity of quad-precision arithmetic in a complex FFT. The same paper also mentions a proposal to use three FFT primes, even avoiding double precision arithmetic in the NTT’s, but imposing severe restriction on the length of convolution possible for machines of that era. More recently Carey Bloodworth’s record-holding programs used eight NTTs and CRT, and were topped in 2004 by the program of Xavier Gourdon [16] for greatest number of digits of π computed on a home computer. Gourdon’s program uses an unspeciﬁed number of NTTs. More recent than our theta computation is the record π computation of Fabrice Bellard [6], using NTTs and a home computer. For out-of-core operations, his computation made use of eight 64 bit moduli, however for in-core components he made use of ﬂoating point arithmetic and unproven, heuristically chosen error bounds on the precision required.

4

The Power Series Multiplication

For any method using FFTs, optimised for out-of-core operation, the main bottleneck becomes disk I/O. To minimise this, it is not only important to minimise the number of passes over the data, but also to minimise the amount of data that must be traversed.

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Two issues arise. Firstly, techniques such as the Sch¨ onhage-Strassen technique are diﬃcult to optimise for convolution lengths which are not a power of two, in the worst case increasing the disk I/O by a factor of two. Secondly, when performing a large FFT or a small number of very large NTTs that do not ﬁt into memory, even when combined with Bailey’s technique, truncation of the polynomial multiplications occurs after each large FFT computation. In other words, the disk I/O occurs for the entire untruncated FFT computation. For multiplication of integers of n bits, these methods require a total disk I/O of 12n bits with a peak usage of 8n bits. Our technique reduces this to a total disk I/O of just over 6n bits with a peak usage of just over 4n bits. This is achieved by eﬃcient multimodular reduction and CRT recombination using a large number of small primes p with truncation occurring in-core. One advantage of using NTTs is that the primes p can be chosen in such a way that reduction modulo p can be performed very eﬃciently. E.g. for primes p of the form 2K + 1 reduction modulo p can be performed with subtractions rather than expensive divisions. More generally, many primes of the form p = m2K + 1 for small values of m can be used. Reduction modulo p can still be computed relatively eﬃciently. For our computation we chose to use many general word sized primes p and an alternative method of performing polynomial multiplications over Z/pZ. For the largest polynomial multiplications, in the 1 (mod 8) and 3 (mod 8) cases, we used just over 500 primes. The main reason for this choice was the existence of well-tested, high performance packages for doing such computations, such as FLINT [18] and zn poly [22]. There was also an advantage in having two separate implementations of arithmetic in Z/pZ[x] in that comparisons could be made between the two implementations whilst testing. The implementation of multiplication in Z/pZ[x] in zn poly is highly optimised. It oﬀers a thread-safe, cache-eﬃcient, truncated, Sch¨onhageNussbaumer convolution [21], which performs signiﬁcantly better than other implementations for general primes p. In contrast, Victor Shoup’s NTL package [34] was the only library we were aware of with asymptotically fast NTTs. However NTL is not threadsafe. Also, numerous recent improvements in polynomial arithmetic are not reﬂected in NTL, which is no longer under active development. Our implementation made use of 16 CPU cores. The data for all 16 threads must be in memory simultaneously, and thus to beneﬁt from the disk-to-memory ratio of the multimodular approach it was necessary to use a number of primes signiﬁcantly larger than this. One disadvantage of using so many primes is that multimodular reduction and CRT reconstruction constitute a signiﬁcant part of the runtime. The naive approach is to reduce the large coeﬃcients of the polynomials in Z[x] modulo each of the primes p in turn and to similarly reconstruct each coeﬃcient one prime at a time. However for n1 coeﬃcients in Z of n2 bits, reconstruction using this approach will take time O(n1 n22 ). This is asymptotically much worse than the time required to do the actual polynomial multiplications over Z/pZ.

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In order to avoid this, a divide-and-conquer approach was used for the multimodular reduction and recombination phases. This completes the CRT recombination in time O(n2 log2 n1 n2 ) ignoring smaller log log factors. Note that this is asymptotically a log factor greater than the time for the multiplications, however the running time is still quasilinear in the input size. The extra theoretical complexity of our approach is oﬀset by the “embarrasingly parallel” nature of the multiplications, multimodular reduction and recombinations and the large saving in disk I/O (by far the bottleneck for our computation). For a straightforward description of the divide-and-conquer approach to the CRT algorithm see [41], pages 57–58. Similar preconditioning and a divide-andconquer approach was of course applied to the multimodular reduction phase. A slight adjustment was also made to both the reduction and CRT phases to cope with a number of primes which is not a power of 2. 4.1

The Algorithm in Pseudocode

We now describe our algorithm in full. We make use of two sets of disk ﬁles, F = {Fi : i = 0, 1, .., F ILES − 1} and G = {Gj : j = 0, 1, .., F ILES − 1} . In our implementation we used FILES = 500 for the 1 (mod 8) and 3 (mod 8) computations and half that in the 2 (mod 16) and 10 (mod 16) computations. We also set: LIMIT (the length of the theta functions), BLOCK (number of theta coeﬃcients computed at a time), BUNDLE (number of theta coeﬃcients bundled, using Kronecker Segmentation, into each large polynomial coeﬃcient) and THREADS (number of threads used), PRIMES (number of primes used in multimodular reduction and CRT). We experimented with various values for BUNDLE from 500 to 1000. To simplify the computation, PRIMES was rounded up to a multiple THREADS. The value LIMIT, (1012 /8 in the 1, 3 (mod 8) cases and 1012 /16 in the 2, 10 (mod 16) cases), was chosen to be a multiple of FILES×BUNDLE, and a multiple of FILES×BLOCK. Coeﬃcients of the product of our θ-series comfortably ﬁt into 16 signed bits. Thus the Kronecker Segmentation phase used zero-padded ﬁelds of 16 bits. Throughout the following we write FOR i = 0 to A and similar expressions, by which we mean i in 0 0, each vertex in Vi has exactly one edge leading to a vertex in Vi−1 , and every edge not on the crater is of this form. (b) For i < h, each vertex in Vi has degree + 1. We call the level Vh the ﬂoor of the volcano. Vertices lying on the ﬂoor have degree 1. The following proposition [23] follows essentially from [14, Prop. 23]. Proposition 1. Let p be a prime number, q = pr , and dπ = t2 − 4q. Take = p another prime number. Let G be the undirected graph with vertex set Ellt (Fq ) and edges -isogenies deﬁned over Fq . We denote by h the largest power of dividing the conductor of dπ . Then the connected components of G that do not contain curves with j-invariant 0 or 1728 are -volcanoes of height h and for each component V , we have : (a) The elliptic curve whose j-invariants lie in V0 have endomorphism rings isomorphic to some Od0 ⊇ Odπ whose conductor is not divisible by . (b) The elliptic curve whose j-invariants lie in Vi have endomorphism rings isomorphic to Odi , where di = 2i d0 . Elliptic curves are determined by their j-invariant, up to a twist1 . Throughout the paper, we refer to a vertex in a volcano by giving the curve or its j-invariant. Exploring the volcano. Given a curve E on an -volcano, two methods are known to ﬁnd its neighbours. The ﬁrst method relies on the use of modular polynomials. The -th modular polynomial, denoted by Φ (X, Y ) is a polynomial with integer coeﬃcients. It satisﬁes the following property: given two elliptic curves E and E with j-invariants j(E) and j(E ) in Fq , there is an -isogeny deﬁned over Fq , if and only if, #E(Fq ) = #E (Fq ) and Φ (j(E), j(E )) = 0. As a consequence, the curves related to E via an -isogeny can be found by solving Φ (X, j(E)) = 0. As stated in [20], this polynomial2 may have 0, 1, 2 or + 1 roots in Fq . In order to ﬁnd an edge on the volcano, it suﬃces to ﬁnd a root j of this polynomial. Finally, if we need the equation of the curve E with j-invariant j , we may use the formula in [20]. The second method to build -isogenous curves constructs, given a point P of order on E, the -isogeny I : E → E whose kernel G is generated by P using 1 2

For a deﬁnition of twists of elliptic curves, refer to [21]. The case where the modular polynomial does not have any root corresponds to a degenerate case of isogeny volcanoes containing a single curve and no -isogenies.

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V´elu’s classical formulae [24] in an extension ﬁeld Fqr . To use this approach, we need the explicit coordinates of points of order on E. We denote by Gi , 1 ≤ i ≤ + 1, the + 1 subgroups of order of E. In [17], Miret and al. give the degree ri of the smallest extension ﬁeld of Fq such that Gi ⊂ Fqri , 1 ≤ i ≤ + 1. This degree is related to the order of q in the group F∗ , that we denote by ord (q). Proposition 2. Let E deﬁned over Fq be an elliptic curve with k rational isogenies, > 2, and let Gi , 1 ≤ i ≤ k, be their kernels, and let ri be the minimum value for which Gi ⊂ E(Fqri ). (a) If k = 1 then r1 = ord (q) or r1 = 2ord (q). (b) If k = + 1 then either ri = ord (q) for all i, or ri = 2ord (q) for all i. (c) If k = 2 then ri | − 1 for i = 1, 2. We also need the following corollary [17]. ˜ its twist. If E/Fq Corollary 1. Let E/Fq be an elliptic curve over Fq and E ˜ has 1 or + 1 rational -isogenies, then #E(Fqord q ) or #E(Fqord q ) is a multiple of . Moreover, if there are + 1 rational isogenies, then it is a multiple of 2 .

Z n 1 Z

×

Z n 2 Z

Z n1 +1 Z

×

Z n2 −1 Z

Z n1 +n2 −1 Z

×

Z Z

Z n1 +n2 Z

Fig. 1. A regular volcano

The group structure of the elliptic curve on the volcano. Lenstra [13] relates the group structure of an elliptic curve to its endomorphism ring by proving that E(Fq ) OE /(π − 1) as OE -modules. It is thus natural to see how this structure relates to the isogeny volcano. From Lenstra’s equation, we can deduce that E(Fq ) Z/M Z × Z/N Z. We write π = a + gω, with: 1+√d K (t − g)/2 if dK ≡ 1 (mod 4) a= and ω = √ 2 t/2 dK if dK ≡ 2, 3 (mod 4) where dK is the discriminant of the quadratic imaginary ﬁeld containing OE . Note that N is maximal such that E[N ] ⊂ E(Fq ) and by [19, Lemma 1] we get that N = gcd(a − 1, g/f ). Note moreover that N |M , N |(q − 1) and M N = #E(Fq ). This implies that on a -volcano the structure of all the curves in a given level is the same.

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Let E be a curve on the isogeny volcano such that v (N ) < v (M ). As explained in [16] (in the case = 2, but the result is general), a is such that v (a − 1) ≥ min {v (g), v (#E(Fq ))/2} . Since N = gcd(a−1, g/f ) and v (N ) ≤ v (#E(Fq ))/2, it follows that v (N ) = v (g/f ). As we descend, the valuation at of the conductor f increases by 1 at each level (by proposition 1b). This implies that the -valuation of N for curves at each level decreases by 1 and is equal to 0 for curves lying on the ﬂoor. Note that if v (#E(Fq )) is even and the height h of the volcano is greater than v (#E(Fq )), the structure of the -torsion group is unaltered from the crater down to the level h − v (#E(Fq ))/2. From this level down, the structure of the -torsion groups starts changing as explained above. In the sequel, we call this level the ﬁrst stability level.3 A volcano with ﬁrst stability level equal to 0, i.e. on the crater, is called regular. Notations. Let n ≥ 0. We denote by E[n ] the n -torsion subgroup, i.e. the ¯ q ), by E[n ](Fqk ) the subgroup subgroup of points of order n on the curve E(F n of points of order deﬁned over an extension ﬁeld of Fq and by E[∞ ](Fq ) the -Sylow subgroup of E(Fq ). Given a point P ∈ E[n ](Fq ), we also need to know the degree of the smallest extension ﬁeld containing an n+1 -torsion point such that P˜ = P . The following result is taken from [7]. Proposition 3. Let E/Fq be an elliptic curve which lies on a -volcano whose height h(V ) is diﬀerent from 0. Then the height of V , the -volcano of the curve E/Fqs is h(V ) = h(V ) + v (s). From this proposition, it follows easily that if the structure of -torsion on the curve E/Fq is Z/n1 Z × Z/n2 Z, then the smallest extension in which the structure of the -torsion changes is Fq . We sketch here the proof in the case n1 = n2 = n, which is the only case in which we consider volcanoes over extension ﬁelds in this paper4 . First of all, note that E lies on a -volcano V /Fq of height at least n. We consider a curve E lying on the ﬂoor of V /Fq such that there is a descending path of isogenies between E and E . Obviously, we have E [∞ ](Fq ) Z/2n Z. By proposition 3, V /Fq has one extra down level, which means that the curve E is no longer on the ﬂoor, but on the level just above the ﬂoor. Consequently, we have that E [] ⊂ E (Fq ) and, moreover, E [∞ ](Fq ) Z/2n+Δ Z × Z/Z. By ascending on the volcano from E to E, we deduce that the structure of the -torsion of E over Fq is necessarily E[∞ ](Fq ) Z/n+Δ Z × Z/n+1 Z. Moreover, Δ ≥ 1, because if it were 0, the height of V /Fq would be n.

3 4

Miret et al. call it simply the stability level. For the proof in the general case, see [11].

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Background on Pairings

Let E be an elliptic curve deﬁned over some ﬁnite ﬁeld Fq , m a number such that m| gcd(#E(Fq ), q − 1). Let P ∈ E[m](Fq ) and Q ∈ E(Fq )/mE(Fq ). Let fm,P be the function whose divisor5 is m(P ) − m(O), where O is the point at inﬁnity of the curve E. Take R a random point in E(Fq ) such as the support of the divisor D = (Q + R) − (R) is disjoint from the support of fm,P . Then we can deﬁne the Tate pairing as follows: tm : E[m] × E(Fq )/mE(Fq ) → F∗q /(F∗q )m (P, Q) → fm,P (Q + R)/fm,P (R). The Tate pairing is a bilinear non-degenerate application, i.e. for all P ∈ E[m](Fq ) diﬀerent from O there is a Q ∈ E(Fq )/mE(Fq ) such that Tm (P, Q) = 1. The output of the pairing is only deﬁned up to a coset of (F∗q )m . However, for implementation purposes, it is useful to have a uniquely deﬁned value and to use the reduced Tate pairing, i.e. Tm (P, Q) = tm (P, Q)(q−1)/m ∈ μm , where μm denotes the group of m-th roots of unity. Pairing computation can be done in time O(log m) using Miller’s algorithm [15]. For more details and properties of pairings, the reader can refer to [9]. Note that in the recent years, in view of cryptographic applications, many implementation techniques have been developed and pairings on elliptic curves can be computed very eﬃciently6 . Suppose now that m = n , with n ≥ 1 and prime. Now let P and Q be two n -torsion points on E. We deﬁne the following symmetric pairing [12] 1

S(P, Q) = (Tn (P, Q) Tn (Q, P )) 2 .

(1)

Note that for any point P , Tn (P, P ) = S(P, P ). In the remainder of this paper, we call S(P, P ) the self-pairing of P . We focus on the case where the pairing S is non-constant. Suppose now that P and Q are two linearly independent n torsion points. Then all n -torsion points R can be expressed as R = aP + bQ. Using bilinearity and symmetry of the S-pairing, we get log(S(R, R)) = a2 log(S(P, P )) + 2ab log(S(P, Q)) + b2 log(S(Q, Q)) (mod n ), where log is a discrete logarithm function in μn . We denote by k the largest integer such that the polynomial P(a, b) = a2 log(S(P, P )) + 2ab log(S(P, Q)) + b2 log(S(Q, Q))

(2)

is identically zero modulo k and nonzero modulo k+1 . Obviously, since S is non-constant we have 0 ≤ k < n. Dividing by k , we may thus view P as a polynomial in F [a, b]. When we want to emphasize the choice of E and n , we write PE,n instead of P. 5 6

For background on divisors, see [21]. See [10] for a fast recent implementation.

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Since P is a non-zero quadratic polynomial, it has at most two homogeneous roots, which means that that from all the + 1 subgroups of E[n ]/E[n−1 ]

(Z/Z)2 , at most 2 have self-pairings in μk (see also [12]). In the remainder of this paper, we denote by NE,n the number of zeros of PE,n . Note that this number does not depend on the choice of the two generators P and Q of the n -torsion subgroup E[n ]. Moreover, we say that a n -torsion point R has degenerate self-pairing if Tn (R, R) is a k -th root of unity and that R has nondegenerate self-pairing if Tn (R, R) is a primitive k+1 -th root of unity. Also, if Tn (R, R) is a primitive n -th root of unity, we say that R has primitive selfpairing.

4

Determining Directions on the Volcano

In this section, we explain how we can distinguish between diﬀerent directions on the volcano by making use of pairings. We give some lemmas explaining the relations between pairings on two isogenous curves. Lemma 1. Suppose E/Fq is an elliptic curve and P, Q are points in E(Fq ) of ˜ ∈ E[F ¯ q ] the points such that P˜ = P and order n , n ≥ 1. Denote by P˜ , Q ˜ Q = Q. We have the following relations for the Tate pairing ˜ 2 = Tn (P, Q). ˜ ∈ E[Fq ], then Tn+1 (P˜ , Q) (a) If P˜ , Q ˜ ∈ E[Fq ]\E[Fq ], then Tn+1 (P˜ , Q) ˜ = Tn (P, Q). (b) Suppose ≥ 3. If Q Proof. a. By writing down the divisors of the functions fn+1 ,P˜ , fn ,P˜ , fn ,P , one can easily check that n fn+1 ,P˜ = (f,P˜ ) · fn ,P . We evaluate these functions at some points Q + R and R (where R is carefully chosen) and raise the equality to the power (q − 1)/n . b. Due to the equality on divisors div(fn+1 ,P ) = div(fn ,P ), we have ) q ˜ = T (F ˜ Tn+1 (P˜ , Q) (P, Q), n (F

where Tn q show that

)

is the n -Tate pairing for E deﬁned over Fq . It suﬃces then to

(F ) ˜ Tn q (P, Q)

= Tn (P, Q). We have (1+q+···+q−1 )(q−1)

(F ) ˜ = fn ,P ([Q ˜ + R] − [R]) n Tn q (P, Q) ˜ + R) + (π(Q) ˜ + R) + (π 2 (Q) ˜ + R) + . . . = fn ,P ((Q

˜ + R) − (R)) + (π −1 (Q)

(q−1) n

(3)

where R is a random point deﬁned over Fq . It is now easy to see that for ≥ 3, ˜ + . . . + π −1 (Q) ˜ = Q ˜ = Q, ˜ + π(Q) ˜ + π 2 (Q) Q

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˜ = Q ˜ + T , where T is a point of order . By applying Weil’s because π(Q) reciprocity law [21, Ex. II.2.11], it follows that the equation (3) becomes: (F ) ˜ Tn q (P, Q)

=

fn ,P (Q + R) fn ,P (R)

q−1 n

f ((P ) − (O))q−1 ,

˜ + R) + (π(Q) ˜ + R) + (π 2 (Q) ˜ + R) + ... + where f is such that div(f ) = (Q −1 ˜ (π (Q) + R) − (Q + R) − ( − 1)(R). Note that this divisor is Fq -rational, so f ((P ) − (O))q−1 = 1. This concludes the proof. Lemma 2. (a) Let φ : E → E be a separable isogeny of degree d deﬁned over Fq , P a -torsion on the curve E such that φ(P ) is a -torsion point on E , and Q a point on E. Then we have T (φ(P ), φ(Q)) = T (P, Q)d . (b) Let φ : E → E be a separable isogeny of degree deﬁned over Fq , P a -torsion point such that Ker φ = P and Q a point on the curve E. Then we have T (φ(P ), φ(Q)) = T (P, Q) . Proof. Proof omitted for lack of space. See [3, Th. IX.9.4] for (a), [11] for (b). Proposition 4. Let E be an elliptic curve deﬁned a ﬁnite ﬁeld Fq and assume that E[∞ ](Fp ) is isomorphic to Z/n1 Z × Z/n2 Z (with n1 ≥ n2 ). Suppose that there is a n2 -torsion point P such that Tn2 (P, P ) is a primitive n2 -th root of unity. Then the -isogeny whose kernel is generated by n2 −1 P is descending. Moreover, the curve E does not lie above the ﬁrst stability level of the corresponding -volcano. Proof. Let I1 : E → E1 be the isogeny whose kernel is generated by n2 −1 P and suppose this isogeny is ascending or horizontal. This means that E1 [n2 ] is deﬁned over Fq . Take Q another n2 -torsion point on E, such that E[n2 ] = P, Q and denote by Q1 = I1 (Q). One can easily check that the dual of I1 has kernel generated by n2 −1 Q1 . It follows that there is a point P1 ∈ E1 [n2 ] such that P = Iˆ1 (P1 ). By Lemma 2 this means that T (P, P ) ∈ μn2 −1 , which is false. This proves not only that the isogeny is descending, but also that the structure of the -torsion is diﬀerent at the level of E1 . Hence E cannot be above the stability level. Proposition 5. Let ≥ 3 a prime number and suppose that E/Fq is a curve which lies in a -volcano and on the ﬁrst stability level. Suppose E[∞ ](Fq )

Z/n1 Z×Z/n2 Z, n1 ≥ n2 . Then there is at least one n2 -torsion point R ∈ E(Fq ) with primitive self-pairing. Proof. Let P be a n1 -torsion point and Q be a n2 -torsion point such that {P, Q} generates E[∞ ](Fq ). I

1 E1 be a descending -isogeny and Case 1. Suppose n1 ≥ n2 ≥ 2. Let E −→ n1 +1 n2 −1 denote by P1 and Q1 the and -torsion points generating E1 [∞ ](Fp ). Moreover, without loss of generality, we may assume that I1 (P ) = P1 and I1 (Q) = Q1 . If Tn2 −1 (Q1 , Q1 ) is a primitive n2 −1 -th root of unity, Tn2 (Q, Q) is

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a primitive n2 -th root of unity by Lemma 2. If not, from the non-degeneration of the pairing, we deduce that Tn2 −1 (Q1 , P1 ) is a primitive n2 −1 -th root of unity, which means that Tn2 −1 (Q1 , P1 ) is a n2 −2 -th primitive root of unity. By applying Lemma 2, we get Tn2 (Q, P ) ∈ μn2 −1 at best. It follows that Tn2 (Q, Q) ∈ μn2 by the non-degeneracy of the pairing. Case 2. If n2 = 1, then consider the volcano deﬁned over the extension ﬁeld Fq . ˜ ∈ E(Fq ) with Q = Q. ˜ We obviously have 2 |q − 1 There is a 2 -torsion point Q and from Lemma 1, we get T2 (P˜ , P˜ ) = T (P, P ). By applying Case 1, we get that T2 (P˜ , P˜ ) is a primitive 2 -th root of unity, so T (P, P ) is a primitive -th root of unity. Two stability levels. Remember that in any irregular volcano, v (#E(Fq )) is even and the height h of the volcano is greater than v (#E(Fq )). Moreover, all curves at the top of the volcano have E[∞ ](Fq ) Z/n2 Z × Z/n2 Z with n2 = v (#E(Fq )). The existence of a primitive self-pairing of a n2 -torsion point on any curve lying on the ﬁrst stability level implies that the polynomial P is nonzero at every level from the ﬁrst stability level up to the level max(h + 1 − 2n2 , 0) (by Lemma 2). We call this level the second level of stability. On the second stability level there is at least one point of order n2 with pairing equal to a primitive -th root of unity. At every level above the second stability level all polynomials PE,n2 may be zero7 . Consider now E a curve on the second stability level and I : E → E1 an ascending isogeny. Let P be a n2 -torsion point on E and assume that Tn2 (P, P ) ∈ μ∗ . We denote by P˜ ∈ E(Fq )\E(Fq ) the point such that P˜ = P . By Lemma 1 we get Tn2 +1 (P˜ , P˜ ) is a primitive 2 -th root of unity. It follows by Lemma 2 that Tn2 (I(P ), I(P )) is a primitive -th root of unity. We deduce that PE1 ,n2 +1 corresponding to E1 /Fq is non-zero. Applying this reasoning repeatedly, we conclude that for every curve E above the second stability level there is an extension ﬁeld Fqs such that the polynomial PE,n2 +s associated to the curve deﬁned over Fqs is non-zero. When the second stability level of a volcano is 0, we say that the volcano is almost regular. We now make use of a result on the representation of ideal classes of orders in imaginary quadratic ﬁelds. This is Corollary 7.17 from [5]. Lemma 3. Let O be an order in an imaginary quadratic ﬁeld. Given a nonzero integer M , then every ideal class in Cl(O) contains a proper O-ideal whose norm is relatively prime to M . Proposition 6. We use the notations and assumptions from Proposition 1. Furthermore, we assume that for all curves Ei lying at a ﬁxed level i in V the curve structure is Z/n1 Z × Z/n2 Z, with n1 ≥ n2 . The value of NEi ,n2 , the number of zeros of the polynomial deﬁned at 2, is constant for all curves lying at level i in the volcano. Proof. Let E1 and E2 be two curves lying at level i in the volcano V . Then by Proposition 1 they both have endomorphism ring isomorphic to some order Odi . 7

In all the examples we considered for this case, P is always 0.

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Now by taking into account the fact that the action of Cl(Odi ) on Ell di (Fq ) is transitive, we consider an isogeny φ : E1 → E2 of degree 1 . By applying Lemma 3, we may assume that (1 , ) = 1. Take now P and Q two independent n2 -torsion points on E1 and denote by PE1 ,n2 the quadratic polynomial corresponding to the n2 -torsion on E1 as in (2). We use Lemma 2 to compute S(φ(P ), φ(P )), S(φ(P ), φ(Q)) and S(φ(Q), φ(Q)) and deduce that a polynomial PE2 ,n2 (a, b) on the curve E2 computed from φ(P ) and φ(Q) is such that PE1 ,n2 (a, b) = PE2 ,n2 (a, b). This means that NE1 ,n2 and NE2 ,n2 coincide, which concludes the proof. Moreover, we have showed that the value of k for two curves lying on the same level of a volcano is the same. Proposition 7. Let E be an elliptic curve deﬁned a ﬁnite ﬁeld Fq and let E[∞ ](Fq ) be isomorphic to Z/n1 Z × Z/n2 Z with ≥ 3 and n1 ≥ n2 ≥ 1. Suppose NE,n2 ∈ {1, 2} and let P be a n2 -torsion point with degenerate selfpairing. Then the -isogeny whose kernel is generated by n2 −1 P is either ascending or horizontal. Moreover, for any n2 -torsion point Q whose self-pairing is non-degenerate, the isogeny with kernel spanned by n2 −1 Q is descending. Proof. Case 1. Suppose Tn2 (P, P ) ∈ μk , k ≥ 1 and that Tn2 (Q, Q) ∈ μk+1 \μk . Denote by I1 : E → E1 the isogeny whose kernel is generated by n2 −1 P and I2 : E → E2 the isogeny whose kernel is generated by n2 −1 Q. By repeatedly applying Lemmas 1 and 2, we get the following relations for points generating the n2 −1 -torsion on E1 and E2 : Tn2 −1 (I1 (P ), I1 (P )) ∈ μk−1 , Tn2 −1 (I1 (Q), I1 (Q)) ∈ μk−2 \μk−3 Tn2 −1 (I2 (P ), I2 (P )) ∈ μk−3 , Tn2 −1 (I2 (Q), I2 (Q)) ∈ μk \μk−1 with the convention that μh = ∅ whenever h ≤ 0. From the relations above, we deduce that on the -volcano having E, E1 and E2 as vertices, E1 and E2 do not lie at the same level. Given the fact that there are at least − 1 descending rational -isogenies parting from E and that Q is any of the − 1 (or more) n2 torsion points with non-degenerate self-pairing, we conclude that I1 is horizontal or ascending and that I2 is descending. Case 2. Suppose now that k = 0. Note that the case n2 = 1 was already treated in proposition 4. Otherwise, consider the curve E deﬁned over Fq . By lemma 1 we have k = 1 for points on E/Fq , and we may apply Case 1. A special case. If E is a curve lying under the ﬁrst stability level and that E[∞ ](Fq ) Z/n1 Z × Z/n2 Z, with n1 > n2 , then it suﬃces to ﬁnd a point P1 of order n1 and the point n1 −1 P1 generates the kernel of an horizontal or ascending isogeny (P1 has degenerate self-pairing). Crater detection. Assume that P = 0. When is split in OE , there are two horizontal isogenies from E and this is equivalent, by propositions 6 and 7, to NE,n2 = 2. Similarly, when is inert in OE , there are neither ascending nor

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horizontal isogenies and NE,n2 = 0. In these two cases, we easily detect that the curve E is on the crater. Note. All statements in the proof of Case 1 are true for = 2 also. The statement in Proposition 4 is also true for = 2. The only case that is not clear is what happens when k = 0 and n2 ≥ 1. We did not ﬁnd a proof for the statement in proposition 5 for = 2, but in our computations with MAGMA we did not ﬁnd any counterexamples either. We conclude this section by presenting an algorithm which determines the group structure of the ∞ -torsion group of a curve E and also an algorithm which outputs the kernel of an horizontal (ascending) isogeny from E, when E[∞ ](Fq ) is given. Algorithm 1. Computing the structure of the ∞ -torsion of E over Fq (assuming volcano height ≥ 1) Require: A curve E deﬁned over Fq , a prime Compute: Structure Z/n1 Z × Z/n2 Z, generators P1 and P2 1: Check that q ≡ 1 (mod ) (if not need to move to extension ﬁeld: abort) 2: Let t be the trace of E(Fq ) 3: Check q + 1 − t ≡ 0 (mod ) (if not consider twist or abort) 4: Let dπ = t2 − 4q, let z be the largest integer such that z |dπ and h = z2 5: Let n be the largest integer such that n |q + 1 − t and N = q+1−t n 6: Take a random point R1 on E(Fq ), let P1 = N · R1 7: Let n1 be the smallest integer such that n1 P1 = 0 8: if n1 = n then 9: Output: Structure is nZZ , generator P1 . Exit (E is on the ﬂoor, ascending isogeny with kernel n−1 P1 ) 10: end if 11: Take a random point R2 on E(Fq ), let P2 = N · R2 and n2 = n − n1 12: Let α = logn2 P1 (n2 P2 ) (mod n1 −n2 ) 13: if α is undeﬁned then 14: Goto 6 (n2 P2 does not belong to n2 P1 ) 15: end if 16: Let P2 = P2 − αP1 17: If WeilPairing (n1 −1 P1 , n2 −1 P2 ) = 1 goto 6 (This checks linear independence) 18: Output: Structure is nZ1 Z × nZ2 Z , generators (P1 , P2 )

We assume that the height of the volcano is h ≤ 2n2 + 1, or, equivalently, that the curve E lies on or below the second stability level, which implies that the polynomial P is non-zero at every level in the volcano. This allows us to distinguish between diﬀerent directions of -isogenies parting from E. Of course, similar algorithms can be given for curves lying above the second stability level, but in this case we are compelled to consider the volcano over an extension ﬁeld Fqs . Since computing points deﬁned over extension ﬁelds of degree greater than is expensive, our complexity analysis in section 5 will show that it is more eﬃcient to use Kohel’s and Fouquet-Morain algorithms to explore the volcano until the second level of stability is reached and to use algorithms 1 and 2

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Algorithm 2. Finding the kernel of ascending or horizontal isogenies (Assuming curve not on ﬂoor and below the second stability level) Require: A curve E, its structure nZ1 Z × nZ2 Z and generators (P1 , P2 ) 1: if n1 > n2 then 2: The isogeny with kernel n1 −1 P1 is ascending or horizontal 3: To check whether there is another, continue the algorithm 4: end if 5: Let g be a primitive -th root of unity in Fq 6: Let Q1 = n1 −n2 P1 7: Let a = Tn2 (Q1 , Q1 ), b = Tn2 (Q1 , P2 ) · Tn2 (P2 , Q1 ) and c = Tn2 (P2 , P2 ) 8: If (a, b, c) = (1, 1, 1) abort (Above the second stability level) 9: repeat 10: Let a = a, b = b and c = c 11: Let a = a , b = b and c = c 12: until a = 1 and b = 1 and c = 1 13: Let La = log g (a ), Lb = logg (b ) and Lc = logg (c ) (mod ) 14: Let P(x, y) = La x2 + Lb xy + Lc y 2 (mod ) 15: If P has no roots modulo , Output: No isogeny (a single point on the crater) 16: If single root (x1 , x2 ) Output: One isogeny with kernel n2 −1 (x1 Q1 + x2 P2 ) 17: if P has two roots (x1 , x2 ) and (y1 , y2 ) then 18: Two isogenies with kernel n2 −1 (x1 Q1 + x2 P2 ) and n2 −1 (y1 Q1 + y2 P2 ) 19: end if

afterwards. We assume ≥ 3, even though in many cases these methods work also for = 2.

5

Walking the Volcano: Modified Algorithms

As mentioned in the introduction, several applications of isogeny volcanoes have recently been proposed. These applications require the ability to walk descending and ascending paths on the volcano and also to walk on the crater of the volcano. We recall that a path is a sequence of isogenies that never backtracks. We start this section with a brief description of existing algorithms for these tasks, based on methods given by Kohel [14] and by Fouquet and Morain in [8]. We present modiﬁed algorithms, which rely on the method presented in Algorithm 2 to ﬁnd ascending or horizontal isogenies. Then, we give complexity analysis for these algorithms and show that in many cases our method is competitive. Finally, we give two concrete examples in which the new algorithms can walk the crater of an isogeny volcano very eﬃciently compared to existing algorithms. A brief description of existing algorithms. Existing algorithms rely on three essential properties in isogeny volcanoes. Firstly, it is easy to detect that a curve lies on the ﬂoor of a volcano, since in that case, there is a single isogeny from this curve. Moreover, this isogeny can only be ascending (or horizontal if the height is 0). Secondly, if in an arbitrary path in a volcano there is a descending isogeny,

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then all the subsequent isogenies in the path are also descending. Thirdly, from a given curve, there is either exactly one ascending isogeny or at most two horizontal ones. As a consequence, ﬁnding a descending isogeny from any curve is easy: it suﬃces to walk three paths in parallel until one path reaches the ﬂoor. This shortest path is necessarily descending and its length gives the level of the starting curve in the volcano. To ﬁnd an ascending or horizontal isogeny, the classical algorithms try all possible isogenies until they ﬁnd one which leads to a curve either at the same level or above the starting curve. This property is tested by contructing descending paths from the all the neighbours of the initial curve and picking the curve which gave the longest path. Note that alternatively, one could walk in parallel all of the +1 paths starting from the initial curve and keep the (two) longest as horizontal or ascending. As far as we know, this has not been proposed in the literature, but this variant of existing algorithms oﬀers a slightly better asymptotic time complexity. For completeness, we give a pseudo-code description of this parallel variant of Kohel and Fouquet-Morain algorithms as Algorithm 3.

Algorithm 3. Parallel variant of ascending/horizontal step (using modular polynomials) Require: A j-invariant j0 in Fq , a prime , the modular polynomial Φ (X, Y ). 1: Let f (x) = Φ (X, j0 ) 2: Compute J0 the list of roots of f (x) in Fq 3: If #J0 = 0 Output: “Trivial volcano” Exit 4: If #J0 = 1 Output: “On the ﬂoor, step leads to:”, J0 [1] Exit 5: If #J0 = 2 Output: “On the ﬂoor, two horizontal steps to:”, J0 [1] and J0 [2] Exit 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

Let J = J0 . Let J and K be empty lists. Let Done = false. repeat Perform multipoint evaluation of Φ (X, j), for each j ∈ J. Store in list F for i from 1 to + 1 do Perform partial factorization of F [i], computing at most two roots r1 and r2 if F [i] has less than two roots then Let Done = true. Append ⊥ to K (Reaching ﬂoor) else If r1 ∈ J then append r1 to K else append r2 to K. (Don’t backtrack) end if end for Let J = J, J = K and K be the empty list until Done for each i from 1 to + 1 such that J[i] = ⊥ append J0 [i] to K Output: “Possible step(s) lead to:” K (One or two outputs)

Basic idea of the modiﬁed algorithms. In our algorithms, we ﬁrst need to choose a large enough extension ﬁeld to guarantee that the kernels of all required isogenies are spanned by -torsion points deﬁned on this extension ﬁeld. As explained in

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Corollary 1, the degree r of this extension ﬁeld is the order of q modulo and it can be computed very quickly after factoring q − 1. As usual, we choose an arbitrary irreducible polynomial of degree r to represent Fqr . The necessary points of ∞ -torsion are computed in Algorithm 1, multiplying random points over Fqr by the cardinality of the curve divided by the highest possible power of . Once this is done, assuming that we are starting from a curve below the second level of stability, we use Algorithms 1 and 2 to ﬁnd all ascending or horizontal isogenies from the initial curve. In order to walk a descending path, it suﬃces to choose any other isogeny. Note that, in the subsequent steps of a descending path, in the cases where the group structure satisﬁes n1 > n2 , it is not necessary to run Algorithm 2 as a whole. Indeed, since we know that we are not on the crater, there is a single ascending isogeny and it is spanned by n1 −1 P1 . Finally, above the second stability level, we have two options. In theory, we can consider curves over larger extension ﬁelds (in order to get polynomials P = 0. Note that this is too costly in practice. Therefore, we use preexisting algorithms, but it is not necessary to follow descending paths all the way to the ﬂoor. Instead, we can stop these paths at the second stabilty level, where our methods can be used. 5.1

Complexity Analysis

Computing a single isogeny. Before analyzing the complete algorithms, we ﬁrst compare the costs of taking a single step on a volcano by using the two methods existing in the literature: modular polynomials and classical V´elu’s formulae. Suppose that we wish to take a step from a curve E. With the modular polynomial approach, we have to evaluate the polynomial f (X) = Φ (X, j(E)) and ﬁnd its roots in Fq . Assuming that the modular polynomial (modulo the characteristic of Fq ) is given as input and using asymptotically fast algorithms to factor f (X), the cost of a step in terms of arithmetic operations in Fq is O(2 + M () log q), where M () denotes the operation count of multiplying polynomials of degree . In this formula, the ﬁrst term corresponds to evaluation of Φ (X, j(Ei−1 )) and the second term to root ﬁnding8 . With V´elu’s formulae, we need to take into account the fact that the required -torsion points are not necessarily deﬁned over Fq . Let r denotes the smallest integer such that the required points are all deﬁned over Fqr . We know that 1 ≤ r ≤ − 1. Using asymptotically eﬃcient algorithms to perform arithmetic operations in Fqr , multiplications in Fqr cost M (r) Fq -operations. Given an torsion point P in E(Fqr ), the cost of using V´elu’s formulae is O() operations in Fqr . As a consequence, in terms of Fq operations, each isogeny costs O(M (r)) operations. As a consequence, when q is not too large and r is close to , using V´elu formulae is more expensive by a logarithmic factor. 8

Completely splitting f (X) to ﬁnd all its roots would cost O(M () log log q), but this is reduced to O(M () log q) because we only need a constant number of roots for each polynomial f (X).

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Computing an ascending or horizontal path. With the classical algorithms, each step in an ascending or horizontal path requires to try O() steps and test each by walking descending paths of height bounded by h. The cost of each descending path is O(h(2 + M () log q)) and the total cost is O(h(3 + M () log q)) (see [14,23]). When >> log q, this cost is dominated by the evaluations of the polynomial Φ at each j-invariant. Thus, by walking in parallel + 1 paths from the original curve, we can amortize the evaluation of Φ (X, j) over many j-invariants using fast multipoint evaluation, see [18, Section 3.7] or [25], thus replacing 3 by M () log and reducing the complexity of a step to O(h M ()(log + log q)). However, this increases the memory requirements. With our modiﬁed algorithms, we need to ﬁnd the structure of each curve, compute some discrete logarithms in -groups, perform a small number of pairing computations and compute the roots of PE,n2 . Except for the computation of discrete logarithms, it is clear that all these additional operations are polynomial in n2 and log and they take negligible time in practice √ (see Section 5.2). Using generic algorithms, the discrete logarithms cost O( ) operations, and this can be reduced to log by storing a sorted table of precomputed logarithms. After this is done, we have to compute at most two isogenies, ignoring the one that backtracks. Thus, the computation of one ascending or horizontal step is dominated by the computation of isogenies and costs O(M (r)). For completeness, we also mention the complexity analysis of Algorithm 1. The dominating step here is the multiplication by N of randomly chosen points. When we consider the curve over an extension ﬁeld Fqr , this costs O(r log q) operations in Fqr , i.e. O(rM (r) log q) operations in Fq . Finally, comparing the two approaches on a regular volcano, we see that even in the less favorable case, we gain a factor h compared to the classical algorithms. More precisely, the two are comparable, when the height h is small and r is close to . In all the other cases, our modiﬁed algorithms are more eﬃcient. This analysis is summarized in Table 1. For compactness O(·)s are omitted from the table. Table 1. Walking the volcano: Order of the cost per step

[14,8] Parallel evaluation Regular volcanoes Best case Worst case r ≈ /2 Regular volcanoes Best case Worst case r ≈ /2 Irregular volcanoes (worst case)

Descending path Ascending/Horizontal One step Many steps h(2 + M () log q) (2 + M () log q) h(3 + M () log q) – – h M ()(log + log q) Structure determination log q log q r M (r) log q r M (r) log q Isogeny construction r M (r) r M (r) No improvement

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Irregular volcanoes. Consider a ﬁxed value of q and let s = v (q − 1). First of all, note that all curves lying on irregular volcanoes satisfy 2s |q + 1 − t and 2s+2 |t2 − 4q. For traces that satisfy only the ﬁrst condition, we obtain a regular volcano. We estimate the total number of diﬀerent traces of elliptic curves lying √ √ √ 4 q on -volcanoes by #{t s.t. 2s |q + 1 − t and t ∈ [−2 q, 2 q]} ∼ 2s . Next, we estimate traces of curves lying on irregular volcanoes by √ √ √ 4 q #{t s.t. 2s |q + 1 − t , 2s+2 |t2 − 4q and t ∈ [−2 q, 2 q]} ∼ 2s+2 . Indeed, by writing q = 1 + γs and t = 2 + γs + μ2s , and imposing the condition 2s+2 |t2 − 4q, we ﬁnd that t ∼ = t0 (γ, μ)(mod 2s+2 ). Thus, we estimate the probability of picking a curve whose volcano is not regular, among curves lying on volcanoes of height greater than 0, by 12 . (This is a crude estimate because the number of curves for each trace is proportional to the Hurwitz class number9 H(t2 − 4q)). This probability is not negligible for small values of . However, since our method also works everywhere on almost regular volcano, the probability of ﬁnding a volcano where we need to combine our modiﬁed algorithm with the classical algorithms is even lower. Furthermore, in some applications, it is possible to restrict ourselves to regular volcanoes. 5.2

Two Practical Examples

A favorable case. In order to demonstrate the potential of the modiﬁed algorithm, we consider the favorable case of a volcano of height 2, where all the necessary -torsion points are deﬁned over the base ﬁeld Fp , where p = 619074283342666852501391 is prime. We choose = 100003. Let E be the elliptic curve whose Weierstrass equation is y 2 = x3 + 198950713578094615678321 x + 32044133215969807107747. The group E[∞ ] over Fp has structure 4ZZ . It is spanned by the point P = (110646719734315214798587, 521505339992224627932173). Taking the -isogeny I1 with kernel 3 P , we obtain the curve E1 : y 2 = x3 + 476298723694969288644436 x + 260540808216901292162091, with structure of the ∞ -torsion Z3 × Z and generators P1 = (22630045752997075604069, 207694187789705800930332) and Q1 = (304782745358080727058129, 193904829837168032791973). The -isogeny I2 with kernel 2 P1 leads to the curve E2 : y 2 = x3 + 21207599576300038652790 x + 471086215466928725193841, on the volcano’s crater and with structure 2ZZ × 2ZZ and generators P2 = (545333002760803067576755, 367548280448276783133614) and Q2 = (401515368371004856400951, 225420044066280025495795). Using pairings on these points, we construct the polynomial: P(x, y) = 97540 x2 + 68114 x y + 38120 y 2, having homogeneous roots (x, y) = (26568, 1) and (72407, 1). As a consequence, we have two horizontal isogenies with kernels (26568 P2 + Q2 ) and (72407 P2 + Q2 ). We can continue and make a complete walk around the 9

See [5, Th. 14.18] for q prime.

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crater which contains 22 diﬀerent curves. Using a simple implementation under Magma 2.15-15, a typical execution takes about 134 seconds10 on a single core of an Intel Core 2 Duo at 2.66 GHz. Most of the time is taken by the computation of V´elu’s formulas (132 seconds) and the computation of discrete logarithms (1.5 seconds) which are not tabulated in the implementation. The computation of pairings only takes 20 milliseconds. A less favorable example. We have also implemented the computation for = 1009 using an elliptic curve with j-invariant j = 34098711889917 in the prime ﬁeld deﬁned by p = 953202937996763. The -torsion appears in a extension ﬁeld of degree 84. The -volcano has height two and the crater contains 19 curves. Our implementation walks the crater in 20 minutes. More precisely, 750 seconds are needed to generate the curves’ structures, 450 to compute V´elu’s formulas, 28 seconds for the pairings and 2 seconds for the discrete logarithms.

6

Conclusion and Perspectives

In this paper, we have proposed a method which allows, in the regular part of an isogeny volcano, to determine, given a curve E and a -torsion point P , the type of the -isogeny whose kernel is spanned by P . In addition, this method also permits, given a basis for the -torsion, to ﬁnd the ascending isogeny (or horizontal isogenies) from E. We expect that this method can be used to improve the performance of several volcano-based algorithms, such as the computation of the Hilbert class polynomial [23] or of modular polynomials [4]. Acknowledgments. The authors thank Jean-Marc Couveignes for the idea in the proof of Lemma 1 and two anonymous reviewers for their helpful comments. The ﬁrst author is grateful to Ariane M´ezard for many discussions on number theory and isogeny volcanoes, prior to this work.

References 1. Belding, J., Broker, R., Enge, A., Lauter, K.: Computing Hilbert Class Polynomials. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 282–295. Springer, Heidelberg (2008) 2. Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a ﬁnite ﬁeld. Journal of Number Theory (to appear 2010) 3. Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005) 4. Broker, R., Lauter, K., Sutherland, A.: Computing modular polynomials with the chinese remainder theorem (2009), http://arxiv.org/abs/1001.0402 10

This timing varies between executions. The reason that we ﬁrst try one root of P, if it backtracks on the crater, we need to try the other one. On average, 1.5 root is tried for each step, but this varies depending on the random choices.

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5. Cox, D.A.: Primes of the Form x2 + ny 2 : Fermat, class ﬁeld theory, and complex multiplication. John Wiley & Sons, Inc., Chichester (1989) 6. Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkorper. Abh. Math. Sem. Hansischen Univ., vol. 14 (1941) 7. Fouquet, M.: Anneau d’endomorphismes et cardinalit´e des courbes elliptiques: aspects algorithmiques. PhD thesis, Ecole Polytechnique (2001) 8. Fouquet, M., Morain, F.: Isogeny Volcanoes and the SEA Algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002) 9. Frey, G.: Applications of arithmetical geometry to cryptographic constructions. In: Proceedings of the Fifth International Conference on Finite Fields and Applications, pp. 128–161. Springer, Heidelberg (2001) 10. Grabher, P., Großsch¨ adl, J., Page, D.: On software parallel implementation of cryptographic pairings. In: Avanzi, R.M., Keliher, L., Sica, F. (eds.) SAC 2008. LNCS, vol. 5381, pp. 35–50. Springer, Heidelberg (2009) 11. Ionica, S.: Algorithmique des couplages et cryptographie. PhD thesis, Universit´e de Versailles St-Quentin-en-Yvelines (2010) 12. Joux, A., Nguyen, K.: Separating decision Diﬃe–Hellman from computational Diﬃe–Hellman in cryptographic groups. Journal of Cryptology 16(4), 239–247 (2003) 13. Lenstra Jr., H.W.: Complex multiplication structure of elliptic curves. Journal of Number Theory 56(2), 227–241 (1996) 14. Kohel, D.: Endomorphism rings of elliptic curves over ﬁnite ﬁelds. PhD thesis, University of California, Berkeley (1996) 15. Miller, V.S.: The Weil pairing, and its eﬃcient calculation. Journal of Cryptology 17(4), 235–261 (2004) 16. Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: An algorithm to compute volcanoes of 2-isogenies of elliptic curves over ﬁnite ﬁelds. Applied Mathematics and Computation 176(2), 739–750 (2006) 17. Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: Computing the height of volcanoes of l-isogenies of elliptic curves over ﬁnite ﬁelds. Applied Mathematics and Computation 196(1), 67–76 (2008) 18. Montgomery, P.L.: A FFT extension of the elliptic curve method of factorization. PhD thesis, University of California (1992) 19. Ruck, H.-G.: A note on elliptic curves over ﬁnite ﬁelds. Mathematics of Computation 179, 301–304 (1987) 20. Schoof, R.: Counting points on elliptic curves over ﬁnite ﬁelds. Journal de Theorie des Nombres de Bordeaux 7, 219–254 (1995) 21. Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986) 22. Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, Heidelberg (1994) 23. Sutherland, A.: Computing Hilbert Class Polynomials with the Chinese Remainder Theorem. Mathematics of Computation (2010) 24. V´elu, J.: Isogenies entre courbes elliptiques. Comptes Rendus De L’Academie Des Sciences Paris, Serie I-Mathematique, Serie A. 273, 238–241 (1971) 25. von zur Gathen, J., Shoup, V.: Computing Frobenius maps and factoring polynomials. Computational Complexity 2, 187–224 (1992)

A Subexponential Algorithm for Evaluating Large Degree Isogenies David Jao and Vladimir Soukharev Department of Combinatorics and Optimization University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada {djao,vsoukhar}@math.uwaterloo.ca

Abstract. An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves eﬃciently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.

1

Introduction

A well known theorem of Tate [29] states that two elliptic curves deﬁned over the same ﬁnite ﬁeld Fq are isogenous (i.e. admit an isogeny between them) if and only if they have the same number of points over Fq . Using fast point counting algorithms such as Schoof’s algorithm and others [9,25], it is very easy to check whether this condition holds, and thus whether or not the curves are isogenous. However, constructing the actual isogeny itself is believed to be a hard problem due to the nonconstructive nature of Tate’s theorem. Indeed, given an ordinary curve E/Fq and an ideal of norm n in the endomorphism ring, the fastest previously known algorithm for constructing the unique (up to isomorphism) isogeny having this ideal as kernel has a running time of O(n3+ε ), except in a certain very small number of special cases [4,16,17]. In this paper, we present a new probabilistic algorithm for evaluating such isogenies, which in the vast majority of cases runs (heuristically) in subexponential time. Speciﬁcally, we show that for ordinary curves, one can evaluate isogenies of degree n between curves of √ nearly equal endomorphism ring over Fq in time less than Lq ( 12 , 23 ) log(n), provided n has no large prime divisors in common with the endomorphism ring discriminant. Although this running time is not polynomial in the input length, our algorithm is still much faster than the (exponential) previous best known algorithm, and in practice allows for the evaluation of isogenies of cryptographically sized degrees, some examples of which we present here. We emphasize that, G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 219–233, 2010. c Springer-Verlag Berlin Heidelberg 2010

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in contrast with the previous results of Br¨ oker et al. [4], our algorithm is not limited to special curves such as pairing friendly curves with small discriminant. If an explicit equation for the isogeny as a rational function is desired, our approach in combination with the algorithm of Bostan et al. [3] can produce the equation in time O(n1+ε ) given E and an ideal of norm n, which is quasi-optimal in the sense that (up to log factors) it is equal to the size of the output. To our knowledge, this method is the only known algorithm for computing rational function expressions of large degree isogenies in quasi-optimal time in the general case, given only the starting curve and the kernel. Apart from playing a central role in the implementation of the point counting algorithms mentioned above, isogenies have been used in cryptography to transfer the discrete logarithm problem from one elliptic curve to another [9,16,17,20,23,30]. In many of these applications, our algorithm cannot be used directly, since in cryptography one is usually given two isogenous curves, rather than one curve together with the isogeny degree. However, earlier results [16,17,20] have shown that the problem of computing isogenies between a given pair of curves can be reduced to the problem of computing isogenies of prime degree starting from a given curve. It is therefore likely that the previous best isogeny construction algorithms in the cryptographic setting can be improved or extended in light of the work that we present here.

2

Background

Let E and E be elliptic curves deﬁned over a ﬁnite ﬁeld Fq of characteristic p. An isogeny φ : E → E deﬁned over Fq is a non-constant rational map deﬁned over Fq which is also a group homomorphism from E(Fq ) to E (Fq ). This deﬁnition diﬀers slightly from the standard deﬁnition in that it excludes constant maps [27, §III.4]. The degree of an isogeny is its degree as a rational map, and an isogeny of degree is called an -isogeny. Every isogeny of degree greater than 1 can be ¯ q [11]. factored into a composition of isogenies of prime degree deﬁned over F For any elliptic curve E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 deﬁned over Fq , the Frobenius endomorphism is the isogeny πq : E → E of degree q given by the equation πq (x, y) = (xq , y q ). The characteristic polynomial of πq is X 2 − tX + q where t = q + 1 − #E(Fq ) is the trace of E. An endomorphism of E is an isogeny E → E deﬁned over the algebraic closure ¯ q of Fq . The set of endomorphisms of E together with the zero map forms F a ring under the operations of pointwise addition and composition; this ring is called the endomorphism ring of E and denoted End(E). The ring End(E) is isomorphic either to an order in a quaternion algebra or to an order in an imaginary quadratic ﬁeld [27, V.3.1]; in the ﬁrst case we say E is supersingular and in the second case we say E is ordinary. Two elliptic curves E and E deﬁned over Fq are said to be isogenous over Fq if there exists an isogeny φ : E → E deﬁned over Fq . A theorem of Tate states that two curves E and E are isogenous over Fq if and only if #E(Fq ) = #E (Fq ) [29, §3]. Since every isogeny has a dual isogeny [27, III.6.1], the property of being

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¯ q -isomorphism isogenous over Fq is an equivalence relation on the ﬁnite set of F classes of elliptic curves deﬁned over Fq . Moreover, isomorphisms between elliptic curves can be classiﬁed completely and computed eﬃciently in all cases [16]. Accordingly, we deﬁne an isogeny class to be an equivalence class of elliptic ¯ q -isomorphism, under this equivalence relation. curves, taken up to F Curves in the same isogeny class are either all supersingular or all ordinary. The vast majority of curves are ordinary, and indeed the number of isomorphism classes of supersingular curves is ﬁnite for each characteristic. Also, ordinary curves form the majority of the curves of interest in applications such as cryptography. Hence, we assume for the remainder of this paper that we are in the ordinary case. Let K denote the imaginary quadratic ﬁeld containing End(E), with maximal order OK . For any order O ⊆ OK , the conductor of O is deﬁned to be the integer [OK : O]. The ﬁeld K is called the CM ﬁeld of E. We write cE for the conductor of End(E) and cπ for the conductor of Z[πq ]. It follows from [12, §7] that End(E) = Z + cE OK and Δ = c2E ΔK , where Δ (respectively, ΔK ) is the discriminant of the imaginary quadratic order End(E) (respectively, OK ). Furthermore, the characteristic polynomial has discriminant Δπ = t2 − 4q = disc(Z[πq ]) = c2π ΔK , with cπ = cE · [End(E) : Z[πq ]]. Following [14] and [16], we say that an isogeny φ : E → E of prime degree deﬁned over Fq is “down” if [End(E) : End(E )] = , “up” if [End(E ) : End(E)] = , and “horizontal” if End(E) = End(E). Two curves in an isogeny class are said to “have the same level” if their endomorphism rings are equal. Within each isogeny class, the property of having the same level is an equivalence relation. A horizontal isogeny always goes between two curves of the same level; likewise, an up isogeny enlarges the endomorphism ring and a down isogeny reduces it. Since there are fewer elliptic curves at higher levels than at lower levels, the collection of elliptic curves in an isogeny class visually resembles a “pyramid” or a “volcano” [14], with up isogenies ascending the structure and down isogenies descending. If we restrict to the graph of -isogenies for a single , then in general the -isogeny graph is disconnected, having one -volcano for each intermediate order Z[πq ] ⊂ O ⊂ OK such that O is maximal at (meaning [OK : O]). The “top level” of the class consists of curves E with End(E) = OK , and the “bottom level” consists of curves with End(E) = Z[πq ]. We say that is an Elkies prime [2, p. 119] if cE and Δ = −1, or equivalently if and only if E admits a horizontal isogeny of degree . The number of -isogenies of each type can easily be determined explicitly [14,16,21]. In particular, for all but the ﬁnitely many primes dividing [OK : Z[πq ]], we have that every rational -isogeny admitted by E is horizontal.

3

The Br¨ oker-Charles-Lauter Algorithm

Our algorithm is an extension of the algorithm developed by Br¨ oker, Charles, and Lauter [4] to evaluate large degree isogenies over ordinary elliptic curves with

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endomorphism rings of small class number, such as pairing-friendly curves [15]. In this section we provide a summary of their results. The following notation corresponds to that of [4]. Let E/Fq be an ordinary elliptic curve with endomorphism ring End(E) isomorphic to an imaginary quadratic order OΔ of discriminant Δ < 0. Identify End(E) with OΔ via the unique isomorphism ι such that ι∗ (x)ω = xω for all invariant diﬀerentials ω and all x ∈ OΔ . Then every horizontal separable isogeny on E of prime degree corresponds (up to isomorphism) to a unique prime ideal L ⊂ OΔ of norm for some Elkies prime . We denote the kernel of this isogeny by E[L]. Any two distinct isomorphic horizontal isogenies (i.e., pairs of isogenies where one is equal to the composition of the other with an isomorphism) induce diﬀerent maps on the space of diﬀerentials of E, and a separable isogeny is uniquely determined by the combination of its kernel and the induced map on the space of diﬀerentials. A normalized isogeny is an isogeny φ : E → E for which φ∗ (ωE ) = ωE where ωE denotes the invariant diﬀerential of E. Algorithm 1 (identical to Algorithm 4.1 in [4]) evaluates, up to automorphisms of E, the unique normalized horizontal isogeny of degree corresponding to a given kernel ideal L ⊂ OΔ . The following theorem, taken verbatim from [4], shows that the running time of Algorithm 1 is polynomial in the quantities log(), log(q), n, and |Δ|. Theorem 3.1. Let E/Fq be an ordinary elliptic curve with Frobenius πq , given by a Weierstrass equation, and let P ∈ E(Fqn ) be a point on E. Let Δ = disc(End(E)) be given. Assume that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and let L = (, c + dπq ) be an End(E)-ideal of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Algorithm 1 computes the unique elliptic curve E such that there exists a normalized isogeny φ : E → E with kernel E[L]. Furthermore, it computes the x-coordinate of φ(P ) if End(E) does not equal Z[i] or Z[ζ3 ] and the square, respectively cube, of the x-coordinate of φ(P ) otherwise. The running time of the algorithm is polynomial in log(), log(q), n and |Δ|.

4

A Subexponential Algorithm for Evaluating Horizontal Isogenies

As was shown in Sections 2 and 3, any horizontal isogeny can be expressed as a composition of prime degree isogenies, one for each prime factor of the kernel, and any prime degree isogeny is a composition of a normalized isogeny and an isomorphism. Therefore, to evaluate a horizontal isogeny given its kernel, it suﬃces to treat the case of horizontal normalized prime degree isogenies. Our objective is to evaluate the unique horizontal normalized isogeny on a given elliptic curve E/Fq whose kernel ideal is given as L = (, c + dπq ), at a given point P ∈ E(Fqn ), where is an Elkies prime. As in [4], we must also impose the additional restriction that [End(E) : Z[πq ]]; for Elkies primes, an equivalent restriction is that [OK : Z[πq ]], but we retain the original formulation for consistency with [4].

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Algorithm 1. The Br¨oker-Charles-Lauter algorithm Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ and a point P ∈ E(Fqn ) such that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and an End(E)-ideal L = (, c+dπq ) of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Output: The unique elliptic curve E admitting a normalized isogeny φ : E → E with kernel E[L], and the x-coordinate of φ(P ) for Δ = −3, −4 and the square (resp. cube) of the x-coordinate otherwise. 1: Compute the direct sum decomposition Pic(OΔ ) = [Ii ] of Pic(OΔ ) into cyclic groups generated by the degree 1 prime ideals Ii of smallest norm that are coprime to the product p · #E(Fqn ) · [End(E) : Z[πq ]]. 2: Using brute force1 , ﬁnd e1 , e2 , . . . , ek such that [L] = [I1e1 ] · [I2e2 ] · · · [Ikek ]. e 3: Find α (using Cornacchia’s algorithm) and express L = I1e1 · I2e2 · · · Ikk · (α). 4: Compute a sequence of isogenies (φ1 , . . . , φs ) such that the composition φc : E → Ec has kernel E[I1e1 · I2e2 · · · Ikek ] using the method of [4, § 3]. 5: Evaluate φc (P ) ∈ Ec (Fqn ). ∼ 6: Write α = (u + vπq )/(zm). Compute the isomorphism η : Ec → E with η ∗ (ωE ) = (u/zm)ωEc . Compute Q = η(φc (P )). 7: Compute (zm)−1 mod #E(Fqn ), and compute R = ((zm)−1 (u + vπq ))(Q). ∗ 8: Put r = x(R)|OΔ | /2 and return (E , r).

In practice, one is typically given instead of L, but since it is easy to calculate the list of (at most two) possible primes L lying over (cf. [6]), these two interpretations are for all practical purposes equivalent, and we switch freely between them when convenient. When is small, one can use modular polynomial based techniques [4, §3.1], which have running time O(3 log()4+ε ) [13]. However, for isogeny degrees of cryptographic size (e.g. 2160 ), this approach is impractical. The Br¨oker-Charles-Lauter algorithm sidesteps this problem, by using an alternative factorization of L. However, the running time of Br¨ oker-Charles-Lauter is polynomial in |Δ|, and therefore even this method only works for small values of |Δ|. In this section we present a modiﬁed version of the Br¨ oker-Charles-Lauter algorithm which is suitable for large values of |Δ|. We begin by giving an overview of our approach. In order to handle large values of |Δ|, there are two main problems to overcome. One problem is that we need a fast way to produce a factorization L = I1e1 I2e2 · · · Ikek · (α)

(1)

as in lines 2 and 3 of Algorithm 1. The other problem is that the exponents ei in Equation (1) need to be kept small, since the running times of lines 3 and 4 of Algorithm 1 are proportional to i |ei | Norm(Ii )2 . The ﬁrst problem, that of ﬁnding a factorization of L, can be solved in subexponential time using the index calculus algorithm of Hafner and McCurley [18] (see also [6, Chap. 11]). 1

Br¨ oker, Charles, and Lauter mention that this computation can be done in “various ways” [4, p. 107], but the only explicit method given in [4] is brute force. The use of brute force limits the algorithm to elliptic curves for which |Δ| is small, such as pairing-friendly curves.

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Algorithm 2. Computing a factor base Input: A discriminant Δ, a bound N . Output: The set I consisting of split prime ideals of norm less than N , together with the corresponding set F of quadratic forms. 1: Set F ← ∅. 2: Set I ← ∅. 3: Find all primes p < N such that ( Δ ) = 1. Call this set P . Let k = |P |. p 4: For each prime pi ∈ P , ﬁnd an ideal pi of norm pi (using Cornacchia’s algorithm). 5: For each i, ﬁnd a quadratic form fi = [(pi , bi , ci )] corresponding to pi in Cl(OΔ ), using the technique of [26, §3]. 6: Output I = {p1 , p2 , . . . , pk } and F = {f1 , f2 , . . . , fk }.

To resolve the second problem, we turn to an idea which was ﬁrst introduced by Galbraith et. al [17], and recently further reﬁned by Bisson and Sutherland [1]. The idea is that, in the process of sieving for smooth norms, one can arbitrarily restrict the input exponent vectors to sparse vectors (e1 , e2 , ..., ek ) such that 2 i |ei |N (Ii ) is kept small. This restriction is implemented in line 6 of Algorithm 3. As in [1], one then assumes heuristically that the imposition of this restriction does not aﬀect the eventual probability of obtaining a smooth norm in the Hafner and McCurley algorithm. Note that, unlike the input exponents, the exponents appearing in the factorizations of the ensuing smooth norms (that is, the values of yi in Algorithm 3) are always small, since the norm in question is derived from a reduced quadratic form. We now describe the individual components of our algorithm in detail. 4.1

Finding a Factor Base

Let Cl(OΔ ) denote the ideal class group of OΔ . Algorithm 2 produces a factor base consisting of split primes in OΔ of norm less than some bound N . The optimal value of N will be determined in Section 4.4. 4.2

“Factoring” Large Prime Degree Ideals

Algorithm 3, based on the algorithm of Hafner and McCurley, takes as input a discriminant Δ, a curve E, a prime ideal L of prime norm in OΔ , a smoothness bound N , and an extension degree n. It outputs a factorization L = I1e1 I2e2 · · · Ikek · (α) as in Equation 1, where the Ii ’s are as in Algorithm 1, the exponents ei are positive, sparse, and small (i.e., polynomial in N ), and the ideal (α) is a principal fractional ideal generated by α. 4.3

Algorithm for Evaluating Prime Degree Isogenies

The overall algorithm for evaluating prime degree isogenies is given in Algorithm 4. This algorithm is identical to Algorithm 1, except that the factorization of L is performed using Algorithm 3. To maintain consistency with [4], we

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Algorithm 3. “Factoring” a prime ideal Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ , a smoothness bound N , a prime ideal L of norm inOΔ , an extension degree n. Output: Relation of the form L = (α) · ki=1 Iiei , where (α) is a fractional ideal, Ii are as in Algorithm 1, and ei > 0 are small and sparse. 1: Run Algorithm 2 on input Δ and N to obtain I = {p1 , p2 , . . . , pk } and F = {f1 , f2 , . . . , fk }. Discard any primes dividing p · #E(Fqn ) · [End(E) : Z[πq ]]. 2: Set pi ← Norm(pi ). (These values are also calculated in Algorithm 2.) 3: Obtain the reduced quadratic form [L] corresponding to the ideal class of L. 4: repeat 5: for i = 1, . . . , k do 6: Pick exponents xi in the range [0, (N/pi )2 ] such that at most k0 are nonzero, where k0 is a global absolute constant (in practice, k0 = 3 suﬃces). 7: end for 8: Compute the reduced quadratic form a = (a, b, c) for which the ideal class [a] is equivalent to [L] · ki=1 fixi . 9: until The integer primes pi , and the relation derived a factors completely into the from [a] = [L] · ki=1 fixi contains fewer than log(|Δ|/3)/z nonzero exponents. 10: Write a = ki=1 pui i . 11: for i=1, . . . , k do 12: Using the technique of Seysen ([26, Theorem 3.1]), determine the signs of the exponents yi = ±ui for which a = ki=1 fiyi . 13: Let ei = yi − xi . (These exponents satisfy [L] = ki=1 fiei .) 14: if ei ≥ 0 then 15: Set Ii ← ¯ pi 16: else 17: Set Ii ← pi 18: end if 19: end for |e | 20: Compute the principal ideal I = L · ki=1 Ii i . 21: Using Cornacchia’s algorithm, ﬁnd a generator β ∈ OΔ of I. |e | β . 22: Set m ← ki=1 pi i and α ← m |e | |e | |e | 23: Output L = (α) · I¯1 1 · I¯2 2 · · · I¯k k .

have included the quantities Δ and End(E) as part of the input to the algorithm. However, we remark that these quantities can be computed from E/Fq √ in Lq ( 12 , 23 ) operations using the algorithm of Bisson and Sutherland [1], even if they are not provided as input. 4.4

Running Time Analysis

In this section, we determine the theoretical running time of Algorithm 4, as well as the optimal value of the smoothness bound N to use in line 1 of the algorithm. As is typical for subexponential time factorization algorithms involving a factor base, these two quantities depend on each other, and hence both are calculated simultaneously.

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Algorithm 4. Evaluating prime degree isogenies Input: A discriminant Δ, an elliptic curve E/Fq with End(E) = OΔ and a point P ∈ E(Fqn ) such that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and an End(E)-ideal L = (, c+dπq ) of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Output: The unique elliptic curve E admitting a normalized isogeny φ : E → E with kernel E[L], and the x-coordinate of φ(P ) for Δ = −3, −4 and the square (resp. cube) of the x-coordinate otherwise. 1: Choose a smoothness bound N (see Section 4.4). 2: Using Algorithm 3 on input (Δ, E, N, L, n), obtain a factorization of the form L = I1e1 · I2e2 · · · Ikek · (α). 3: Compute a sequence of isogenies (φ1 , . . . , φs ) such that the composition φc : E → e Ec has kernel E[I1e1 · I2e2 · · · Ikk ] using the method of [4, § 3]. 4: Evaluate φc (P ) ∈ Ec (Fqn ). ∼ 5: Write α = (u + vπq )/(zm). Compute the isomorphism η : Ec → E with η ∗ (ωE ) = (u/zm)ωEc . Compute Q = η(φc (P )). 6: Compute (zm)−1 mod #E(Fqn ), and compute R = ((zm)−1 (u + vπq ))(Q). ∗ 7: Put r = x(R)|OΔ | /2 and return (E , r).

As in [9], we deﬁne2 Ln (α, c) by Ln (α, c) = O(exp((c + o(1))(log(n))α (log(log(n)))1−α )). The quantity Ln (α, c) interpolates between polynomial and exponential size as α ranges from 0 to 1. We set N = L|Δ| ( 12 , z) for an unspeciﬁed value of z, and in the following paragraphs we determine the optimal value of z which minimizes the running time of Algorithm 4. (The fact that α = 12 is optimal is clear from the below analysis, as well as from prior experience with integer factorization algorithms.) For convenience, we will abbreviate L|Δ| (α, c) to L(α, c) throughout. Line 2 of Algorithm 4 involves running Algorithm 3, which in turn calls Algorithm 2. As it turns out, Algorithm 2 is almost the same as Algorithm 11.1 from [6], which requires L( 21 , z) time, as shown in [6]. The only diﬀerence is that we add an additional step where we obtain the quadratic form corresponding to each prime ideal in the factor base. This extra step requires O(log(Norm(I))1+ε ) time for a prime ideal I, using Cornacchia’s Algorithm [19]. Thus, the overall running time for Algorithm 2 is bounded above by L( 12 , z) · log(L( 12 , z))1+ε = L( 12 , z). Line 2 of Algorithm 3 takes log() time using standard algorithms [12]. The loop in lines 4–9 of Algorithm 3 is very similar to the FindRelation algorithm in [1], except that we only use one discriminant, and we omit the requirement that #R/D1 > #R/D2 (which in any case is meaningless when there is only one discriminant). Needless to say, this change can only speed up the algorithm. 2

The deﬁnition of Ln (α, c) in [6] diﬀers from that of [9] in the o(1) term. We account for this discrepancy in our text.

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√ Taking μ = 2z in [1, Prop. 6], we ﬁnd that the (heuristic) expected running 1 time of the loop in lines 4–9 of Algorithm 3 is L( 21 , 4z ). The next step in Algorithm 3 having nontrivial running time is the computation of the ideal product in line 20. To exponentiate an element of an arbitrary semigroup to a power e requires O(log e) semigroup multiplication operations [10, §1.2]. To multiply two ideals I and J in an imaginary quadratic order (via composition of quadratic forms) requires O(max(log(Norm(I)), log(Norm(J)))1+ε ) bit operations using fast multiplication [24, §6]. Each of the expressions |Ii ||ei | therefore requires O(log |ei |) ideal multiplication operations to compute, with each individual multiplication requiring ⎛

1+ε ⎞ 2 N ⎠ = O(N 2+ε ) O((|ei | log(Norm(Ii )))1+ε ) = O ⎝ log(pi ) pi bit operations, for a total running time of (log ei )O(N 2+ε ) = L( 12 , 2z) for each i. This calculation must be performed once for each nonzero exponent ei . By line 9, the number of nonzero exponents appearing in the relation is at|emost log(|Δ|/3)/z, so the amount of time required to compute all of the |Ii | i | for all i is ( log(|Δ|/3)/z)L( 21 , 2z) = L( 12 , 2z). Afterward, the values |Ii ||ei | must all be multiplied together, a calculation which entails at most log(|Δ|/3)/z ideal multiplications where the log-norms of the input multiplicands are bounded above by |ei |

log Norm(Ii

) = |ei | log Norm(Ii ) ≤

N pi

2 log pi ≤ N 2 = L( 12 , 2z),

and thus each of the (at most) log(|Δ|/3)/z multiplications in the ensuing product can be completed in time at most ( log(|Δ|/3)/z)L( 21 , 2z) = L( 12 , 2z). Finally, we must multiply this end result by L, an operation which requires O(max(log , L( 21 , 2z))1+ε ) time. All together, the running time of step 20 is L( 12 , 2z) + O(max(log , L( 21 , 2z))1+ε ) = max((log )1+ε , L( 12 , 2z)), and the norm of the resulting ideal I is bounded above by · exp(L( 12 , 2z)). Obtaining the generator β of I in line 21 of Algorithm 3 using Cornacchia’s algorithm requires O(log(Norm(I))1+ε ) = (log + L( 12 , 2z))1+ε time. We remark that ﬁnding β given I is substantially easier than the usual Cornacchia’s algorithm, which entails ﬁnding β given only Norm(I). The usual algorithm requires ﬁnding all the square roots of Δ modulo Norm(I), which is very slow when Norm(I) has a large number of prime divisors. This time-consuming step is unnecessary when the ideal I itself is given, since the embedding of the ideal I in End(E) already provides (up to sign) the correct square root of Δ mod I. A detailed description of this portion of Cornacchia’s algorithm in the context of the full algorithm, together with running time ﬁgures speciﬁc to each

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sub-step, is given by Hardy et al. [19]; for our purposes, the running time of a single iteration of Step 6 in [19, §4] is the relevant ﬁgure. This concludes our analysis of Algorithm 3. Returning to Algorithm 4, we ﬁnd that (as in [4]) the computation of the individual isogenies φi in line 3 of Algorithm 4 is limited by the time required to compute the modular polynomials Φn (x, y). Using the Chinese remainder theorembased method of Br¨oker et al. [5], these polynomials can be computed mod q in time O(n3 log3+ε (n)), and the resulting polynomials require O(n2 (log2 n+log q)) space. For each ideal Ii , the corresponding modular polynomial of level pi only needs to be computed once, but the polynomial once computed must be evaluated, diﬀerentiated, and otherwise manipulated ei times, at a cost of O(p2+ε ) i ﬁeld operations in Fq per manipulation, or O(p2+ε )(log q)1+ε bit operations using i fast multiplication. The total running time of line 3 is therefore

N 2 3+ε 2+ε 1+ε 3+ε p2+ε |ei |pi (log q) ≤ O(N )+ (log q)1+ε O(pi ) + i p i i i log(|Δ|/3) 2+ε ≤ O(N 3+ε ) + (log q)1+ε = L( 21 , 3z) + L( 12 , 2z)(log q)1+ε . N z Similarly, the evaluation of φc in line 4 requires |ei |p2+ε = L( 21 , 2z) i i

ﬁeld operations in Fqn , which corresponds to L( 12 , 2z)(log q n )1+ε bit operations using fast multiplication. Combining all the above quantities, we obtain a total running time of L( 12 , z) +

(algorithm 2)

1 ) L( 12 , 4z

(lines 4–9, algorithm 3)

+ max((log )

1+ε

+ + +

, L( 21 , 2z)) 1+ε

(log + L( 12 , 2z)) L( 12 , 3z) + L( 12 , 2z)(log q)1+ε L( 12 , 2z)(log q n )1+ε

(line 20, algorithm 3) (line 21, algorithm 3) (line 3, algorithm 4) (line 4, algorithm 4)

1 ) + (log + L( 12 , 2z))1+ε + L( 12 , 3z) + L( 21 , 2z)(log q n )1+ε . = L( 12 , 4z

When |Δ| is large, we may impose the reasonable assumption that log() L( 12 , z) and log(q n ) L( 12 , z). In this case, the running time of Algorithm 4 is 1 dominated by the expression L( 12 , 4z ) + L( 12 , 3z), which attains a minimum at 1 z = 2√3 . Taking this value of z, we ﬁnd that the running time of Algorithm 4 is equal to L|Δ| ( 12 ,

√ 3 2 ).

Since the maximum value of |Δ| ≤ |Δπ | = 4q − t2 is 4q,

we can alternatively express this running time as simply Lq ( 12 ,

√

3 2 ).

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In the general case, log() and log(q n ) might be non-negligible compared to L( 12 , z). This can happen in one of two ways: either |Δ| is small, or (less likely) is very large and/or n is large. When this happens, we can still bound the 1 running time of Algorithm 4 by taking z = 2√ in the foregoing calculation, 3 although such a choice may fail to be optimal. We then ﬁnd that the running time of Algorithm 4 is bounded above by (log() + L( 12 , √13 ))1+ε + L( 12 ,

√

3 2 )

+ L( 12 , √13 )(log q n )1+ε .

We summarize our results in the following theorem. Theorem 4.1. Let E/Fq be an ordinary elliptic curve with Frobenius πq , given by a Weierstrass equation, and let P ∈ E(Fqn ) be a point on E. Let Δ = disc(End(E)) be given. Assume that [End(E) : Z[πq ]] and #E(Fqn ) are coprime, and let L = (, c + dπq ) be an End(E)-ideal of prime norm = char(Fq ) not dividing the index [End(E) : Z[πq ]]. Under the heuristics of [1, §4], Algorithm 4 computes the unique elliptic curve E such that there exists a normalized isogeny φ : E → E with kernel E[L]. Furthermore, it computes the x-coordinate of φ(P ) if End(E) does not equal Z[i] or Z[ζ3 ] and the square, respectively cube, of the x-coordinate of φ(P ) otherwise. The running time of the algorithm is bounded above by (log() + L( 12 , √13 ))1+ε + L( 12 ,

√

3 2 )

+ L( 12 , √13 )(log q n )1+ε .

The running time of the algorithm is subexponential in log |Δ|, and polynomial in log(), log(q), and n.

5 5.1

Examples Small Example

Let p = 1010 +19 and let E/Fp be the curve y 2 = x3 +15x+129. Then E(Fp ) has cardinality 10000036491 = 3 · 3333345497 and trace t = −36471. To avoid any bias in the selection of the prime , we set to be the smallest Elkies prime of E larger than p/2, namely = 5000000029. We will evaluate the x-coordinate of φ(P ), where φ is an isogeny of degree , and P is chosen arbitrarily to be the point (5940782169, 2162385016) ∈ E(Fp ). We remark that, although this example is designed to be artiﬁcially small for illustration purposes, the evaluation of this isogeny would already be infeasible if we were using prior techniques based on modular functions of level . √ The discriminant Δ of E is Δ = t2 − 4p = −38669866235. Set w = 1+2 Δ and O = OΔ . The quadratic form (5000000029, −2326859861, 270713841) represents a prime ideal L of norm , and we show how to calculate the isogeny φ having kernel corresponding to E[L]. Using an implementation of Algorithm 3 β in MAGMA [22], we ﬁnd immediately the relation L = ( m ) · p19 · p24 31

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where β = 588048307603210005w − 235788727470005542279904, m = 19 · 3124 , p19 = (19, 2w + 7), and p31 = (31, 2w + 5). Using this factorization, we can then evaluate φ : E → E using the latter portion of Algorithm 4. We ﬁnd that E is the curve with Weierstrass equation y 2 = x3 + 3565469415x + 7170659769, and φ(P ) = (7889337683, ±3662693258). We omit the details of these steps, since this portion of the algorithm is identical to the algorithm of Br¨ oker, Charles and Lauter, and the necessary steps are already extensively detailed in their article [4]. We can check our computations for consistency by performing a second computation, starting from the curve E : y 2 = x3 + 3565469415x + 7170659769, the ¯ which point P = (7889337683, 3662693258) ∈ E (Fp ), and the conjugate ideal L, is represented by the quadratic form (5000000029, 2326859861, 270713841). Let ¯ Up to φ¯ : E → E denote the unique normalized isogeny with kernel E [L]. ¯ a normalization isomorphism ι : E → E , the isogeny φ should equal the dual ¯ isogeny φˆ of φ, and the composition φ(φ(P )) should yield ι(P ). Indeed, upon performing the computation, we ﬁnd that E has equation y 2 = x3 + (15/4 )x + (129/6), which is isomorphic to E via the isomorphism ι : E → E deﬁned by ι(x, y) = (x/2 , y/3 ), and ¯ φ(φ(P )) = (3163843645, 8210361642) = (5551543736/2, 6305164567/3), in agreement with the value of P , which is (5551543736, 6305164567). 5.2

Medium Example

Let E be the ECCp-109 curve [8] from the Certicom ECC Challenge [7], with equation y 2 = x3 + ax + b over Fp where p = 564538252084441556247016902735257 a = 321094768129147601892514872825668 b = 430782315140218274262276694323197 As before, to avoid any bias in the choice √ of , we set to be the least Elkies prime greater than p/2, and we deﬁne w = 1+2 Δ where Δ = disc(End(E)). Let L be the prime ideal of norm in End(E) corresponding to the reduced quadratic form (, b, c) of discriminant Δ, where b = −105137660734123120905310489472471. For each Elkies prime p, let pp denote the unique prime ideal corresponding to the reduced quadratic form (p, b, c) where b ≥ 0. Our smoothness bound in this 1 case is N = L( 12 , 2√ ) ≈ 200. Using Sutherland’s smoothrelation package [28], 3 which implements the FindRelation algorithm of [1], one ﬁnds in a few seconds β I, where (using an initial seed of 0) the relation L = m p100 p14 p247 p¯273 ¯ p103 p179 p191 I=¯ p72 7 ¯ 13 ¯ 23 ¯ m = 772 13100 2314 472 732 1031 1791 1911

A Subexponential Algorithm for Evaluating Large Degree Isogenies

231

and β = 3383947601020121267815309931891893555677440374614137047492987151\ 2226041731462264847144426019711849448354422205800884837 − 1713152334033312180094376774440754045496152167352278262491589014\ 097167238827239427644476075704890979685 · w We ﬁnd that the codomain E of the normalized isogeny φ : E → E of kernel E[L] has equation y 2 = x3 + a x + b where a = 84081262962164770032033494307976 b = 506928585427238387307510041944828 and that the base point P = (97339010987059066523156133908935, 149670372846169285760682371978898)

of E given in the Certicom ECC challenge has image (450689656718652268803536868496211, ±345608697871189839292674734567941).

under φ. As with the ﬁrst example, we checked the computation for consistency by using the conjugate ideal. 5.3

Large Example

Let E be the ECCp-239 curve [8] from the Certicom ECC Challenge [7]. Then E has equation y 2 = x3 + ax + b over Fp where p = 862591559561497151050143615844796924047865589835498401307522524859467869 a = 820125117492400602839381236756362453725976037283079104527317913759073622 b = 545482459632327583111433582031095022426858572446976004219654298705912499

Let L be the prime ideal whose norm is the least Elkies prime greater than p/2 and whose ideal class is represented by the quadratic form (, b, c) with 1 b ≥ 0. We have N = L( 12 , 2√ ) ≈ 5000, and one ﬁnds in a few hours using 3 smoothrelation [28] that L is equivalent to I=¯ p27 p11 p19 p237 ¯ p271 ¯ p131 p211 ¯ p389 ¯ p433 ¯ p467 ¯ p18 p1019 ¯p1151 ¯p1597 ¯p62143 ¯p52207 ¯p3359 859 p863 ¯ where each ideal pp is represented by the reduced quadratic form (p, b, c) having b ≥ 0 (this computation can be reconstructed with [28] using the seed 7). The quotient L/I is generated by β/m where m = Norm(I) and β is −923525986803059652225406070265439117913488592374741428959120914067053307\ 4585317 − 917552768623818156695534742084359293432646189962935478129227909w.

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D. Jao and V. Soukharev

Given this relation, evaluating isogenies of degree is a tedious but routine computation using Elkies-Atkin techniques [4, §3.1]. Although we do not complete it here, the computation is well within the reach of present technology; indeed, Br¨ oker et al. [5] have computed classical modular polynomials mod p of level up to 20000, well beyond the largest prime of 3389 appearing in our relation.

6

Related Work

Bisson and Sutherland [1] have developed an algorithm to compute the endomorphism ring of an elliptic curve in subexponential time, using relation-ﬁnding techniques which largely overlap with ours. Although our main results were obtained independently, we have incorporated their ideas into our algorithm in several places, resulting in a simpler presentation as well as a large speedup compared to the original version of our work. Given two elliptic curves E and E over Fq admitting a normalized isogeny φ : E → E of degree , the equation of φ as a rational function contains O() coeﬃcients. Bostan et al. [3] have published an algorithm which produces this equation, given E, E , and . Their algorithm has running time O(1+ε ), which is quasi-optimal given the size of the output. Using our algorithm, it is possible √ to compute E from E and in time log()L|Δ| ( 12 , 23 ) for large . Hence the combination of the two algorithms can produce the equation of φ within a quasioptimal running time of O(1+ε ), given only E and (or E and L), without the need to provide E in the input.

Acknowledgments We thank the anonymous referees for numerous suggestions which led to substantial improvements in our main result.

References 1. Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a ﬁnite ﬁeld. Journal of Number Theory (to appear 2009) 2. Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (2000); Reprint of the 1999 original (1999) ´ Fast algorithms for computing iso3. Bostan, A., Morain, F., Salvy, B., Schost, E.: genies between elliptic curves. Math. Comp. 77(263), 1755–1778 (2008) 4. Br¨ oker, R., Charles, D., Lauter, K.: Evaluating large degree isogenies and applications to pairing based cryptography. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 100–112. Springer, Heidelberg (2008) 5. Br¨ oker, R., Lauter, K., Sutherland, A.: Modular polynomials via isogeny volcanoes (2010) 6. Buchmann, J., Vollmer, U.: Binary quadratic forms. Algorithms and Computation in Mathematics, vol. 20. Springer, Berlin (2007); An algorithmic approach 7. Certicom ECC Challenge, http://www.certicom.com/images/pdfs/cert_ecc_challenge.pdf.

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8. Certicom ECC Curves List, http://www.certicom.com/index.php/curves-list 9. Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications. Chapman & Hall/CRC (2006) 10. Cohen, H.: A course in computational algebraic number theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993) 11. Couveignes, J.-M., Morain, F.: Schoof’s algorithm and isogeny cycles. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 43–58. Springer, Heidelberg (1994) 12. Cox, D.A.: Primes of the form x2 + ny 2 . A Wiley-Interscience Publication, John Wiley & Sons Inc., New York (1989); Fermat, class ﬁeld theory and complex multiplication 13. Enge, A.: Computing modular polynomials in quasi-linear time. Math. Comp. 78(267), 1809–1824 (2009) 14. Fouquet, M., Morain, F.: Isogeny volcanoes and the SEA algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002) 15. Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptology (to appear 2010) 16. Galbraith, S.D.: Constructing isogenies between elliptic curves over ﬁnite ﬁelds. LMS J. Comput. Math. 2, 118–138 (1999) (electronic) 17. Galbraith, S.D., Hess, F., Smart, N.P.: Extending the GHS Weil descent attack. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 29–44. Springer, Heidelberg (2002) 18. Hafner, J., McCurley, K.: A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2(4), 837–850 (1989) 19. Hardy, K., Muskat, J.B., Williams, K.S.: A deterministic algorithm for solving n = f u2 + gv 2 in coprime integers u and v. Math. Comp. 55(191), 327–343 (1990) 20. Jao, D., Miller, S.D., Venkatesan, R.: Do all elliptic curves of the same order have the same diﬃculty of discrete log? In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 21–40. Springer, Heidelberg (2005) 21. Kohel, D.: Endomorphism rings of elliptic curves over ﬁnite ﬁelds. PhD thesis, University of California, Berkeley (1996) 22. MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/ 23. Menezes, A., Teske, E., Weng, A.: Weak ﬁelds for ECC. In: Okamoto, T. (ed.) CT-RSA 2004. LNCS, vol. 2964, pp. 366–386. Springer, Heidelberg (2004) 24. Sch¨ onhage, A.: Fast reduction and composition of binary quadratic forms. In: ISSAC 1991: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, pp. 128–133. ACM, New York (1991) 25. Schoof, R.: Counting points on elliptic curves over ﬁnite ﬁelds. J. Th´eor. Nombres Bordeaux 7(1), 219–254 (1995); Les Dix-huiti`emes Journ´ees Arithm´etiques (Bordeaux, 1993) 26. Seysen, M.: A probabilistic factorization algorithm with quadratic forms of negative discriminant. Math. Comp. 48(178), 757–780 (1987) 27. Silverman, J.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1992); Corrected reprint of the 1986 original (1986) 28. Sutherland, A.:Smoothrelation, http://math.mit.edu/~ drew/smoothrelation_v1.tar 29. Tate, J.: Endomorphisms of abelian varieties over ﬁnite ﬁelds. Invent. Math. 2, 134–144 (1966) 30. Teske, E.: An elliptic curve trapdoor system. J. Cryptology 19(1), 115–133 (2006)

Huﬀ’s Model for Elliptic Curves Marc Joye1 , Mehdi Tibouchi2, , and Damien Vergnaud2 1 Technicolor, Security & Content Protection Labs 1 avenue de Belle Fontaine, 35576 Cesson-S´evign´e Cedex, France [email protected] 2 ´ Ecole Normale Sup´erieure – C.N.R.S. – I.N.R.I.A. 45, Rue d’Ulm – 75230 Paris CEDEX 05 – France {mehdi.tibouchi,damien.vergnaud}@ens.fr

Abstract. This paper revisits a model for elliptic curves over Q introduced by Huﬀ in 1948 to study a diophantine problem. Huﬀ’s model readily extends over ﬁelds of odd characteristic. Every elliptic curve over such a ﬁeld and containing a copy of Z/4Z × Z/2Z is birationally equivalent to a Huﬀ curve over the original ﬁeld. This paper extends and generalizes Huﬀ’s model. It presents fast explicit formulæ for point addition and doubling on Huﬀ curves. It also addresses the problem of the eﬃcient evaluation of pairings over Huﬀ curves. Remarkably, the so-obtained formulæ feature some useful properties, including completeness and independence of the curve parameters. Keywords: Elliptic curves, Huﬀ’s model, uniﬁed addition law, complete addition law, explicit formulæ, scalar multiplication, Tate pairing, Miller’s algorithm.

1

Introduction

Elliptic curves have been extensively studied in algebraic geometry and number theory since the middle of the nineteenth century. More recently, they have been used to devise eﬃcient algorithms for factoring large integers [19,22] or for primality proving [2,13,23]. They also revealed useful in the construction of cryptosystems [18,20]. In this paper, we develop an elliptic curve model introduced by Huﬀ in 1948 to study a diophantine problem. We present fast explicit formulæ for adding or doubling points on Huﬀ curves. We also devise a couple of extensions and generalizations upon this model. We analyze the impact of these curves in cryptographic applications. Some of our addition formulæ are uniﬁed; i.e., they remain valid for doubling a point. Even better, they achieve completeness (i.e., are valid for all inputs) when restricted to a cyclic subgroup, as is customary in cryptographic settings. We also consider the problem of pairing computation over Huﬀ curves.

This research was completed while the second author was visiting the Okamoto Research Laboratory at the NTT Information Sharing Platform (Tokyo, Japan).

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 234–250, 2010. c Springer-Verlag Berlin Heidelberg 2010

Huﬀ’s Model for Elliptic Curves

1.1

235

Background

Elliptic curves and cryptography. In 1985, Koblitz [18] and Miller [20] independently proposed the use of elliptic curves in public-key cryptography. The main advantage of elliptic curve systems stems from the absence of a subexponentialtime algorithm to compute discrete logarithms on general elliptic curves over ﬁnite ﬁelds. Consequently, one can use an elliptic curve group that is smaller in size compared with systems based on either integer factorization or the discrete log problem in the multiplicative group of a ﬁnite ﬁeld, while maintaining the same (heuristic) level of security (see [17] for a recent survey on elliptic curve cryptography). The use of elliptic curves in cryptography makes the key sizes smaller but the arithmetic of the underlying group is more tedious (for example, with the widely-used Jacobian coordinates, the general addition of two points on an elliptic curve typically requires 16 ﬁeld multiplications). Therefore a huge amount of research has been devoted to the analysis of the performance of various forms of elliptic curves proposed in the mathematical literature: Weierstraß cubics, Jacobi intersections, Hessian curves, Jacobi quartics, or the more recent forms of elliptic curves due to Montgomery, Doche-Icart-Kohel or Edwards (see [6] for an encyclopedic overview of these models). For instance, since 2007, there has been a rapid development of the curves introduced by Edwards in [12] and their use in cryptology. Bernstein and Lange proposed a more general version of these curves in [7] and the inverted Edwards coordinates in [8]. Bernstein, Birkner, Joye, Lange, and Peters studied twisted Edwards curves in [5]. Hisil, Wong, Carter and Dawson proposed extended twisted Edwards coordinates in [14]. Bernstein, Lange, and Farashahi covered the binary case in [9]. The ﬁrst formulæ for computing pairings over Edwards curves were published by Das and Sarkar [11]. They were subsequently improved by Ionica and Joux [16]. The best implementation to date is due to Ar`ene, Lange, Naehrig, and Ritzenhaler [1]. The present paper is aimed at providing a similar study for a forgotten model of elliptic curves hinted by Huﬀ in 1948. A diophantine problem. Huﬀ [15] considered rational distance sets S (i.e., subsets S of the plane R2 such that for all s, t ∈ S, the distance between s and t is a rational number) of the following form: given distinct a, b ∈ Q, S contains the four points (0, ±a) and (0, ±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q. Such a point (x, 0) must then satisfy the equations x2 +a2 = u2 and x2 + b2 = v 2 with u, v ∈ Q. The system of associated homogeneous equations x2 + a2 z 2 = u2 and x2 + b2 z 2 = v 2 deﬁnes a curve of genus 1 in P3 . Huﬀ, and later his student Peeples [24], provided examples where this curve has positive rank over Q, thus exhibiting examples of arbitrarily large rational distance sets of cardinality k > 4 such that exactly k − 4 points are on one line. The above mentioned genus 1 curve is birationally equivalent to the curve ax(y 2 − 1) = by(x2 − 1)

(1)

for some parameters a and b in Q. It is easily seen that, over any ﬁeld K of odd characteristic, Equation (1) deﬁnes an elliptic curve if a2 = b2 and a, b = 0.

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Indeed, if ab = 0, the gradient of the curve F (X, Y, Z) = aX(Y 2 −Z 2 )−bY (X 2 − Z 2 ) in the projective plane P2 (K) is ∂F ∂F ∂F , , = a(Y 2 − Z 2 ) − 2bXY, 2aXY − b(X 2 − Z 2 ), 2(−aX + bY )Z , ∂X ∂Y ∂Z which does not vanish at the three points at inﬁnity (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0) and vanishes at a ﬁnite point (x : y : 1) if and only if ax = by, which together with Eq. (1) implies that x2 = y 2 and therefore a2 = b2 . It is worth noting that in characteristic 2, the point (1 : 1 : 1) is always singular and therefore the family of curves deﬁned by (1) does not contain any smooth curve. As will be shown in Section 3, we can extend our study to even characteristic by considering a generalized model. 1.2

Contributions of the Paper

Our ﬁrst contribution is a detailed study of Huﬀ’s form for elliptic curves over ﬁnite ﬁelds of odd characteristic and a statement of the addition law in these groups. We show in particular that all elliptic curves over non-binary ﬁnite ﬁelds with a subgroup isomorphic to Z/4Z × Z/2Z can be transformed to Huﬀ’s form. We then analyze their arithmetic and investigate several generalizations and extensions. In particular, we present explicit formulæ (i.e., as a series of ﬁeld operations) that – – – –

compute compute compute compute

a a a a

complete addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : Z2 ) using 12m; uniﬁed addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : Z2 ) using 11m; mixed addition (X1 : Y1 : Z1 ) ⊕ (X2 : Y2 : 1) using 10m; doubling [2](X1 : Y1 : Z1 ) using 6m + 5s

where m and s denote multiplications and squarings in the base ﬁeld K. As a further contribution, since bilinear pairings have found numerous applications in cryptography, we also present formulæ for computing Tate pairings using Huﬀ’s form. Speciﬁcally, we present explicit formulæ that – compute a full Miller addition using 1M + (k + 15)m; – compute a mixed Miller addition using 1M + (k + 13)m; – compute a Miller doubling using 1M + 1S + (k + 11)m + 6s on a Huﬀ curve over K = Fq of embedding degree k. M and S denote multiplications and squarings in the larger ﬁeld Fqk while m and s are operations in Fq as before. Outline. The rest of this paper is organized as follows. The next section introduces Huﬀ’s model. We develop eﬃcient uniﬁed addition formulæ and discuss the applicability of the model. We explicit the class of elliptic curves covered by Huﬀ’s model. In Section 3, we present several generalizations and extensions. We oﬀer dedicated addition formulæ. We generalize Huﬀ’s model to cover a larger class of elliptic curves. We also extend the model to the case of binary ﬁelds. Section 4 deals with pairings over Huﬀ curves. We exploit the relative simplicity of the underlying group law to devise eﬃcient formulæ for the evaluation of the Tate pairing. Finally, we conclude in Section 5.

Huﬀ’s Model for Elliptic Curves

2

237

Huﬀ’s Model

Let K denote a ﬁeld of characteristic = 2. Consider the set of projective points (X : Y : Z) ∈ P2 (K) satisfying the equation E/K : aX(Y 2 − Z 2 ) = bY (X 2 − Z 2 )

(2)

where a, b ∈ K× and a2 = b2 . This form is referred to as Huﬀ ’s model of an elliptic curve.

Fig. 1. Example of a Huﬀ curve (over R)

The tangent line at (0 : 0 : 1) is aX = bY , which intersects the curve with multiplicity 3, so that O = (0 : 0 : 1) is an inﬂection point of E. (E, O) is therefore an elliptic curve with O as neutral element and whose group law, denoted ⊕, has the following property: for any line intersecting the cubic curve E at the three points P1 , P2 and P3 (counting multiplicities), we have P1 ⊕ P2 ⊕ P3 = O. In particular, the inverse of point P1 = (X1 : Y1 : Z1 ) is P1 = (X1 : Y1 : −Z1 ) and the sum of P1 and P2 is P1 ⊕ P2 = P3 . We note that a point at inﬁnity is its own inverse. Hence, the three points at inﬁnity (i.e., on the line Z = 0 in P2 ) — namely, (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0), are exactly the three primitive 2-torsion points of E. The sum of any two of them is equal to the third one. More generally, (X1 : Y1 : Z1 ) ⊕ (1 : 0 : 0) is the inverse of the point of intersection of the “horizontal” line passing through (X1 : Y1 : Z1 ) with E. When Z1 = 0, we have (X1 : Y1 : Z1 ) ⊕ (1 : 0 : 0) = (Z1 2 : −X1 Y1 : X1 Z1 ) , and analogously, (X1 : Y1 : Z1 ) ⊕ (0 : 1 : 0) = (−X1 Y1 : Z1 2 : Y1 Z1 ) .

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From (a : b : 0) = (1 : 0 : 0) ⊕ (0 : 1 : 0), when Z1 = 0, we get (X1 : Y1 : Z1 ) + (a : b : 0) = (Z1 2 : −X1 Y1 : X1 Z1 ) ⊕ (0 : 1 : 0) and therefore (a : b : 0) if (X1 : Y1 : Z1 ) = (0 : 0 : 1) . (X1 : Y1 : Z1 ) ⊕ (a : b : 0) = (Y1 Z1 : X1 Z1 : −X1 Y1 ) otherwise We remark that adding (a : b : 0) to any of the points (±1 : ±1 : 1) transforms it into its inverse. It follows that these four points are the four solutions to the equation [2]P = (a : b : 0) and so are primitive 4-torsion points. The eight remarkable points we identiﬁed form a subgroup isomorphic to Z/4Z × Z/2Z. When K = Q, this must be the full torsion since, according to a theorem by Mazur, the torsion subgroup is of order at most 12 (and thus exactly 8 here). Remark 1. In [15, p. 445], it is noted that the inverse projective transformations Υ : P2 (K) → P2 (K) :

(X : Y : Z) → (U : V : W ) = ab(bX − aY ) : ab(b2 − a2 )Z : −aX + bY

and Υ −1 : P2 (K) → P2 (K) :

(U : V : W ) → (X : Y : Z) = b(U + a2 W ) : a(U + b2 W ) : V

induce a correspondence between Eq. (2) and the Weierstraß equation V 2 W = U (U + a2 W )(U + b2 W ) . Observe that point at inﬁnity (0 : 1 : 0) on the Weierstraß curve is mapped to (0 : 0 : 1) on the Huﬀ curve through Υ −1 . Observe also that map Υ −1 is a line-preserving transformation. This is another way to see that the group law on a Huﬀ curve E follows the chord-and-tangent rule [25, § 2] with O = (0 : 0 : 1) as neutral element. 2.1

Aﬃne Formulæ

We give explicit formulæ for the group law. Excluding the 2-torsion, we use the non-homogeneous form ax(y 2 − 1) = by(x2 − 1). Let y = λ x + μ denote the secant line passing through two diﬀerent points P1 = (x1 , y1 ) and P2 = (x2 , y2 ). This line intersects the curve at a third point P3 = (−x3 , −y3 ). Plugging the line equation into the curve equation, we get ax (λx+ μ)2 − 1 = b(λx+ μ)(x2 − 1) =⇒ λ(aλ− b)x3 + μ(2aλ− b)x2 + · · · = 0 . Whenever deﬁned, we so obtain ⎧ ⎨x = x + x + μ(2aλ − b) 3 1 2 λ(aλ − b) ⎩ y3 = λx3 − μ

Huﬀ’s Model for Elliptic Curves

with λ =

239

y1 − y2 and μ = y1 − λx1 . After simpliﬁcation, we have x1 − x2 (x1 y2 − x2 y1 ) 2a(y1 − y2 ) − b(x1 − x2 ) x3 = x1 + x2 + (y1 − y2 ) a(y1 − y2 ) − b(x1 − x2 ) (x1 − x2 ) a(y1 2 − y2 2 ) − b(x1 y1 − x2 y2 ) = (y1 − y2 ) a(y1 − y2 ) − b(x1 − x2 )

and (y1 − y2 ) b(x1 2 − x2 2 ) − a(x1 y1 − x2 y2 ) y3 = − . (x1 − x2 ) a(y1 − y2 ) − b(x1 − x2 ) The above formulæ can be further simpliﬁed by reusing the curve equation. A simple calculation shows that a(y1 − y2 ) − b(x1 − x2 ) (x1 + x2 )y1 y2 = a(x2 y1 − x1 y2 )(y1 y2 − 1) . Hence, we can write

2a(y1 − y2 ) − b(x1 − x2 ) (x1 + x2 )y1 y2 (y1 − y2 )a(y1 y2 − 1) x2 y1 − x1 y2 (x1 + x2 )y1 y2 − = x1 + x2 − y1 − y2 y1 y2 − 1 (x1 + x2 )y1 y2 x1 y1 − x2 y2 . − = y1 − y2 y1 y2 − 1

x3 = x1 + x2 −

Furthermore, as easily shown b(x1 y1 − x2 y2 )(x1 x2 + 1) = (y1 − y2 ) ax1 x2 (y1 + y2 ) + b(x1 + x2 ) , it thus follows that ax1 x2 (y1 + y2 ) + b(x1 + x2 ) (x1 + x2 )y1 y2 − b(x1 x2 + 1) y1 y2 − 1 (x1 + x2 )(1 + y1 y2 ) , = (1 + x1 x2 )(1 − y1 y2 )

x3 =

(3)

since ax1 x2 (y1 + y2 )(1 − y1 y2 ) = by1 y2 (x1 + x2 )(1 − x1 x2 ). Likewise, by symmetry, we have y3 =

(y1 + y2 )(1 + x1 x2 ) . (1 − x1 x2 )(1 + y1 y2 )

(4)

Equations (3) and (4) are deﬁned whenever x1 x2 = ±1 and y1 y2 = ±1. Advantageously, curve parameters are not involved. Moreover, this addition law is uniﬁed : it can be used to double a point (i.e., when P2 = P1 ).

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Projective Formulæ

Previous aﬃne formulæ involve inversions in K. To avoid these operations and get faster arithmetic, projective coordinates may be preferred. We let m and s represent the cost of a multiplication and of a squaring in K, respectively. The projective form of Eqs (3) and (4) is ⎧ 2 ⎪ ⎨X3 = (X1 Z2 + X2 Z1 )(Y1 Y2 + Z1 Z2 ) (Z1 Z2 − X1 X2 ) . (5) Y3 = (Y1 Z2 + Y2 Z1 )(X1 X2 + Z1 Z2 )2 (Z1 Z2 − Y1 Y2 ) ⎪ ⎩ 2 2 2 2 2 2 2 2 Z3 = (Z1 Z2 − X1 X2 )(Z1 Z2 − Y1 Y2 ) In more detail, this can be evaluated as m1 = X1 X2 , m2 = Y1 Y2 , m3 = Z1 Z2 , m4 = (X1 + Z1 )(X2 + Z2 ) − m1 − m3 , m5 = (Y1 + Z1 )(Y2 + Z2 ) − m2 − m3 , m6 = (m2 + m3 )(m3 − m1 ), m7 = (m1 + m3 )(m3 − m2 ), m8 = m4 (m2 + m3 ), m9 = m5 (m1 + m3 ), X3 = m8 m6 , Y3 = m9 m7 , Z3 = m6 m7 , that is, with 12m. 2.3

Applicability

If (x1 , y1 ) = (0, 0) then (x1 , y1 ) ⊕ (a : b : 0) = −( x11 , y11 ). Observe that Equation (5) remains valid for doubling point (a : b : 0) or for adding point (a : b : 0) to another ﬁnite point (i.e., which is not at inﬁnity) diﬀerent from O; we get (X1 : Y1 : Z1 ) ⊕ (a : b : 0) = (−Y1 Z1 : −X1 Z1 : X1 Y1 ) as expected. The addition formula is however not valid for adding (0 : 1 : 0) or (1 : 0 : 0). More generally, we have: Theorem 1. Let K be a ﬁeld of characteristic = 2. Let P1 = (X1 : Y1 : Z1 ) and P2 = (X2 : Y2 : Z2 ) be two points on a Huﬀ curve over K. Then the addition formula given by Eq. (5) is valid provided that X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 . Proof. If P1 and P2 are ﬁnite, we can write P1 = (x1 , y1 ) and P2 = (x2 , y2 ). The above aﬃne formula for (x3 , y3 ) as given by Eqs (3) and (4) is deﬁned whenever x1 x2 = ±1 and y1 y2 = ±1. This translates into X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 for their projective coordinates. It remains to analyze points at inﬁnity. The points with their Z-coordinate equal to 0 are (1 : 0 : 0), (0 : 1 : 0) and (a : b : 0). If P1 or P2 ∈ {(1 : 0 : 0), (0 : 1 : 0)}, the condition X1 X2 = ±Z1 Z2 and Y1 Y2 = ±Z1 Z2 is not satisﬁed. Suppose now P2 = (a : b : 0). The condition becomes X1 = 0 and Y1 = 0, which corresponds to P1 ∈ / {O, (1 : 0 : 0), (0 : 1 : 0)}. As aforementioned, the addition law is then valid for adding P1 to (a : b : 0).

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The previous theorem says that the addition on a Huﬀ curve is almost complete. However, the exceptional inputs are easily prevented in practice. Cryptographic applications typically involve (large) prime-order subgroups. More speciﬁcally, we state: Corollary 1. Let E be a Huﬀ curve over a ﬁeld K of odd characteristic. Let also P ∈ E(K) be a point of odd order. Then the addition law in the subgroup generated by P is complete. Proof. All points in P are of odd order and thus are ﬁnite (remember that points at inﬁnity are of order 2). It remains to show that for any points P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ P , we have x1 x2 = ±1 and y1 y2 = ±1. Note that x1 , y1 , x2 , y2 = ±1 since this corresponds to points of order 4 (and thus not in P ). Suppose that x1 x2 = ±1. Then ax1 (y1 2 − 1) = by1 (x1 2 − 1) =⇒ a x11 (y1 2 − 1) = by1 (1 − x11 2 ) =⇒ ±ax2 (y1 2 − 1) = −by1 (x2 2 − 1). Hence, since ax2 (y2 2 − 1) = by2 (x2 2 − 1), it follows that ∓y2 (y1 2 − 1) = y1 (y2 2 − 1) =⇒ (y1 ± y2 )(1 ∓ y1 y2 ) = 0 =⇒ y2 = ∓y1 or y1 y2 = ±1. As a result, when x1 x2 = ±1, we have (x2 , y2 ) ∈ ( x11 , −y1 ), ( x11 , y11 ), (− x11 , y1 ), (− x11 , − y11 ) . In all cases, one of (x1 , y1 ) ⊕ (x2 , y2 ) or (x1 , y1 ) (x2 , y2 ) is a 2-torsion point, a contradiction. Likewise, it can be veriﬁed that the case y1 y2 = ±1 leads to a contradiction, which concludes the proof.

The completeness of the addition law is very useful as it yields a natural protection against certain side-channel attacks (e.g., see [10]). Another useful feature is that the addition law is independent of the curve parameters. 2.4

Universality of the Model

The next theorem states that every elliptic curve over a ﬁeld of characteristic = 2 containing a copy of Z/4Z×Z/2Z can be put in Huﬀ’s form. Generalizations and extensions are discussed in Section 3. Theorem 2. Any elliptic curve (E, O) over a perfect ﬁeld K of characteristic = 2 such that E(K) contains a subgroup G isomorphic to Z/4Z × Z/2Z is birationally equivalent over K to a Huﬀ curve. Proof. The Riemann-Roch theorem implies that if D = a1 P1 + · · · + ar Pr is a divisor of degree 0 on E then the dimension of the vector space L (D) = {f ∈ K(E)× | div(f ) −D} ∪ {0} is equal to 1 when a1 P1 ⊕ · · · ⊕ ar Pr = O, and to 0 otherwise. Let H++ , H+− , H−+ and H−− denote the four points of G of order exactly 4 (with the convention H++ ⊕ H−− = O). Doubling these points produces a unique primitive 2-torsion point that we denote R. We further let P and Q denote the other two 2-torsion points; say, P = H++ ⊕ H+− and Q = H++ ⊕ H+− . We have P ⊕ R Q O = O; so there exists a nonzero

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rational function x with divisor exactly Q + O − P − R. In particular, x is well-deﬁned and nonzero at H++ and thus without loss of generality we may assume that x(H++ ) = 1. Similarly, there exists a rational function y with divisor P + O − Q − R such that y(H++ ) = 1. The rational function x − 1 has the same poles as x and vanishes at H++ . Its divisor div(x − 1) is thus given by H++ + X − P − R for some point X. Since this divisor is principal, we have H++ ⊕ X P R = O. Hence, it follows that X = P ⊕ R H++ = H++ ⊕ H+− ⊕ R H++ = H+− . Consequently, we have x(H+− ) = 1. Likewise, it is veriﬁed that y(H−+ ) = 1. Now, consider the map ι taking a rational function f to ιf : M → f (M). This is an endomorphism of the vector space L (P + R − Q − O). Indeed, the poles of ιf are P = P and R = R and its zeros are Q = Q and O = O. Moreover, since ι2 = id and since L (P + R − Q − O) is a onedimensional vector space, ι is the multiplication map by 1 or −1. The equality ιx = x would imply x(H−− ) = x(H++ ) = 1, which contradicts the previous calculation of div(x − 1). As a result, we must have ιx = −x. In particular, noting that H−+ = H+− , we obtain x(H−+ ) = ιx(H+− ) = −x(H+− ) = −1 , and similarly for H−− . Since x + 1 has the same poles as x, its divisor is then given by div(x+1) = H−+ +H−− −P −R. Analogously, we obtain div(y +1) = H+− + H−− − Q − R. Finally, consider the rational functions u = x(y 2 − 1) and v = y(x2 − 1). We have: div(u) = div(x) + div(y − 1) + div(y + 1) = (Q + O − P − R) + (H++ + H−+ − Q − R) + (H+− + H−− − Q − R) = H++ + H+− + H−+ + H−− + O − P − Q − 3R and div(v) = div(y) + div(x − 1) + div(x + 1) = (P + O − Q − R) + (H++ + H+− − P − R) + (H−+ + H−− − P − R) = H++ + H+− + H−+ + H−− + O − P − Q − 3R . But the vector space L (P + Q + 3R − O − H++ − H+− − H−+ − H−−) is of dimension 1, so there exists a linear relation between u and v. In other words, there exist a, b ∈ K× such that au = bv; i.e., such that ax(y 2 − 1) = by(x2 − 1). The rational map E → P2 (K) given by M → (x(M ) : y(M ) : 1) extends to a morphism deﬁned on all of E, and its image is contained in Ea,b in view of the previous relation (and Ea,b itself is a smooth irreducible curve as seen in §1.1). We therefore have a non-constant — and hence surjective — morphism of curves E → Ea,b . Moreover, its degree is at most 1: indeed, if a point (x0 : y0 : 1) ∈ Ea,b (K) has two distinct pre-images M = M ∈ E(K), the functions x − x0 and

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y − y0 vanish at M and M . Since they have the same poles as x and y, their divisors are respectively M + M − P − R and M + M − Q − R, which yields P ⊕ R = M ⊕ M = Q ⊕ R, a contradiction. As a surjective morphism of degree 1, the map E → Ea,b is thus an isomorphism.

3

Generalizations and Extensions

This section presents dedicated addition formulæ. It also presents a generalization of the model as originally introduced by Huﬀ so that it covers more curves and extends to binary ﬁelds. 3.1

Faster Computations

Dedicated doubling. The doubling formula can be sped up by evaluating squarings in K with a specialized implementation. The cost of a point doubling then becomes 7m + 5s. When s > 34 m, an even faster way for doubling a point is given by m1 = X1 Y1 , m2 = X1 Z1 , m3 = Y1 Z1 , s1 = Z1 2 , m4 = (m2 − m3 )(m2 + m3 ), m5 = (m1 − s1 )(m1 + s1 ), m6 = (m1 − s1 )(m2 − m3 ), m7 = (m1 + s1 )(m2 + m3 ), X([2]P1 ) = (m6 − m7 )(m4 + m5 ), Y ([2]P1 ) = (m6 + m7 )(m4 − m5 ), Z([2]P1 ) = (m4 + m5 )(m4 − m5 ), that is, with 10m + 1s. Moving the origin. Choosing O = (0 : 1 : 0) as the neutral element results in translating the group law. If we let ⊕ denote the corresponding point addition, we have P1 ⊕ P2 = (P1 O ) ⊕ (P2 O ) ⊕ O = P1 ⊕ P2 ⊕ O . Hence, we get ⎧ ⎪ ⎨X3 = (X1 Z2 + X2 Z1 )(Y1 Y2 + Z1 Z2 )(Y1 Z2 + Y2 Z1 ) Y3 = (X1 X2 − Z1 Z2 )(Z1 2 Z2 2 − Y1 2 Y2 2 ) ⎪ ⎩ Z3 = (Y1 Z2 + Y2 Z1 )(X1 X2 + Z1 Z2 )(Y1 Y2 − Z1 Z2 )

.

This can be evaluated with 11m as m1 = X1 X2 , m2 = Y1 Y2 , m3 = Z1 Z2 , m4 = (X1 + Z1 )(X2 + Z2 ) − m1 − m3 , m5 = (Y1 + Z1 )(Y2 + Z2 ) − m2 − m3 , X3 = m4 (m2 + m3 )m5 , Y3 = (m1 − m3 )(m3 − m2 )(m3 + m2 ), Z3 = m5 (m1 + m3 )(m2 − m3 ) . (6) This addition formula is uniﬁed: it can be used for doubling as well.

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For a mixed point addition (i.e., when Z2 = 1), we have m3 = Z1 and the number of required multiplications drops to 10m. When used for dedicated doubling, the above addition formula requires 6m + 5s, which can equivalently be obtained as s1 = X1 2 , s2 = Y1 2 , s3 = Z1 2 , s4 = (X1 + Y1 )2 − s1 − s2 , s5 = (Y1 + Z1 )2 − s2 − s3 , X([2]P1 ) = 2s3 s4 (s2 + s3 ), Y ([2]P1 ) = (s1 − s3 )(s3 − s2 )(s3 + s2 ),

(7)

Z([2]P1 ) = s5 (s1 + s3 )(s2 − s3 ) . Note that the expression for the inverse of point P1 is unchanged: P1 = (P1 O ) ⊕ O = P1 = (X1 : Y1 : −Z1 ). 3.2

More Formulæ

Alternative addition formulæ can be derived using the curve equation. For example, whenever deﬁned, we can write (x3 , y3 ) = (x1 , y1 ) ⊕ (x2 , y2 ) with x3 =

(x1 − x2 )(y1 + y2 ) (y1 − y2 )(1 − x1 x2 )

and y3 =

(y1 − y2 )(x1 + x2 ) . (x1 − x2 )(1 − y1 y2 )

In projective coordinates, this gives ⎧ 2 ⎪ ⎨X3 = (X1 Z2 − X2 Z1 ) (Y1 Z2 + Y2 Z1 )(Z1 Z2 − Y1 Y2 ) Y3 = (Y1 Z2 − Y2 Z1 )2 (X1 Z2 + X2 Z1 )(Z1 Z2 − X1 X2 ) ⎪ ⎩ Z3 = (X1 Z2 − X2 Z1 )(Y1 Z2 − Y2 Z1 )(Z1 Z2 − X1 X2 )(Z1 Z2 − Y1 Y2 )

,

which can be evaluated with 13m as m1 = X1 Z2 , m2 = X2 Z1 , m3 = Y1 Z2 , m4 = Y2 Z1 , m5 = (Z1 − X1 )(Z2 + X2 ) + m1 − m2 , m6 = (Z1 − Y1 )(Z2 + Y2 ) + m3 − m4 , m7 = (m1 − m2 )m6 , m8 = (m3 − m4 )m5 , X3 = (m1 − m2 )(m3 + m4 )m7 , Y3 = (m1 + m2 )(m3 − m4 )m8 , Z3 = m7 m8 . Although not as eﬃcient as the usual addition, this alternative formula is useful in some pairing computations (see Section 4.2). 3.3

Twisted Curves

As shown in Theorem 1, the group of points of a Huﬀ elliptic curve contains a copy of Z/4Z×Z/2Z. This implies that the curve order is a multiple of 8. Several cryptographic standards, however, require elliptic curves with group order of the form h n where h ∈ {1, 2, 3, 4} and n is a prime. We can generalize Huﬀ’s model to accommodate the case h = 4. Let P ∈ K[t] denote a monic polynomial of degree 2, with non-zero discriminant, and such that P(0) = 0. We can then introduce the cubic curve axP(y) = byP(x)

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where a, b ∈ K× . The set of points {(0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0), (a : b : 0)} ∼ = Z/2Z × Z/2Z belongs to the curve. Moreover, when P factors in K — i.e., when P(t) = (t − ω1 )(t − ω2 ) with ω1 , ω2 ∈ K× , the four points (±ω1 : ±ω2 : 1) are also on the curve. When Char K = 2, we consider P(t) = t2 − d for some d ∈ K× . So we deal with the set of projective points (X : Y : Z) ∈ P2 (K) satisfying the non-singular cubic equation (8) Eˆd : aX(Y 2 − dZ 2 ) = bY (X 2 − dZ 2 ) where a, b, d ∈ K× and a2 = b2 . This equation corresponds to Weierstraß equa2 2 (X : Y : tion V 2 W = U (U + ad W )(U + bd W ) under the inverse transformations Z) = b(dU + a2 W ) : a(dU + b2 W ) : dV and (U : V : W ) = ab(bX − aY ) : √ ab(b2 − a2 )Z : d(−aX + bY ) . The transformation (X : Y : Z) ← (X : Y : Z d) √ ˆd over K( d). Curves Eˆd are therefore induces an isomorphism from E = Eˆ1 to E quadratic twists of Huﬀ curves. In aﬃne coordinates, we consider the curve equation ax(y 2 − d) = by(x2 − d). The sum of two ﬁnite points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) such that x1 x2 = ±d and y1 y2 = ±d is given by (x3 , y3 ) where x3 =

d(x1 + x2 )(d + y1 y2 ) (d + x1 x2 )(d − y1 y2 )

and y3 =

d(y1 + y2 )(d + x1 x2 ) . (d − x1 x2 )(d + y1 y2 )

(9)

Extending the computations of § 2.2, it is readily veriﬁed that the sum of two points can be evaluated with 12m (plus a couple of multiplications by constant d) using projective coordinates. The faster computations of the previous section also generalize to twisted curves. 3.4

Binary Fields

Huﬀ’s form can be extended to a binary ﬁeld as ax(y 2 + y + 1) = by(x2 + x + 1) . This curve is birationally equivalent to Weierstraß curve v(v + (a + b)u) = u(u + a2 )(u + b2 ) under the inverse maps b(u + a2 ) a(u + b2 ) (x, y) = , v v + (a + b)u

and (u, v) =

ab ab(axy + b) , xy x2 y

.

The neutral element is O = (0, 0).

4 4.1

Pairings Preliminaries

Let (E, O) be an elliptic curve over K = Fq , with q odd. Suppose that #E(Fq ) = hn where n is a prime such that gcd(n, q) = 1. Let further k denote the

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embedding degree with respect to n, namely the smallest extension Fqk of Fq containing all n-th roots of unity. In other words, k is the smallest positive integer k such that n | q k − 1. For better eﬃciency, we further assume that k > 1 is even. For any point P ∈ E(Fq )[n], we let fP denote a rational function on E deﬁned over Fq such that div(fP ) = nP − nO; it exists and is unique up to a multiplicative constant, according to the Riemann-Roch theorem. The group of n-th roots of unity in Fqk is denoted by μn . The (reduced) Tate pairing is then deﬁned as Tn : E(Fq )[n] × E(Fqk )/[n]E(Fqk ) → μn : (P , Q) → fP (Q)(q

k

−1)/n

.

This deﬁnition does not depend on the choice of fP with the appropriate divisor, nor on the class of Q mod [n]E(Fqk ). In practice, Tn can be computed using a technique due to Miller [21], in terms of rational functions gR,P depending on P and on a variable point R. Function gR,P is the so-called line function with divisor R + P − O − (R ⊕ P ), which arises in addition formulæ when E is represented as a plane cubic. The core idea is to derive function fP iteratively. Letting fi,P be the function with divisor div(fi,P ) = iP − ([i]P ) − (i − 1)O, it is easily veriﬁed that fi+j,P = fi,P · fj,P · g[i]P ,[j]P . Observe that f1,P = 1 and fn,P = fP . Hence, if n = n−1 n−1 · · · n0 2 is the binary representation of n, the Tate pairing can be computed as follows.

Algorithm 1. Miller’s algorithm 1: 2: 3: 4: 5: 6: 7: 8:

f ← 1; R ← P for i = − 2 down to 0 do f ← f 2 · gR,R (Q); R ← [2]R if (ni = 1) then f ← f · gR,P (Q); R ← R ⊕ P end if end for k return f (q −1)/n

Contrary to Edwards curves or Jacobi quartics, Huﬀ curves are represented as plane cubics. This makes Miller’s algorithm, along with a number of improvements proposed for Weierstraß curves (e.g., as presented in [3]), directly applicable to the computation of pairings over Huﬀ curves. 4.2

Pairing Formulæ for Huﬀ Curves

Throughout the for-loop of Algorithm 1, the line function is always evaluated at the same point Q ∈ E(Fqk ) \ E(Fq ). It is therefore customary to represent

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this point in aﬃne coordinates. In our case, it is most convenient to choose the coordinates of Q as Q = (y, z) = (1 : y : z). Indeed, since the embedding degree k is even, the ﬁeld Fqk can be represented as Fqk/2 (α), where α is any quadratic non-residue in Fqk/2 . As a result, Q can be chosen of the form Q = (yQ , zQ α) with yQ , zQ ∈ Fqk/2 [4]. To do so, it suﬃces to pick a point on a quadratic twist of E over Fqk/2 and take its image under the isomorphism over Fqk . Now, for any two points R, P in E(Fq ), let R,P denote the rational function vanishing on the line through R and P . In general, we have R,P (Q) =

(zXP − ZP ) − λ(yXP − YP ) YP

where λ is the “(y, z)-slope” of the line through R and P . Then, the divisor of R,P is div(R,P ) = R + P + T − (1 : 0 : 0) − (0 : 1 : 0) − (a : b : 0) where T is the third point of intersection (counting multiplicities) of the line through R and P with the elliptic curve. In particular, if the neutral element of the group law ⊕ is denoted by U , the line function gR,P can be written as gR,P =

R,P R⊕P ,U

.

We concentrate on the case when U = O = (0 : 0 : 1). Then for any Q = (yQ , zQ α), we have R⊕P ,O (Q) = yQ −

YR⊕P ∈ Fqk/2 . XR⊕P

Since this quantity lies in a proper subﬁeld of Fqk , it goes to 1 after the ﬁnal exponentiation in Miller’s algorithm, which means that it can be discarded altogether. Similarly, divisions by XP can be omitted, and denominators in the expression of λ can be canceled. In other words, if λ = A/B, we can compute the line function as gR,P (Q) = (zXP − ZP ) · B − (yXP − YP ) · A and get the required result. We can now detail precise formulæ for the addition and doubling steps in the so-called Miller loop (i.e., the main for-loop in Algorithm 1). We let M and S represent the cost of a multiplication and of a squaring in Fqk while m and s are operations in Fq as before. Addition step. In the case of addition, the (y, z)-slope of the line through R = (XR : YR : ZR ) and P = (XP : YP : ZP ) is λ=

ZR XP − ZP XR . YR XP − YP XR

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Therefore, the line function to be evaluated is of the form gR,P (Q) = (zQ α·XP −ZP )(YR XP −YP XR )−(yQ ·XP −YP )(ZR XP −ZP XR ) . Since P and Q are constant throughout the loop, the values depending only on P and Q — in this case yQ = yQ · XP − YP and z Q = zQ α · XP , can be precomputed. Then, each Miller addition step requires computing R ⊕ P (one addition on the curve over Fq ), evaluating gR,P (Q), and computing f · gR,P (Q) (one multiplication in the ﬁeld Fqk ). We consider two types of Miller addition steps: full addition, for which no assumption is made on the representation of P , and mixed addition, for which we further assume that P is given in aﬃne coordinates (i.e., XP = 1). Both steps start with computing R ⊕ P , including all intermediate results. Full addition. Computing R ⊕ P requires 13m using the dedicated addition formula from §3.1, including all intermediate results m1 , . . . , m8 . Compute further m9 = (XR + YR )(XP − YP ). We then have gR,P (Q) = (z Q − ZP )(m9 + m5 − m6 ) − yQ (m1 − m2 ) where the ﬁrst term requires ( k2 + 1)m and the second term k2 m. With the ﬁnal multiplication over Fqk , the total cost of full addition is thus of 1M + (k + 15)m. Mixed addition. Now that XP = 1, computing R ⊕ P using the formula from §2.2, including all the intermediate results m1 , . . . , m9 , only requires 11m, since the computation of m1 is free. We then have gR,P (Q) = (z Q − ZP )(YR − YP XR ) − y Q (2ZR − m4 ) where both terms require the same number of multiplications as before, plus one for YP XR . The total cost of mixed addition is thus of 1M + (k + 13)m. Doubling step. In the case of doubling, the (y, z)-slope of the tangent line at R = (XR : YR : ZR ) is λ=

a(ZR )2 − 2bYR ZR − a(XR )2 A . = b(YR )2 − 2aYR ZR − b(XR )2 B

Thus, the line function is of the form gR,R (Q) = zQ α · XR B − ZR B − yQ · XR A + YR A . Miller’s doubling involves computing the point [2]R, which we do using the formulæ from §2.2 in 7m + 5s. Then the quantities A and B are obtained by computing the additional product m10 = 2YR ZR = (YR + ZR )2 − m2 − m3 using a single squaring. Computing gR,R (Q) requires multiplying those two values by XR and YR (resp. XR and ZR ), hence an additional 4m. And ﬁnally, multiplications by yQ and zQ α both require k2 m. Taking into account the multiplication and the squaring in Fqk needed to complete the doubling step, the total cost of Miller doubling is thus of 1M + 1S + (k + 11)m + 6s.

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249

Conclusion

This paper introduced and studied Huﬀ’s model, a new representation of elliptic curves to be considered alongside previous models such as Montgomery, Doche-Icart-Kohel and Edwards. This new model provides eﬃcient arithmetic, competitive with some of the fastest known implementations (although not quite as fast as “inverted Edwards” for now). Moreover, it has a number of additional desirable properties, including uniﬁed/complete addition laws and formulæ that do not depend on curve parameters (both properties are useful in cryptographic applications to thwart certain implementation attacks). It is also suitable to other computations on elliptic curves, such as the evaluation of pairings. We believe that this model is worthy of consideration by the community, and hope our contribution might spark further research into eﬃcient implementations of elliptic curve arithmetic. Acknowledgments. We are grateful to an anonymous referee for useful comments. This work was partly supported by the French ANR-07-TCOM-013-04 PACE Project and by the European Commission through the IST Program under Contract ICT-2007-216646 ECRYPT II.

References 1. Ar`ene, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster computation of the Tate pairing. In: Cryptology ePrint Archive, Report 2009/155 (2009), http://eprint.iacr.org/ 2. Atkin, A.O.L., Morain, F.: Elliptic curves and primality proving. Math. Comp. 61(203), 29–68 (1993) 3. Barreto, P.S.L.M., Lynn, B., Scott, M.: Eﬃcient implementation of pairing-based cryptosystems. J. Cryptology 17(4), 321–334 (2004) 4. Barreto, P.S.L.M., Lynn, B., Scott, M.: On the selection of pairing-friendly groups. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 17–25. Springer, Heidelberg (2004) 5. Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389– 405. Springer, Heidelberg (2008) 6. Bernstein, D.J., Lange, T.: Explicit-formulas database, http://www.hyperelliptic.org/EFD/ 7. Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007) 8. Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: Bozta¸s, S., Lu, H.F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007) 9. Bernstein, D.J., Lange, T., Farashahi, R.R.: Binary Edwards curves. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 244–265. Springer, Heidelberg (2008) 10. Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography, ch. V. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005)

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11. Das, M.P.L., Sarkar, P.: Pairing computation on twisted Edwards form elliptic curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 192–210. Springer, Heidelberg (2008) 12. Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc., New Ser. 44(3), 393–422 (2007) 13. Goldwasser, S., Kilian, J.: Primality testing using elliptic curves. J. ACM 46(4), 450–472 (1999) 14. Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards curves revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326–343. Springer, Heidelberg (2008) 15. Huﬀ, G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15, 443–453 (1948) 16. Ionica, S., Joux, A.: Another approach to pairing computation in Edwards coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008) 17. Koblitz, A.H., Koblitz, N., Menezes, A.: Elliptic curve cryptography: The serpentine course of a paradigm shift. J. Number Theory (to appear) 18. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987) 19. Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. 126(2), 649– 673 (1987) 20. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986) 21. Miller, V.S.: The Weil paring, and its eﬃcient implementation. J. Cryptology 17(1), 235–261 (2004) 22. Montgomery, P.L.: Speeding up the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48(177), 243–264 (1987) 23. Morain, F.: Primality proving using elliptic curves: An update. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 111–127. Springer, Heidelberg (1998) 24. Peeples Jr., W.D.: Elliptic curves and rational distance sets. Proc. Am. Math. Soc. 5, 29–33 (1954) 25. Silverman, J.H.: The Arithmetic of Elliptic Curves, ch III. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986)

Eﬃcient Pairing Computation with Theta Functions David Lubicz1,2 and Damien Robert3 1

2

DGA-MI, BP 7419, F-35174 Bruz IRMAR, Universt´e de Rennes 1, Campus de Beaulieu, F-35042 Rennes 3 LORIA, CARAMEL Project, Campus Scientiﬁque, BP 239, 54506 Vandoeuvre-l`es-Nancy Cedex

Abstract. In this paper, we present a new approach based on theta functions to compute Weil and Tate pairings. A beneﬁt of our method, which does not rely on the classical Miller’s algorithm, is its generality since it extends to all abelian varieties the classical Weil and Tate pairing formulas. In the case of dimension 1 and 2 abelian varieties our algorithms lead to implementations which are eﬃcient and naturally deterministic. We also introduce symmetric Weil and Tate pairings on Kummer varieties and explain how to compute them eﬃciently. We exhibit a nice algorithmic compatibility between some algebraic groups quotiented by the action of the automorphism −1, where the Z-action can be computed eﬃciently with a Montgomery ladder type algorithm.

1

Introduction

In recent years, many new and interesting cryptographic protocols have been proposed which use the existence of pairings on abelian varieties. In order to obtain eﬃcient and secure implementations of these protocols it is important to be able to compute quickly these pairings. Miller has proposed a method (see for instance [2]) to compute the function on an algebraic curve given up to a constant factor by the data of a principal divisor. This method is a key ingredient of all known algorithms to compute pairings. In this paper, we propose a diﬀerent approach based on theta functions. We ﬁrst make explicit the link between Weil and Tate pairings and the intersection pairing on the degree 1 homology of an abelian variety. Our method appears to be a very natural and straightforward way to compute the pairing associated to the Riemann form (or its arithmetic counterpart the commutator pairing) of an abelian variety. It is then easy to deduce practical formulas to compute Weil and Tate pairings. A ﬁrst beneﬁt of our approach is its generality: where Miller’s algorithm rely on the representation of an abelian variety as the Jacobian of an algebraic curve, our method works with any abelian varieties. The case of the Tate pairing is noticeable: while the original deﬁnition of Tate [8] deals with any abelian varieties, the formula of Lichtenbaum [9] used in cryptographic applications is restricted to Jacobian of curves. This restriction does not appear in our formulas. Our algorithm also G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 251–269, 2010. c Springer-Verlag Berlin Heidelberg 2010

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expand the algorithmic toolbox based on theta functions to compute with abelian varieties. For the complexity analysis of our algorithm we focus on the case of level 2 and 4 theta functions in order to obtain the best running time and memory consumption. The only diﬀerence between the two cases lies in the initialisation phase of the algorithm: in level 4 one can recover enough information from the data of two points to compute the pairings. This is not possible with the level 2 embedding since it does not distinguish a point and its opposite. Nonetheless it is possible to deﬁne a “symmetric pairing” on the quotient of an abelian variety by the action of the automorphism −1. These notions extend the deﬁnition of the trace pairing proposed in [3]. We have chosen to present all the formulas of this paper using the classical analytic theory of theta functions. In order to consider also rationality problems which are essential to the deﬁnition of the Tate pairing, we make the assumption that all the abelian varieties that we consider are deﬁned over a number ﬁeld K and we suppose given a ﬁxed embedding of K in its algebraic closure C. Nonetheless, it should be understood that all our algorithms apply to the case of abelian varieties deﬁned over any ﬁeld of characteristic not equal to 2. To see this one can invoke the Lefschetz’s principle or use Mumford’s theory of algebraic theta functions. We refer to [10] for proofs of the main formulas of this paper in the theory of Mumford. Our paper in organized as follows: in Section 2 we recall some basic deﬁnitions about theta functions. The Section 3 we give a method to compute the usual pairings by using a double and add algorithm based a theta addition formula. In Section 5 we make a precise assessment about the complexity of our algorithm. We also introduce symmetric pairings on Kummer varieties and explain how to adapt our algorithms to compute them eﬃciently. We end the paper with an example of computation in Section 6.

2

Some Notations and Basic Facts

In this section, in order to ﬁx the notations, we recall some well known facts on analytic theta functions (see for instance [14,6]). Let Hg be the g dimensional Siegel upper-half space which is the set of g × g symmetric matrices Ω whose imaginary part is positive deﬁnite. For Ω ∈ Hg , we denote by ΛΩ = ΩZg + Zg the lattice of Cg deﬁned by Ω. If A is an abelian variety of dimension g over the number ﬁeld K with a principal polarisation then A is analytically isomorphic to Cg /ΛΩ for a certain Ω ∈ Hg . In the rest of this paper, we denote by π : Cg → Cg /ΛΩ = A the canonical projection. The classical theory of theta functions gives a lot of functions on Cg that are pseudo-periodic with respect to ΛΩ and can be used as a projective coordinate system for A. More precisely, for a, b ∈ Qg , the theta function with rational characteristics (a, b) is an analytic function on Cg × Hg given by: θ [ ab ] (z, Ω) = exp πit (n + a).Ω.(n + a) + 2πit (n + a).(z + b) . (1) n∈Zg

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In order to write the pseudo-periodicity relations veriﬁed by the theta functions it is convenient to introduce a certain pairing on Cg . First we identify Cg to R2g via the isomorphism R2g → Cg , (x1 , x2 ) → Ωx1 + x2 . Then for α, β ∈ R2g with α = (α1 , α2 ) and β = (β1 , β2 ), we put eΩ (α, β) = exp(2πi(α1 β2 − α2 β1 )). The pseudo-periodicity of θ [ ab ] is given by t

θ [ ab ] (z +Ω.m+n, Ω) = eΩ (Ω.a+b, Ω.m+n)e−πi

m.Ω.m−2πit m.z

θ [ ab ] (z, Ω). (2)

We say that a function f on Cg is ΛΩ -quasi-periodic of level ∈ N if for all z ∈ Cg and m ∈ Zg , we have:f (z + m) = f (z), f (z + Ω.m) = exp(−πit m.Ω.m − 2πit z.m)f (z). For any ∈ N∗ , the set HΩ, of ΛΩ -quasi-periodic functions of level is a ﬁnite dimensional C-vector space basis can be given by the 0 whose theta functions with characteristics: (θ b/ (z, −1 .Ω))b∈[0,...,−1]g . If = k 2 , g then an alternative basis of HΩ, is (θ a/k b/k (kz, Ω))a,b∈[0,...,k−1] . A theorem of Lefschetz tells that if ≥ 3, the functions in HΩ, give a projective embedding g of A in P −1 , the projective space over C of dimension g − 1. For = 2, the functions in HΩ,2 do not give a projective embedding of A. It is easy to check that for all f ∈ HΩ,2 , we have f (−z) = f (z). Under some well known general conditions [7, cor 4.5.2], the image of the embedding deﬁned by HΩ,2 in 2 P −1 is the Kummer variety associated to A, which is the quotient of A by the automorphism −1. Once we have chosen a level ∈ N, for the rest of this paper, we adopt g g the following conventions: we let Z() 0 = (Z/Z) and for 2a point zP ∈ C and i ∈ Z() we put θi (zP ) = θ i/ (zP , Ω/). If = k , for i, j ∈ Z(k), g we let θi,j (zP ) = θ i/k (k.zP , Ω). We denote by P the element of A (C) j/k

with coordinates Pi = θi (zP ) and let P be the associated point of A that g we consider depending on the situation as embedded in P −1 or as a point on the analytic variety Cg /ΛΩ . In this paper, for n, ∈ N, such that n divides we will implicitly consider Z(n) as a subgroup of Z() via the morphism x → (/n).x. We denote by Ξ the theta divisor of level on A which is the divisor of zero of θ [ 00 ] (z, −1 .Ω). There is an isogeny ϕ : A → Aˆ = Pic0A , deﬁned by x → τx∗ Ξ − Ξ where τx is the translation by x morphism on A. The kernel of ϕ is A[]. For = 1 we let Ξ1 = Ξ . We denote by K(A) the function ﬁeld of A and if f ∈ K(A), we denote (f ) the divisor of the function f . Let Z 0 (A) be the group of 0-cycles of A group over the set of closed that is the free commutative points of A. If D = ni Pi is an element of Z 0 (A) and f ∈ K(A) then we put f (D) = i f (Pi )ni .

3

Weil and Tate Pairings and Theta Functions

In this section, we present formulas to compute Weil and Tate pairings from the knowledge of the theta coordinates of some points.

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The Weil Pairing

For Ω ∈ Hg , let A = Cg /ΛΩ be the associated complex abelian variety and denote by π : Cg → A the natural projection. Let be a positive integer, we denote by μ the subgroup of C∗ of th roots of unity. For zP , zQ ∈ Cg , let P, Q be the associated points of A, we consider the pairing: eW : A[] × A[] → μ , (P, Q) → eΩ (zP , zQ ) . It is clear that eW does not depend on the choice of zP and zQ representing P and Q respectively and that eW is a non-degenerate skew linear form. The following proposition gives an expression of this pairing in term of the values of certain theta functions. Lemma 1. Let Ω ∈ Hg . Let a, b ∈ Qg , let be a positive integer and let zP , zQ ∈ Cg be such that .zP = .zQ = 0 mod ΛΩ . Set zP = Ω.zP 1 + zP 2 and zQ = Ω.zQ1 + zQ2 with for i = 1, 2, zP i , zQi ∈ Rg . Let P = π(zP ) and Q = π(zQ ). For all z ∈ Cg , we have: a+zQ1 (z, Ω) θ b+zQ2 θ [ ab ] (z + .zP , Ω) . (3) eW (P, Q) = a+z Q1 θ [ ab ] (z, Ω) (z + .zP , Ω) θ b+zQ2

Proof. By (2), we have: a+zQ1 θ b+zQ2 (z + .zP , Ω) = eΩ (Ω.(a + zQ1 ) + (b + zQ2 ), Ω.zP 1 + zP 2 ) a+zQ1 exp[(πi2 (t zP 1 .Ω.zP 1 ) − 2πit zP 1 .z]θ b+zQ2 (z, Ω), θ [ ab ] (z + .zP , Ω) = eΩ (Ω.a + b, Ω.zP 1 + zP 2 ) exp[−πi2 (t zP 1 .Ω.zP 1 ) − 2πit zP 1 .z]θ [ ab ] (z, Ω). The lemma follows immediately. Let eW : A[]×A[] → μ be the usual Weil pairing. We recall a possible deﬁnition ∗ for eW [13, p. 184]. Let P, Q ∈ A[]. Let D = τQ Ξ − Ξ , then D represents a 0 ˆ point of A[] = PicA []. As a consequence, there exists a function fQ ∈ K(A) such that (fQ ) = .D. In the same way, there exists a function gQ ∈ K(A) such that (gQ ) = []∗ (D). As []∗ (fQ ) = .[]∗ D = (gQ ) there exists a constant c ∈ C∗ such that []∗ fQ = c.gQ . Thus for X a general point of A, of μ which is equal to eW (P, Q).

gQ (X) gQ (X+P )

is an element

Proposition 1. Keeping the notations from above, let zP = Ω.zP 1 + zP 2 and zQ = Ω.zQ1 + zQ2 be elements of Cg such that P = π(zP ) and Q = π(zQ ). For z ∈ Cg , we have the following equalities, up to a multiplication by a constant: z

0 θ zQ1 (.z, Ω) θ [ 0 ](z + zQ ) Q2 −1 , fQ (z) = μQ (z) gQ (z) = , (4) θ [ 00 ] (.z, Ω) θ [ 00 ](z) where μQ (z) : Cg → C is given by μQ (z) =

θ [ 0 ](z+zQ ) 0 . θ[ 0 0 ](z)

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Remark 1. In the preceding equations, the domain of the functions gQ and fQ is Cg but we will see in the course of the proof that gQ and fQ are periodic with respect to ΛΩ and are in fact well deﬁned functions on A. Proof. As π ∗ Ξ is the divisor of zero of θ [ 00 ] (z, Ω), π ∗ D is the divisor of zero of t g (z) = θ [ 00 ] (z + zQ , Ω)/θ [ 00 ] (z, Ω). But g(z) = exp[πitzQ1 Ωz Q1 + 2πi zQ1 (z + zQ1 zQ2 )]g (z) has the same zero divisor as g (z) and g(z) = θ zQ2 (z, Ω)/θ [ 00 ] (z, Ω). : Cg → Cg , z → z. It is clear from its deﬁnition that up to a multiplication Let [l] which gives the left hand of (4). It is easily seen using by a constant gQ = g ◦ [l] (2) that gQ (z) is periodic with respect to ΛΩ and as a consequence descends to a function on A. We turn to the proof of the second equality. As μQ (z) is a non vanishing function, the zero divisor of the function μQ (z)−1 (θ [ 00 ](z + zQ )/θ [ 00 ](z)) is ∗ π (D). Moreover, it is easily seen using (2) that this function is periodic with respect to ΛΩ , and descends to a function on A which up to a multiplication by a constant is fQ (z). Corollary 1. The pairing eW is the Weil pairing. Proof. This is an immediate consequence of Lemma 1 with a = b = 0, ProposigQ (X) tion 1 and the deﬁnition of the Weil pairing as eW (P, Q) = gQ (X+P ). Corollary 2. Let Ω ∈ Hg . Let a, b ∈ Qg , let be a positive integer and let zP , zQ ∈ Cg be such that .zP = .zQ = 0 mod ΛΩ . Let P, Q ∈ A be such that P = π(zP ) and Q = π(zQ ) and let: θ [ ab ] (.zP + zQ , Ω) θ [ ab ] (0, Ω) , θ [ ab ] (zQ , Ω) θ [ ab ] (.zP , Ω) θ [ ab ] (.zQ + zP , Ω) θ [ ab ] (0, Ω) R(zP , zQ ) = . θ [ ab ] (zP , Ω) θ [ ab ] (.zQ , Ω) L(zP , zQ ) =

(5)

If L(zP , zQ ) and R(zP , zQ ) are well deﬁned and non null, we have: eΩ (zP , zQ ) = eW (P, Q) = L(zP , zQ )−1 .R(zP , zQ ).

(6)

Proof. Since Q + P = Q and P = 0, L(zP , zQ ) does not depend on [ ab ] so we can assume that a = b = 0. The corollary can then be proved by a direct computation. But it also follows immediately from Proposition 1 and the formula eW (P, Q) = fP (Q − 0)/fQ (P − 0). In fact, using the notations of Proposition 1, we have fP (Q − 0) μP (zQ )μQ (0) = . fQ (P − 0) μP (0)μQ (zP ) The result follows an immediate computation. Remark 2. One can recognize in (6) a classical formula to compute the ﬁrst Chern class of a line bundle from the knowledge of its factors of automorphy, see for instance [1, Th. 2.1.2].

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The Tate Pairing

Let K be a number ﬁeld and we suppose that A is deﬁned over K. In this section, we suppose that μ ⊂ K and that A[] is rational over K. Let K be the algebraic closure of K and let G = Gal(K/K). Let δ1 : K ∗ /K ∗ → Hom(G, μ ) (resp. δ2 : A(K)/[]A(K) → Hom(G, A[])) be the connecting morphism of the Galois cohomology long exact sequence associated to the Kummer exact sequence (resp. to the exact sequence 0 → A[] → A(K) → A(K) → 0). There exists a bilinear application often referred to as the Tate pairing eT : A(K)/[]A(K) × A[] → K ∗ /K ∗ such that for (P, Q) ∈ A(K)/[]A(K) × A[], eW (δ2 (P ), Q) = δ1 (eT (P, Q)). In the statement of the next proposition, we suppose that the principal polarization L of A deﬁned by the matrix period is deﬁned over K. Thus for any X ∈ A(K) there exits zX ∈ Cg such that π(zX ) = X and θ(zX )/θ(0) ∈ K. In general this rationality condition on L is not veriﬁed but we will see later on in Remark 4 how to adapt the formulas of the next proposition to cover the general case. Proposition 2. Let K be a number ﬁeld and let A be a dimension g abelian variety over K. Let Ω ∈ Hg be such that A is analytically isomorphic to Cg /ΛΩ . Let a, b ∈ Qg , and let be a positive integer. Let P ∈ A(K)/[]A(K) and Q ∈ A[](K) and let zP , zQ ∈ Cg be such that π(zP ) = P and π(zQ ) = Q where π : Cg → A is the natural projection (by abuse of notation we use P, Q to denote the corresponding points of an algebraic and analytic model of A). Suppose that we have chosen zP , zQ and zP +Q such that θ [ 00 ](zP + zQ ) θ [ 00 ](0) ∈ K ∗, θ [ 00 ](zP ) θ [ 00 ](zQ ) then we have eT (P, Q) =

(7)

θ [ 00 ](.zQ + zP ) θ [ 00 ](0) . θ [ 00 ](zP ) θ [ 00 ](.zQ )

(8)

Proof. By Proposition 1, we have θ [ 00 ](.zQ ) θ [ 00 ](zP ) fQ (P − 0) = 0 θ [ 0 ](.zQ + zP ) θ [ 00 ](0)

θ [ 00 ](zP + zQ ) θ [ 00 ](0) θ [ 00 ](zP ) θ [ 00 ](zQ )

.

(9)

Taking care of the fact that eT (P, Q) has value in K ∗ /K ∗ we just have to prove that eT (P, Q) = fQ (0 − P ). The proof follows exactly the same computations as [16, p. 280].

4

Pairing Computations

In this section, we describe a general method to compute Weil or Tate pairings which does not rely on the usual Miller’s loop and prove its correctness. We postpone to the next section the analysis of the running time of these algorithms.

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257

Let n, ∈ N. We suppose that 2 divides n and that and n are relatively prime. Let A be an abelian variety over C with period matrix Ω. We represent g A as a closed subvariety of Pn −1 by the way of level n theta functions and we the pullback of suppose that this embedding is deﬁned over K. Denote by A ng ng −1 A via the natural projection κ : A → P . In the following, we adopt the following convention: if P is a point of A, we denote by P an aﬃne lift of P that g is a point P of An such that κ(P ) = P . An important ingredient of our algorithm is the Riemann addition formulas. The usual form of these formulas works for theta functions of level divisible by 4 (see for instance [6, p. 139]). In this paper we need a slight generalisation of these formulas for working also with level 2 theta functions. We recall that following the convention for the notation of theta functions described 0 at the end of the introduction, we let for all i ∈ Z(n), z ∈ Cg , θi (z) = θ i/n (z, Ω/n). Moreover, we recall that in the following we consider Z(n) (resp. Z(2)) as a subgroup of Z(2n) via the map x → 2x (resp. x → nx). Theorem 1. Let i, j, k, l ∈ Z(2n). We suppose that i + j, i + k and i + l ∈ Z(n). ˆ ˆ Let Z(2) be the dual group of Z(2). For all χ ∈ Z(2) and z1 , z2 ∈ Cg we have ⎛ ⎞⎛ ⎞ ⎝ χ(η)θi+j+η (z1 + z2 )θi−j+η (z1 − z2 )⎠ ⎝ χ(η)θk+l+η (0)θk−l+η (0)⎠ η∈Z(2)

⎛

=⎝

⎞⎛ χ(η)θi+k+η (z1 )θi−k+η (z1 )⎠ ⎝

η∈Z(2)

η∈Z(2)

⎞

χ(η)θj+l+η (z2 )θj−l+η (z2 )⎠

η∈Z(2)

0 (z, Ω/(2n)). Let Proof. For i ∈ Z(2n) and z ∈ Cg , we let θi (z) = θ i/(2n) g i, j ∈ Z(2n) be such that i + j ∈ Z(n) and let z1 , z2 ∈ C . The usual duplication formula [6, p. 139] gives θi+j (z1 +z2 )θi−j (z1 −z2 ) = 21g η∈Z(2) θi+η (z1 )θj+η (z2 ). ˆ For χ ∈ Z(2), using this formula, we compute

1 χ(η1 + η2 )θi+η (z1 )θj+η (z2 ) 1 2 2g η1 ,η2 ∈Z(2) ⎞⎛ ⎞ ⎛ 1 ⎝ = g χ(η)θi+η (z1 )⎠ ⎝ χ(η)θj+η (z2 )⎠ . (10) 2

χ(η)θi+j+η (z1 + z2 )θi−j+η (z1 − z2 ) =

η∈Z(2)

η∈Z(2)

η∈Z(2)

Using this last equation to compute the left and right hand sides of the preceding equation we obtain the result. We suppose that the theta null point 0 = (θi (0))i∈Z(n) is known. We deduce im= mediately from Theorem 1 an algorithm that takes as inputs P = (Pi )i∈Z(n) , Q − Q = ((P − Q)i )i∈Z(n) and outputs P + Q = ((P + Q)i )i∈Z(n) . (Qi )i∈Z(n) and P P We write P + Q = PseudoAdd(P , Q, − Q). Indeed we will see later (Proposition 3) that if n = 4, we can recover the projective point P + Q from P and Q

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using the Riemann addition formulas. It is then easy to see that if we moreover and P know P, Q − Q, then there is a unique aﬃne point P + Q above P + Q that satisfy the addition formulas from Theorem 1. If n = 2, the point P +Q is also unique provided the abelian variety satisﬁes the generic condition from Theorem 3. Chaining the algorithm PseudoAdd in a classical Montgomery ladder [2, alg. = (Q i )i∈Z(n) , P 9.5 p. 148] yields an algorithm that takes as inputs Q +Q = 0 = ( 0i )i∈Z(n) and an integer and outputs ((P + Q)i )i∈Z(n) , P = (Pi )i∈Z(n) , P , P + Q. We write P + Q = ScalarMult(P + Q, Q, 0, ). In particular, we have P = ScalarMult(P , P , 0, 0, ). The following lemma tells that the output of ScalarMult does not depend on the particular chain of PseudoAdd calls it uses. Lemma 2. Let L = {0, 1, . . . , } be a Lucas sequence. Let A0 = P , B0 = 0, A1 = For m ∈ L, m 2, write m = j + k with j, k, j − k ∈ L. Let P + Q and B1 = Q. Bm = PseudoAdd(Bj , Bk , Bj−k ) and Am = PseudoAdd(Aj , Bk , Aj−k ). Then + Q. In other words P + Q does not depend on the Lucas sequence A = P used to compute it. = (θi (zQ ))i∈Z(n) Proof. If there exist zP , zQ ∈ Cg such that P = (θi (zP ))i∈Z(n) , Q and P + Q = (θi (zP + zQ ))i∈Z(n) then by Theorem 1 and a recursion we see that Aj = (θi (zP + jzQ ))i∈Z(n) and Bj = (θi (jzQ ))i∈Z(n) . Hence A = (θi (zP + + Q. zQ )) = P Otherwise there exist λP , λQ and λP +Q in C∗ such that P = λP (θi (zP ))i∈Z(n) , = λQ (θi (zQ ))i∈Z(n) and P + Q = λP +Q (θi (zP + zQ ))i∈Z(n) . Since we have Q λ2 λ2Q P), an PseudoAdd(λP +Q P + Q, λQ Q, λP P) = P +Q PseudoAdd(P + Q, Q, λP 2

j(j−1)

easy recursion shows that Bj = λjQ (θi (jzQ ))i∈Z(n) and Aj = λjP +Q λQ (−1)

(θi (zP + jzQ ))i∈Z(n) . Hence A = λP +Q λQ P + Q. ∗

/λj−1 P ·

/λ−1 · (θj (zP + zQ ))j∈Z(n) = P g

Remark 3. There is a natural action of K on An − {0} by multiplication of g ∗ the coordinates of a point that we denote by α ∗ P for α ∈ K and P ∈ An (K). In the proof of the preceding lemma we have seen the eﬀect of this action on P the output of the algorithm ScalarMult: let P, Q ∈ A(K) and let P, Q, +Q be aﬃne lifts of P , Q and P + Q. Let R = ScalarMult(P + Q, Q, P , 0, ). Let α, β, γ, δ ∈ K, we have γ ∗ P , δ ∗ (11) ScalarMult(α ∗ P + Q, β ∗ Q, 0, ) = (α β (−1) /γ −1 δ (−1) ) ∗ R, 2

α 0, 0, ). (12) ScalarMult(α ∗ P , α ∗ P, δ ∗ 0, δ ∗ 0, ) = 2 −1 ∗ ScalarMult(P, P , δ

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Given P and Q with projective coordinates (θi (zP ))i∈Z(n) and (θi (zQ ))i∈Z(n) for zP , zQ ∈ Cg , we would like to compute eW (P, Q) and eT (P, Q). We can state the main theorem of this section Theorem 2. We suppose that n and are relatively prime. For X, Y ∈ A(K), Y , X denote by X, + Y any aﬃne lifts of X, Y and X + Y . Recall that for i ∈ i the coordinate i of the point X. For ∈ N and i ∈ Z(n), Z(n), we denote by X Y , X let fT (X, + Y , 0, , i) = ScalarMult(X+Y ,X,Y ,0,)i 0i . Then for P, Q ∈ A[] X, 0,0,)i Yi ScalarMult(X, and i ∈ Z(n), we have: P P , P + Q, 0, , i)−1 fT (Q, + Q, 0, , i), eW (P, Q)n = fT (P , Q,

(13)

whenever the right hand side is well deﬁned. and Moreover, for P ∈ A(K)/[]A(K), Q ∈ A[], if we suppose that 0, P , Q P + Q are aﬃne lifts of 0, P , Q and P + Q with coordinates in K, then we have for i ∈ Z(n), P, P + Q, 0, , i), eT (P, Q)n = fT (Q,

(14)

whenever the right hand side is well deﬁned. Proof. Let zP , zQ ∈ Cg such that π(zP ) = P and π(zQ ) = Q (recall that π : = Cg → A = Cg /ΛΩ is the natural projection). Let P = (θi (zP ))i∈Z(n) , Q + Q = (θi (zP + zQ ))i∈Z(n) . Then applying Corollary 2, if (θi (zQ ))i∈Z(n) and P P, Q ∈ A[], we obtain that P P, P eΩ/n (zP , zQ ) = eW (P, Q)n = fT (P , Q, + Q, 0, , i)−1 fT (Q, + Q, 0, , i). In the same way, by Proposition 2 (which apply for i = 0, but it is easy to see that the same result is true for any i ∈ Z(n)), we have for P ∈ A(K)/[]A(K) P, P + Q, 0, , i). Next, let α, β, γ, δ ∈ K. By and Q ∈ A[], eT (P, Q)n = fT (Q, Remark 3, we have γ δ β ∗ Y , γ ∗ X Y , X fT (α ∗ X, + Y,δ ∗ 0, , i) = .fT (X, + Y , 0, , i). αβ

(15)

This shows that the expressions (13) and (14) for the Weil and Tate pairing do not depend on the choice of aﬃne liftings (rational over K in the case of the Tate pairing) of P , Q and P + Q. Remark 4. In this remark we keep the notations of the previous theorem. Let L be a polarization of A associated to Ξn for n ∈ N∗ which is rational over K. Let (θi )i∈Z(n) be a basis of global sections of a trivialisation of π ∗ (L ) (and we rigidify this basis by setting θ0 (0) = 1). In general, it is not true that the polarization deﬁned by the level n classical theta functions is rational over K. Nonetheless we know that there exits a non vanishing function ζ of Cg such that θi = ζθi for i ∈ Z(n) (up to a renumbering of the basis θi ).

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alg = zX ∈ {0, zP , zQ , zP +Q }, if we denote by X ∗ constant factors cX ∈ C such that for X ∈ alg = X. {0, P, Q, P + Q} we have cX ∗ X alg for X ∈ {0, P, Q, P + As we can suppose that the coordinates of the points X Q} are deﬁned over K, we can rewrite (9) as: −

θi (.zQ + zP ) θi (0) cP +Q c0 , eT (P, Q) = cP cQ θi (zP ) θi (.zQ )

Let 0, zP , zQ , zP +Q ∈ Cg . For (θi (zX ))i∈Z(n) , then there exist

alg

alg , P alg , P for i ∈ Z(n). But by (15) we have the equation: fT (Q + Q , 0alg , , i) − − cP +Q c0 cP +Q c0 θi (.zQ +zP ) θi (0) P , P = .fT (Q, + Q, 0, , i) = . ComcP cQ

cP cQ

θi (zP )

θi (.zQ )

paring these formulas, we obtain that we can compute the Tate pairing by taking aﬃne lifts of 0, P , Q and P + Q provided by the coordinates θi . Now using (15) again, we obtain that to compute the Tate pairing we only have to choose aﬃne lifts of 0, P , Q, and P + Q which are rational over K. As we have shown that the formulas of Theorem 2 do not depend on a choice of the aﬃne lifts of the input points of the algorithm (as long as the choices are the same for the computation of the two functions fT in the case of the Weil pairing), from now on we only consider projective points. In order to have a working algorithm to compute Weil and Tate pairings, it remains to explain how to compute P +Q from the knowledge of P and Q. As the formulas to compute the pairings only involve one of the level n theta functions, and since the number of the coordinates used in the computation of ScalarMult is ng , for the sake of eﬃciency it is important to have a small n. As 2 divides n, from now on, we focus on the two interesting cases: n = 2 and n = 4. We ﬁrst treat the case n = 4. Let zP , zQ ∈ Cg and let P = (Pi )i∈Z(n) = (θi (zP ))i∈Z(n) and Q = (Qi )i∈Z(n) = (θi (zQ ))i∈Z(n) . From the knowledge of P and Q, with the addition formula (Theorem 1), one can compute the products: χ(η)θi+j+η (zP + zQ )θi−j+η (zP − zQ ) χ(η)θk+l+η (0)θk−l+η (0) , (16) η∈Z(2)

η∈Z(2)

ˆ for χ ∈ Z(2) and i, j, k, l ∈ Z(2n) such that i + j, i + k, and i + l ∈ Z(n). If we ˆ can prove that for any such choice of i, j, k, l ∈ Z(2n) and χ ∈ Z(2) there exist k ∈ k +Z(n) and l ∈ l+Z(n) such that η∈Z(2) χ(η)θk +l +η (0)θk −l +η (0) = 0, then by summing over the characters the left bracket of (16) one can compute all the products θi (zP + zQ )θj (zP − zQ ), for i, j ∈ Z(n) from which it is easy to recover by taking quotients the projective point (θi (zP + zQ ))i∈Z(n) . Now, using equation (10), we have

χ(η)θk+l+η (0)θk−l+η (0) =

η∈Z(2)

where for k ∈ Z(8), θk (z) = θ

1 χ(η)θk+η (0) χ(η)θl+η (0) , (17) g 2 η∈Z(2)

0 k/8

η∈Z(2)

(z, Ω/8). We have the

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Proposition 3. Let δ ∈ N be such that 4 divides δ. For any a ∈ K(2δ) there exˆ ists an element b 0∈ a + K(δ) such that for all χ ∈ Z(2) we have that η∈Z(2) χ(η)θ (b+η)/(2δ) (0, 1/(2δ).Ω) = 0. Proof. This is just a rephrasing of [11, equation (*) p. 339]. Applying the preceding proposition to the factors of the right hand of equation (17), we obtain that there exists k ∈ k + Z(n) and l ∈ l + Z(n) such that χ(η)θ (0)θ (0) = 0 and we are done. k +l +η k −l +η η∈Z(2) 0 (z, 1/2.Ω). In the case n = 2, as usual, for all i ∈ Z(2), we put θi (z) = θ i/2 ˆ Then by Theorem 1, we have for any χ ∈ Z(2) and for well chosen pairs of quadruples (i, j, k, l), (i , j , k , l ) ∈ Z(2)4 an equation

χ(η)θi+η (zP + zQ )θj+η (zP − zQ ) χ(η)θk+η (0)θl+η (0)

η∈Z(2)

=

η∈Z(2)

χ(η)θi +η (zP )θj +η (zP ) χ(η)θk +η (zQ )θl +η (zQ ) .

η∈Z(2)

(18)

η∈Z(2)

If the kernel of χ does not contain the subgroup of Z(2) generated by k + l then we have η∈Z(2) χ(η)θk+η (0)θl+η (0) = 0, so it is not possible to recover θi+η (zP +zQ ) as before. This is consistent with the fact that for i ∈ Z(2) and z ∈ Cg , θi (z) = θi (−z), the right hand side of (18) is invariant for the transformation zQ → −zQ while it is not the case of the left hand side. The best we can hope is that for almost all period matrices Ω ∈ Hg there exists a k ∈ Z(2) such that ˆ such that k + l is in the kernel of χ, we have for all l ∈ Z(2) and χ ∈ Z(2) the content of Theorem 3. In η∈Z(2) χ(η)θk+η (0)θl+η (0) = 0. This is exactly order to prove this theorem, we let Tk,l,χ = η∈Z(2) χ(η)θk+η (0)θl+η (0) and we state the following lemma: Lemma 3. For Ω ∈ Hg , the two following properties are equivalent: ˆ such that 1. There exists a k ∈ Z(2) such that for all ∈ Z(2) and χ ∈ Z(2) k + l is in the kernel of χ, we have Tk,l,χ = 0. 2. For all i, j ∈ Z(2) such that t i.j = 0, θi,j (0) = 0. t ˆ Proof. For χ ∈ Z(2), let μ ∈ Z(2) be such that χ(η) = (−1) η.μ . Let ρ : Z(4) → Z(2), x → x mod Z(2) be the canonical projection. Then we have (see [14, prop 1.3 p. 124]), for all i ∈ Z(4) η∈Z(2) χ(η)θi+η (0) = 2g .θμ,ρ(i) (0), where 0 θk (z) = θ k/4 (z, 1/4.Ω). Combining this relation together with (17), for all i, j ∈ Z(4) such that i + j ∈ Z(2), let k = i + j, l = i − j, we obtain the equality

Tk,l,χ = Ti+j,i−j,χ = 2g .θμ,ρ(i) (0)θμ,ρ(j) (0) = 2g .θμ,k+l (0)2 . t

Since χ(k + l) = (−1)

(k+l).μ

the lemma follows immediately from (19).

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It is well known that for z ∈ Cg , and k, l ∈ Z(2), we have θk,l (−z) = (−1) k.l θk,l (z). As a consequence, for all k, l ∈ Z(2) such that t k.l = 1 (the odd characteristics), we have θk,l (0) = 0. Denote by M4 the quasi-projective variety over C deﬁned as the locus of zeros of θi,j (0) considered as functions of Ω. It is clear that M4 parametrizes the set of principally polarized abelian varieties together with a level 4 structure since from the knowledge of a point in M4 one can recover the projective embedding of the corresponding abelian variety provided by the Riemann equations. Theorem 3. For all k, l ∈ Z(2) such that t k.l = 0, the function θk,l (0) on M4 is non-trivial and as consequence, its zero locus is a proper subvariety of M4 of codimension 1. Proof. We sketch the proof of the theorem. Suppose on the contrary that for k, l ∈ Z(2) such that t k.l = 0, θk,l (0) is a constant function of Ω. This is a degree 1 relation level 4 theta constants, call it Rk,l . We have for all k ∈ Z(4), 0 for θk (0) = θ (2k)/8 (0, (2Ω)/8). Thus, the level 4 degree 1 relations Rk,l induce degree 1 relations for level 8 theta constants. The hypothesis t k.l = 0 means that these level 8 relations are not a linear combination of the symmetry relations θk (0) = θ−k (0) for all k ∈ Z(8). This is a contradiction with the description of M8 the modular space of level 8 marked abelian varieties given by Mumford in [12, main th. p. 83] as an open subset of the reduced projective variety given by the symmetry relations and the Riemann relations. Remark 5. The preceding theorem shows that the symmetric pairing computation algorithms that we describe in the next section works for a general abelian variety. However, one can ask if the closed proper subset of M4 , given by the cancellation of some even level 4 theta constants contains noticeable abelian varieties. Actually, this is the case since a theorem of Frobenius [15, cor. 6.7 p. 3.102] tells us that the locus of Jacobian of hyperelliptic curves inside M4 can be given by equations of the form θk,l (0) = 0 where (k, l) is an even characteristic. As a consequence, the algorithms of Section 5.2 to compute symmetric pairings don’t apply to Jacobian of hyperelliptic of genus g when g 3. It should be noted however that following [7, cor 4.5.2 and remark (2)], the condition that for all k, l ∈ Z(2) such that t k.l = 0, θk,l (0) = 0 is equivalent to the fact the level 2 theta functions give a projectively normal embedding. Considering this result, the condition of Theorem 3 should be considered as natural.

5

Complexity Analysis

In this section, we explain how to use the results of the preceding section to compute eﬃciently pairings on abelian and Kummer varieties with a special focus on dimension 1 and 2 since these cases are particularly interesting for cryptographic applications.

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5.1

263

Abelian Varieties

We begin with the case of abelian varieties since the main loop of the algorithm can also be used for the computation of symmetric pairings on Kummer varieties. Initialisation phase. The initialisation phase depends on the representation of the points P and Q on the abelian variety A. If P and Q are given by theta coordinates of level 4 we can apply the procedure described in Section 4 to compute the homogeneous coordinates of (θi (P + Q))i∈Z(4) . Suppose that another coordinate system is used to represent P and Q that we denote by (Xi )i∈I where Xi are rational functions on a Zariski open subset of A. Then by deﬁnition there exist formulas to compute θi (P ) and θi (Q) from the knowledge of Xi (P ) and Xi (Q). In practise, the dictionary between some useful coordinate system and the theta coordinates can easily be deduced from well known properties of theta functions. It should be remarked that in order to carry out these computations we might have to do a base ﬁeld extension since in the projective embedding of A provided by the level 4 theta functions the 4-torsion of A is rational over the base ﬁeld, whereas this may not the case with other models of A. The advantage of the level 4 is that no square root extraction is needed for the computation of P + Q,contrarily to the level 2 case as we will see. 0 From the knowledge of θ i/4 (zX , 1/4.Ω), i ∈ Z(4) for X = P, Q, P +Q we can 0 then compute the level 2 coordinates given by ( j∈Z(2) θ i+2j (zX , Ω4 ))i∈Z(2) 4

for the coordinates of the (isogeneous) points X = P, Q, P + Q. Pairing computation phase. As we have seen before, we can carry out the computations of the main loop of the algorithm with level 2 theta functions since at the end we only need one theta coordinate to compute the pairings. This is more eﬃcient because we only need 2g coordinates to represent a point and we can do the computation on the ﬁeld of deﬁnition of the 2-torsion of A. We suppose that we are given the level 2 coordinates of P , Q, P + Q. Rather than considering the formulas of Theorem 1 for the double and add algorithm, we use the level 2 formulas given in [4] for the genus 2 case, and in [5] for the genus 1 case. For instance, let E be an elliptic curve deﬁned by Ω ∈ H1 , let Ω = Ω/2 and put 0 (0, Ω ); A = ϑ [ 00 ] (0, 2Ω ); B = ϑ 1/2 (0, 2Ω ). a = ϑ [ 00 ] (0, Ω ); b = ϑ 1/2 0 The duplication formulas are given by the equalities: 1/2 2 0 0 (z, 2Ω )2 , aϑ 0 0[ 0 ] (z, Ω ) = ϑ [ 00 ] (z, 2Ω )2 + ϑ 1/2 bϑ 1/2 (z, Ω ) = ϑ [ 0 ] (z, 2Ω ) − ϑ 0 (z, 2Ω )2 . 0 2 2Aϑ [ 00] (2z, 2Ω ) = ϑ [ 00 ] (z, Ω )2 + ϑ 1/2 (z, Ω )2 , 1/2 0 2 0 2Bϑ 0 (2z, 2Ω ) = ϑ [ 0 ] (z, Ω ) − ϑ 1/2 (z, Ω ) . 0 Let x = θ [ 00 ] (z, Ω ) and z = θ 1/2 (z, Ω ) using the above formulas yield the following algorithms:

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Doubling Algorithm: Input: A point P = (x : z). Output: The double 2.P = (x : z ). 1. 2. 3. 4. 5.

x0 = (x2 + z 2 )2 ; A2 2 2 2 z0 = B 2 (x − z ) ; x = (x0 + z0 ); z = ab (x0 − z0 ); Return (x : z ).

Diﬀerential Addition Algorithm: Input: Two points P = (x : z) and Q = ( x : z) on E, and R = (x : z) = P −Q, with xz = 0. Output: The point P + Q = (x : z ). 1. 2. 3. 4. 5.

x0 = (x2 + z 2 )( x2 + z2 ); 2 A 2 2 z0 = B x2 − z2 ); 2 (x − z )( x = (x0 + z0 )/x; z = (x0 − z0 )/z; Return (x : z ).

Recall that in order to compute the pairing eT (P, Q), we have to compute P , = ScalarMult(Q, Q, P + Q = ScalarMult(P + Q, Q, 0, ) and Q 0, 0, ). It should be remarked that in the computation of P + Q, we need exactly the Since same values of j.Q for some j ∈ {1, . . . , } as those required to obtain Q. we want to avoid a division in each step, we use a Montgomery ladder so that the diﬀerences in the adding step are always the same points. To speed up the diﬀerential additions, we have renormalised the theta null point (a, b) to (1, b/a). It is easy to see by doing the same computation as in Remark 3 that this does not change the value of the Tate pairing eT (P, Q). Moreover we also have renormalised the theta null point (A, B). Looking back at the proof of 1, we see that this change each aﬃne addition by the constant factor B −2 . This also does not aﬀect the ﬁnal value of the Tate pairing eT (P, Q), since we use the same Lucas and P sequence for computing Q + Q. This give the following steps for the pairing: from (j − 1)Q, jQ and P + jQ we compute 2(j − 1)Q, (2j − 1)Q, P + (2j − 1)Q or (2j − 1)Q, 2jQ and P + 2jQ depending on the binary decomposition of . We remark that at each step we do a doubling and two adding, and that we add the same point to the triple A2 2 2 (j−1)Q, jQ, P +jQ. For instance in genus 1, we only have to compute B 2 (x −z ) once, where (x : z) are the coordinates of the doubled point. The ﬁgure below summarises the cost per bit of computation of the Tate pairing with our algorithm in genus 1 and 2 with the following notations: S is for squaring, M is for general multiplication, m is for multiplication by a constant. Tate pairing First pairing e(P, Q) Following pairings e(P , Q) Dimension 1 8S+4m+4M 2S+1m+2M Dimension 2 13S+12m+11M 4S+3m+4M The algorithms that we have presented in this section are deterministic and generalize immediately to the higher dimension case. Usually when computing a pairing, the ﬁeld of deﬁnition of Q has a smaller degree than the ﬁeld of deﬁnition of P , so that at each step one adding and one doubling is done with points in the smaller ﬁeld. We also remark that if we have to compute several pairings e(P1 , Q), e(P2 , Q), . . . with the same Q, it makes sense to store the

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results of the computations of the jQ so that for the next pairings we only have to compute the Pi + jQ. For instance when g = 1 if we store the log2 () A2 2 2 coordinates (x2 + z 2 , B 2 (x − z )) of each doubling step, we can compute the subsequent pairings with only ﬁve multiplications at each step. 5.2

Kummer Varieties

Let A be a principally polarized abelian variety of dimension g deﬁned by Ω ∈ Hg . As we have seen in the introduction, the level 2 theta functions deﬁned by Ω give a projective embedding of the Kummer variety associated to a A. We recall that the Kummer variety K A of A is the quotient of A by the action of the automorphism −1 of A. Let ζ : A → K A be the natural projection. In the following, if P ∈ A(K) we denote by P its image by ζ. The construction of K A does not preserve the group structure of A. Nonetheless, we remark that from the data of P ∈ K A (K) one can compute 2P without ambiguity, and from the data of P , Q and P − Q one can compute P + Q. As a consequence, K A inherits from A of an action of Z on its points which can be computed by a Montgomery ladder like algorithm. ∗ ∗ Let e be a pairing on A, and let K 0 be the quotient of K by the action of ∗ ∗ the automorphism −1. Let ζ0 : K → K 0 be the natural projection. The pairing ∗ e gives a well deﬁned application e : K A (K) × K A (K) → K 0 , (P , Q) → ∗ ζ0 (e(P, Q)). It is easily seen that the elements of K 0 are in bijection with the ∗ ∗ set S = {x + 1/x, x ∈ K }. Identifying K 0 with S, the application ζ0 is given by ∗ ζ0 (x) = x + 1/x, x ∈ K from which we deduce the expression of e : (P , Q) → e(P, Q) + e(−P, Q). This pairing has been introduced in [3]. In the following, if e is a pairing, we say that e is the symmetric pairing associated to e. The symmetric pairing e can be seen as a version of e for compressed coordinates as it takes as input points with 2g coordinates rather than 4g . Its cryptographic relevance comes from the compatibility of e with the Z-set ∗ structures of K A and K 0 : for all λ, μ ∈ Z, P , Q ∈ K A , we have e(λ.P , μ.Q) = (λμ).e(P , Q). In [3], the authors give an algorithm based on Lucas sequences to compute the action of Z on K 0 for certain ﬁnite ﬁelds. Here we would like to emphasize that the compatibility of the Z-structure of K A and K 0 is also algorithmic. It comes from the fact and on any quotient of an algebraic group by the automorphism −1 there exists a natural Montgomery ladder algorithm to compute the resulting Z-action. In the case of K 0 we obtain very simple and general formulas. For x ∈ K, and i, j ∈ Z, we have 1 2 1 1 1 1 1 ) = (x2i + 2i +2); (xi + i )(xj + j ) = (xi+j + i+j )+(xi−j + i−j ). xi x x x x x We have seen that the codomain of the Tate pairing eT is the multiplicative group K ∗ /K ∗ . Again, we can take the quotient of this group by the action of (−1) on it, denote it by (K ∗ /K ∗ )0 . It is clear that there is a bijection between the set (K ∗ /K ∗ )0 and the set ST = {x + 1/x, x ∈ KT } where KT is a set of representatives of K ∗ /K ∗ . Moreover, one can compute the Z-action on such representatives using the preceding algorithm. (xi +

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Initialisation phase. We suppose that we know the level 2 coordinates θi (zP ) and θi (zQ ), i ∈ Z(2) of P and Q. We may assume (by multiplying by a projective factor) that the values of the projective coordinates (θi (zP ))i∈Z(2) and (θi (zQ ))i∈Z(2) are in K. Using Theorem 1 and Theorem 3, we obtain that for a ˆ it is possible to compute for all i, j ∈ Z(2) and χ ∈ Z(2) general choice of K A , such that χ(i − j) = 1, η∈Z(2) χ(η)θi+η (zP + zQ )θj+η (zP + zQ ) from the inputs. By summing over the characters, we obtain for all i, j ∈ Z(2) κij = θi (zP + zQ )θj (zP − zQ ) + θj (zP + zQ )θi (zP − zQ ).

(20)

We suppose that θ0 (zP +zQ )θ0 (zP −zQ ) = 0, if necessary by replacing the index 0 by another one. By rescaling the projective coordinates, we do our computations as if θ0 (zP − zQ ) = 1 hence we know θ0 (zP + zQ ). θ (z +z ) i0 ii For i ∈ Z(2), let Pi (X) = X 2 −2 κκ00 X + κκ00 . The roots of Pi (X) are θ0i (zPP +zQ , Q) θi (zP −zQ ) θ0 (zP −zQ ) .

If P or Q is a point of 2-torsion, P + Q = P − Q ∈ K A so each Pi (X) has a double root. Otherwise, exist α ∈ Z(2), α = 0

we may suppose that there θ0 (zP + zQ ) θ0 (zP − zQ ) is invertible. such that the matrix M = θα (zP + zQ ) θα (zP − zQ ) We can compute {θα (zP + zQ ), θα (zP − zQ )} by ﬁnding the roots of Pα (X). As by hypothesis, P +Q, P −Q ∈ A(K), we deduce that these roots are in K. We ﬁx an arbitrary ordering (θα (zP + zQ ), θα (zP − zQ )) of these roots (depending on the ordering, we will compute P − Q or P + Q). We can then ﬁnd {θi (zP + zQ ), θi (zP − zQ )} by solving the system

θi (zP − zQ ) κi0 θ0 (zP + zQ ) θ0 (zP − zQ ) = . (21) θα (zP + zQ ) θα (zP − zQ ) θi (zP + zQ ) κiα This method requires one square root. Pairing computation phase. Let P ∈ A(K)/[]A(K) and Q ∈ A[] and denote by P , Q the corresponding points on K A . Denote by θi (z), i ∈ Z(2), the level 2 theta functions associated to Ω. We present two methods to compute the symmetric Tate pairing. A ﬁrst method is to consider the formula eT (P , Q) = eT (P, Q) + eT (P, −Q). We have explained in the last paragraph how to compute the set S = {P + Q, P − Q} at the expence of a square root extraction. By choosing a point in S, we can use the algorithm from Section 5.1 to compute e(P, Q) (resp e(P, −Q)). We can then compute eT (P, Q) = e(P, Q) + e(P, −Q) with a simple division. Another approach is to work in the algebra A = K[X]/(Pα (X)) for α ∈ Z(2) as before. We denote by g the unique automorphism of the algebra of A leaving K invariant and diﬀerent from the identity. For each i ∈ Z(2) by using equation (21) we can express θi (zP + zQ ) = γi X + δi . (We can always compute an inverse of γX + δ except when −δ/γ is a root of Pα . But in this case we have found a root of Pα and we can use the ﬁrst method.) Now, consider the vector (Tj )j∈Z(2) where T0 = 1, Tα = X and Tj = γj X + δj . We compute R = ScalarMult(T, Q, P, 0, )i . Then it is easily seen that

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R + g.R = ScalarMult(P + Q, Q, P, 0, )i + ScalarMult(P − Q, Q, P, 0, )i . By Proposition 2, and using the fact that θi (−zQ ) = θi (zQ ) we have for [θ (.z +z )+θi (−.zQ +zP )]θi (0) i ∈ Z(2) eT (P , Q) = i Q θPi (zP )θ . We can now compute i (.zQ ) eT (P, Q) =

0, )i ]θi (0) [ScalarMult(P +Q, Q, P, 0, )i +ScalarMult(P −Q, Q, P, , θi (zP )ScalarMult(Q, Q, 0, 0, )i

By an application of Lemma 3, the result of the preceding equation is a well deﬁned element of (K ∗ /K ∗ )0 . With this method, we have to compute 1 ScalarMult with value in A and 1 ScalarMult with value in K. It is interesing to note that it avoids the non determinism of the square root computation of the ﬁrst method. In some cryptographic applications, it is important to have a unique value as the result of the Tate pairing. In order to have this property, it is common to compose the Tate pairing with a th root extraction on K which can be done in the case that K is a ﬁnite ﬁeld by an exponentiation in K0∗ . This operation can be performed using the Montgomery ladder type algorithm presented above. The symmetric Weil pairing computation. Since we compute P + Q with the ﬁrst method, we can compute the Weil pairing as in the level 4 case. We explain how to compute it with the second method: let P, Q ∈ A[] and denote by P , Q the corresponding points in K A . Denote by θi (z), i ∈ Z(2) the level 2 theta functions associated to Ω. By Corollary 2, we have: eW (P , Q) =

θi (zQ )θi (.zP ) × θi (zP )θi (.zQ )θi (zQ + .zP )θi (zQ − .zP ) [θi (.zQ + zP )θi (zQ − zP ) + θi (.zQ − zP )θi (zQ + zP )] . (22)

The denominator of this expression can be easily computed from the knowledge of θi (zQ ), θi (.zQ ), θi (zP ) and θi (.zP ) by using the addition formula (1). The numerator can be computed in the algebra A in the following way: keeping the notations from above, we compute R = ScalarMult(T, Q, P, 0, )i .ScalarMult (gT, P, Q, 0, )i . We obtain that R + g.R = ScalarMult(P + Q, Q, P, 0, )i . ScalarMult(P − Q, P, Q, 0, )i + ScalarMult(P − Q, Q, P, 0, )i .ScalarMult(P + Q, P, Q, 0, )i , which gives the numerator of (22).

6

An Example in Dimension 2

In this section we give an example of compution of the pairings on a dimension 2 Jacobian. Let H be the hyperelliptic curve over the prime ﬁeld Fp , p = 331, given by the equation: Y 2 = X 5 + 204X 4 + 198X 3 + 80X 2 + 179X. Let J be the Jacobian of H. The cardinal of J(Fp ) is 26 · 1889 (since we are in level 2, all the 2-torsion points of J are rational), so that we let = 1889, and

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the embedding degree k corresponding to is 4. A theta null point of level 2 associated to J is given by (328 : 213 : 75 : 1). Let P = (255 : 89 : 30 : 1), we have P ∈ J[](Fp ). Let Fpk Fp (t)/(t4 + 3t2 + 290t + 3). We let Q be the Fpk -point of -torsion whose coordinates are: (158t3 + 67t2 + 9t + 293 : 290t3 + 25t2 + 235t + 280 : 155t3 + 84t2 + 15t + 170 : 1).

We compute (and ﬁx an arbitrary ordering): P + Q = (217t3 + 271t2 + 33t + 303 : 308t3 + 140t2 + 216t + 312 : 274t3 + 263t2 + 284t + 302 : 1), P − Q = (62t3 + 16t2 + 255t + 129 : 172t3 + 157t2 + 43t + 222 : 258t3 + 39t2 + 313t + 150 : 1). k

Finally, we let r = p −1 = 6354480 and ζ = tr be a primitive th -root of unity. We then compute using the doubling and diﬀerential addition algorithms: = (12, 141, 31, 327) = 327.0, P = (21t + 280t + 101t + 180, 164t3 + 311t2 + 111t + 129, Q 3

2

0, 137t3 + 282t2 + 123t + 134, 324t3 + 17t2 + 187t + 271) = (324t3 + 17t2 + 187t + 271).

, P, 0, ) = (45t3 + 118t2 + 219t + 308, 152t3 + 97t2 + 166t + 40, ScalarMult(P + Q, Q

, 200t3 + 267t2 + 201t + 192, 117t3 + 42t2 + 106t + 205) = (117t3 + 42t2 + 106t + 205).P

, Q , 0, ) = (50t3 + 31t2 + 84t + 309, 168t3 + 196t2 + 275t + 234, ScalarMult(P + Q, P

. 67t + 186t + 159t + 102, 243t + 320t + 222t + 200) = (243t + 320t + 222t + 200).Q 3

2

3

2

3

2

We then compute (following the previous ordering): eW (P, Q) =

243t3 + 320t2 + 222t + 200 324t3 + 17t2 + 187t + 271 . = ζ −1 , 327 117t3 + 42t2 + 106t + 205

eT (P, Q) =

eT (Q, P ) =

117t3 + 42t2 + 106t + 205 324t3 + 17t2 + 187t + 271

r

243t3 + 320t2 + 222t + 200 327

=ζ

r

1068

,

= ζ 1184 .

Here the Tate pairings are normalized by taking their r = (pk − 1)/-power. The symmetric pairings are then given by eW (P, Q) = 61t3 + 285t2 + 196t + 257 and eT (P, Q) = 194t3 + 163t2 + 97t + 164.

7

Conclusion

In this paper, we have presented an algorithm based on theta functions to compute Weil and Tate pairings. It would be interesting to carry out a ﬁne grained study of the eﬃciency of our algorithm depending on the target implementation (software, hardware etc.) and to compare it with existing implementations based on Miller’s algorithm.

Acknowledgement The authors of this paper would like to thank anonymous referees for their careful reading and helpful comments on an earlier version of the paper.

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References 1. Birkenhake, C., Lange, H.: Complex abelian varieties, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 302. Springer, Berlin (2004) 2. Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications. Chapman & Hall/CRC (2006) 3. Galbraith, S., Lin, X.: Computing pairings using x-coordinates only. Designs, Codes and Cryptography (2008) 4. Gaudry, P.: Fast genus 2 arithmetic based on Theta functions. J. of Mathematical Cryptology 1, 243–265 (2007) 5. Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields Appl. 15(2), 246–260 (2009) 6. Igusa, J.-i.: Theta functions. Springer, New York (1972); Die Grundlehren der mathematischen Wissenschaften, Band 194 7. Koizumi, S.: Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98(4), 865–889 (1976) 8. Lang, S.: Reciprocity and correspondences. Amer. J. Math. 80, 431–440 (1958) 9. Lichtenbaum, S.: Duality theorems for curves over p-adic ﬁelds. Invent. Math. 7, 120–136 (1969) 10. Lubicz, D., Robert, D.: Computing isogenies between abelian varieties (2010), http://arxiv.org/abs/1001.2016 11. Mumford, D.: On the equations deﬁning abelian varieties. I. Invent. Math. 1, 287– 354 (1966) 12. Mumford, D.: On the equations deﬁning abelian varieties. II. Invent. Math. 3, 75– 135 (1967) 13. Mumford, D.: Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay (1970) 14. Mumford, D.: Tata lectures on theta I. Progress in Mathematics, vol. 28. Birkh¨ auser Boston Inc., Boston (1983); With the assistance of Musili, C., Nori, M., Previato E., Stillman, M. 15. Mumford, D.: Tata lectures on theta II. Progress in Mathematics, vol. 43. Birkh¨ auser Boston Inc., Boston (1984); Jacobian theta functions and diﬀerential equations, With the collaboration of Musili, C., Nori, M., Previato, E., Stillman, M., Umemura, H. 16. Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986); Corrected reprint of the 1986 original (1986)

Small-Span Characteristic Polynomials of Integer Symmetric Matrices James McKee Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, England, UK [email protected]

Abstract. Let f (x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd . The span of f (x) is deﬁned to be αd − α1 . Monic irreducible f (x) of span less than 4 are special. In this paper we give a complete classiﬁcation of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we ﬁnd some low-degree polynomials that do not arise as the minimal polynomial of any integer symmetric matrix: these provide lowdegree counterexamples to a conjecture of Estes and Guralnick [6].

1 1.1

Introduction History of the Small Span Problem

Let f (x) ∈ Z[x] be a monic polynomial having only real roots. If these roots are α1 ≤ . . . ≤ αd then we say that f (x) has span αd − α1 . In the case where f (x) is irreducible, the roots are (Galois) conjugates of each other and we then refer to {α1 , . . . , αd } as a conjugate set. If a real interval I has length strictly less than 4, then it is known [19] that I contains only ﬁnitely many conjugate sets of algebraic integers. If I has length greater than 4 then it contains inﬁnitely many such conjugate sets [17]. The problem remains open for intervals of length exactly 4, unless the endpoints are integers, in which case there are inﬁnitely many such sets [11]. Monic f (x) ∈ Z[x] of span less than 4 have therefore attracted some interest: for convenience we shall call these small-span polynomials. The span is unchanged if we replace f (x) by εdeg f f (εx + c) for any choice of ε ∈ {−1, 1} and any integer c: two polynomials related in this way are deemed to be equivalent. The number of equivalence classes of small-span polynomials of any given degree is ﬁnite. Robinson [18] produced a complete list of representatives for degrees up to 6, with conjectured lists for degrees 7 and 8 that were later veriﬁed as complete. Recently Capparelli, Del Fra and Sci` o [2] extended this computation (using new techniques) up to degree 14. For any natural number m, the totally real algebraic integer 2 cos(2π/m) has its conjugate set lying in the interval [−2, 2]; we call the minimal polyomial of such a number a cosine polynomial. Examples of irreducible small-span f (x) not equivalent to one of these cosine polynomials are of special interest. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 270–284, 2010. c Springer-Verlag Berlin Heidelberg 2010

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1.2

271

Characteristic Polynomials of Integer Symmetric Matrices

For any n-by-n integer symmetric matrix A we deﬁne its characteristic polynomial, χA (x), by χA (x) = det(xI − A), where I is the n-by-n identity matrix. Clearly χA (x) is a monic polynomial with integer coeﬃcients; moreover all its roots are real since A is a real symmetric matrix. We deﬁne the span of A to be the span of its characteristic polynomial, and we say that A is a small-span integer symmetric matrix if it has span less than 4. A more usual measure of the size of the eigenvalues of A is its spectral radius, deﬁned to be the largest modulus of any eigenvalue. Plainly the span of A is bounded above by twice its spectral radius. If the spectral radius is at most 2, then the characteristic polynomial is a small-span cosine polynomial (or a product of such polynomials). See [14] for a classiﬁcation of all integer symmetric matrices of spectral radius below 2.019: there are no non-cosine small-span examples. There is a similar list in [14] of all f (x) arising as characteristic polynomials of integer symmetric matrices for which the Mahler measure of xdeg f f (x + 1/x) is below 1.3: if the Mahler measure is 1, then one has a cosine example, and amongst those for which the Mahler measure is close to 1 one ﬁnds some, but not all, non-cosine small-span examples. Petrovi´c [16] classiﬁed all graphs whose characteristic polynomial has span at most 4. From this one can easily deduce which cases give span less than 4. The adjacency matrices of such graphs are special cases of integer symmetric matrices, with the entries restricted to {0, 1}, and with only zero entries on the main diagonal. If f (x) ∈ Z[x] is monic and totally real, then one can sensibly ask whether or not it arises as the characteristic polynomial of an integer symmetric matrix. Not every such f (x) arises in this way: we shall see some examples that do not, below. On the other hand, it is known (see [5], or [1]) that every totally real algebraic integer α is the eigenvalue of some integer symmetric matrix A, so that the minimal polynomial of α divides χA (x). 1.3

Minimal Polynomials of Integer Symmetric Matrices: A Conjecture of Estes and Guralnick

With mystery surrounding the question of which polynomials f (x) arise as χA (x) for some integer symmetric matrix A, Estes and Guralnick [6] turned their attention to the minimal polynomial mA (x), deﬁned as the monic polynomial in Z[x] of minimal degree such that mA (A) = 0. One has that mA (x) divides χA (x), and that every root of χA is a root of mA [9, §11.6]. For an integer symmetric matrix A, the minimal polynomial mA (x) must be separable (i.e., its roots are distinct) since A is diagonalisable. Estes and Guralnick showed [6, Corollary C] that if f (x) ∈ Z[x] has degree n ≤ 4, has all roots real, and is monic and separable, then f (x) is the minimal polynomial of a 2n-by-2n integer symmetric matrix. For example, one can easily show that x2 − 3 is not the characteristic polynomial of an integer symmetric matrix, but it satisﬁes all the hypotheses of the Estes-Guralnick theorem, and sure enough we ﬁnd that

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⎛

−1 ⎜ 1 ⎜ ⎝ 1 0

⎞ 1 1 0 1 0 1⎟ ⎟ 0 1 −1 ⎠ 1 −1 −1

has minimal polynomial x2 − 3. For a less trivial example, we shall see below in §3 that x3 − 4x − 1 is not the characteristic polynomial of any integer symmetric matrix. Yet it is the minimal polynomial of ⎞ ⎛ 1 0 1 1 1 0 ⎜ 0 1 1 −1 0 1 ⎟ ⎟ ⎜ ⎜ 1 1 −1 0 0 0 ⎟ ⎟ ⎜ ⎜ 1 −1 0 −1 0 0 ⎟ . ⎟ ⎜ ⎝1 0 0 0 0 0⎠ 0 1 0 0 0 0 At the end of their paper [6], Estes and Guralnick ask whether or not every monic, separable, totally real f (x) ∈ Z[x] is the minimal polynomial of an integer symmetric matrix: they conjecture that the answer is ‘yes’ (p. 84). This question was answered in the negative by Dobrowolski [4]. He showed that any degree-n irreducible minimal polynomial of an integer symmetric matrix has discriminant at least nn , and then observed that inﬁnitely many cosine polynomials have smaller discriminant than this (for a precise formula for the discriminant of a cosine polynomial see [18, p. 554], derived from a formula in [12]). The smallest degree of any of Dobrowolski’s counterexamples to the conjecture of Estes and Guralnick is 2880; we shall give below some counterexamples of degree 6, for which the discriminant is too large for Dobrowolski’s argument to apply. It remains an open problem as to whether or not there are any counterexamples of degree 5. 1.4

The Contributions of This Paper

In this paper we ask which monic, irreducible, totally real polynomials in Z[x] of span less than 4 arise as characteristic polynomials of integer symmetric matrices. For this restricted class of polynomials, we are able to give a complete classiﬁcation (Theorem 3; more precisely, Theorem 3 classiﬁes the integer symmetric matrices that give rise to small-span characteristic polynomials). As a byproduct of this, we are able to address the conjecture of Estes and Guralnick about minimal polynomials [6, p. 84], and produce some counterexamples with degree as small as 6. In §2 we describe the algorithm for computing the complete list of representatives of equivalence classes of small-span integer symmetric matrices up to any desired degree. This builds on similar algorithms in [13] and [14]. In §3 we detail the results. In §4, we prove a classiﬁcation theorem for the small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. The paper concludes by applying this to the conjecture of Estes and Guralnick.

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2 2.1

273

The Growing Algorithm Equivalence

Let On (Z) be the orthogonal group of n-by-n signed permutation matrices. If A is an n-by-n integer symmetric matrix, and P ∈ On (Z), then we call A and P −1 AP = P T AP strongly equivalent. Strongly equivalent matrices have the same characteristic polynomial. Let A be an n-by-n integer symmetric matrix, and let c be any integer. Then χA+cI (x) = χA (x − c). Also χ−A (x) = (−1)n χA (−x). Thus if f (x) is the characteristic polynomial of an integer symmetric matrix, then so is any polynomial equivalent to f (x) in the sense of §1.1. We deﬁne integer symmetric matrices A and B to be equivalent if A is strongly equivalent to ±B + cI for some integer c. Thus equivalent matrices have equivalent characteristic polynomials. If A has span less than 4, then by adding cI for suitable c we can move to an equivalent matrix B with all eigenvalues in the interval [−2, 3); if B has an eigenvalue greater than 2.5, then it has no eigenvalue smaller than −1.5, and we replace B by the equivalent matrix −B + I. We see that any small-span integer symmetric matrix is equivalent to one with all eigenvalues in the interval [−2, 2.5). Our conclusion is that in order to ﬁnd which monic, totally real polynomials in Z[x] of degree n and span less than 4 arise as characteristic polynomials of integer symmetric matrices, it is enough to ﬁnd all n-by-n integer symmetric matrices up to strong equivalence that satisfy both: (i) the span is less than 4; and (ii) all eigenvalues lie in the interval [−2, 2.5). 2.2

Indecomposable Matrices

An integer symmetric matrix will be called decomposable if one can apply a permutation to the rows, and the same permutation to the columns, to produce a matrix in block diagonal form with more than one block. A matrix that is not decomposable is indecomposable. The characteristic polynomial of a decomposable matrix is the product of the characteristic polynomials of its blocks. In attempting to understand which polynomials arise as characteristic polynomials, it is therefore enough to restrict to indecomposable matrices. There is a nice graph-theoretic description of the property of being indecomposable. The underlying graph of an integer symmetric matrix has vertices labelled by the rows, with an edge between vertex i and vertex j precisely when the (i, j)-entry in the matrix is non-zero. Then a matrix is indecomposable if and only if the underlying graph is connected. We record a standard lemma whose proof is obvious given this interpretation. Lemma 1. Let A be an n-by-n indecomposable matrix, with n ≥ 2. Then there is a choice of i between 1 and n such that deleting row i and column i from A leaves an indecomposable submatrix. When convenient, we shall use the language of graphs to talk about our matrices. We speak of vertices to indicate rows, edges to indicate non-zero matrix entries,

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with natural interpretations of paths, cycles, connectedness, and so on. The distance between two vertices will mean the minimal number of edges on a path from one to the other. If our matrix has a non-zero entry on the diagonal, then we refer to the corresponding vertex as being charged. Lemma 1 is a corollary of the following slightly more precise result, which we shall exploit later. Lemma 2. Let G be a connected graph with at least 2 vertices, and let i and j be vertices for which the distance between i and j is maximal. Then deleting vertex i (and all incident edges) does not disconnect the graph. Proof. Suppose that after deleting i there was a vertex k not in the same component as j. Then every path from k to j in G would have to pass through i, and so the distance from k to j would be strictly greater than that from i to j, giving a contradiction. 2.3

Interlacing

We shall make much use of Cauchy’s interlacing theorem [3] (for more accessible proofs, see [8], [10] or [7]). Theorem 1 (Cauchy, 1829). Let A be an n-by-n integer symmetric matrix, with n ≥ 2, and let B be an (n − 1)-by-(n − 1) submatrix formed by deleting row i and column i from A (for some choice of i between 1 and n). Let λ1 ≤ λ2 ≤ . . . ≤ λn be the eigenvalues of A, and let μ1 ≤ . . . ≤ μn−1 be those of B. Then these two sets of eigenvalues interlace: λ1 ≤ μ1 ≤ λ2 ≤ μ2 ≤ . . . ≤ μn−1 ≤ λn . From this we have an immediate corollary which will be of use in our algorithm for computing small-degree small-span integer symmetric matrices. Corollary 1. Let A be an n-by-n integer symmetric matrix, with n ≥ 2, and let B be an (n − 1)-by-(n − 1) submatrix formed by deleting row i and column i from A (for some choice of i between 1 and n). Then the span of A is at least as large as the span of B. Moreover, if A has all its eigenvalues in the interval [−2, 2.5), then so does B. 2.4

Reduction

Our situation would be considerably more pleasant if for any integer symmetric matrix A we could quickly ﬁnd a canonical representative of its strong equivalence class. Unfortunately this is not the case, and we content ourselves with a quick ‘reduction’ process that gives us a semi-canonical representative, but with the possibility that there are several diﬀerent ‘reduced’ elements in the same strong equivalence class. Some balance must be struck between the speed of reduction and the possible number of strongly-equivalent reduced matrices.

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In practice we used two complementary reduction processes, which for convenience we call fast reduction and slow reduction. The ﬁrst of these is generally much faster and was used to identify quickly many cases of strong equivalence. The slower reduction process was then used to produce further weeding of our lists of matrices. This double reduction was then repeated until no further weeding was achieved. Any matrices in the ﬁnal list having the same characteristic polynomial (and sharing a few other invariants of strong equivalence) were ﬂagged for further inspection: in all such cases, either an equivalence between the two examples was found, or some simple argument established that the two were not equivalent. The principle of fast reduction is to give a ‘score’ to each row of the matrix, such that the multiset of scores is invariant under strong equivalence. The rows and columns would then be ordered according to this score. Finally, if the ﬁrst non-zero entry of any row was negative (and not on the diagonal) then that row (and the corresponding column) would have its sign changed. A more complicated scoring system would take longer to compute but would reduce the number of rows having equal score and thereby reduce the risk of having more than one possible reduced matrix in the same strong equivalence class. The scoring system that we used was to compute the ﬁrst three powers of the matrix A and then rank rows by a linear combination of: (i) the sum of the moduli of the entries in the row; (ii) the same for A2 ; (iii) the same for A3 ; (iv) the size of the diagonal entry. The aim of slow reduction was to attempt to ﬁnd the lexicographically smallest element of a strong equivalence class. If always successful then this would provide a perfect reduction process, but to achieve this perfection would be painfully slow. Instead one deemed a matrix to be reduced if it was ‘locally minimal’ with respect to lexicographical ordering in the sense that: (i) changing the sign of any row (and column) would give a larger matrix (in the sense of the ordering); (ii) swapping any two rows (and the corresponding columns) would give a larger matrix; (iii) cyclically permuting any three rows (and the corresponding columns) would produce a larger matrix. There is no claim that the combination of fast and slow reduction detailed above is optimally eﬃcient, but both reduction methods signiﬁcantly reduced the number of matrices needing to be considered, and enabled the computations to proceed smoothly up to the sizes detailed below. 2.5

Bounds on Entries and Valencies

Using interlacing (Theorem 1 to bound the size of diagonal entries, and Corollary 1 to deal with oﬀ-diagonal entries) we can rapidly restrict the possible entries for integer symmetric matrices that are of interest to us. Lemma 3. Let A be a small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). Then all entries of A have absolute value at most 2, and all oﬀ-diagonal entries have absolute value at most 1.

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Proof. Let a be a diagonal entry in A. Then since (a) has a as an eigenvalue, repeated use of Theorem 1 shows that A has an eigenvalue with modulus at least as large as |a|. Our restriction on the eigenvalues of A shows that |a| ≤ 2. Let b be an oﬀ-diagonal entryof A. Then deleting other rows and columns ab . By repeated use of Corollary 1, this gives a submatrix of the shape bc

submatrix must have span less than 4, giving (a − c)2 + 4b2 < 4. This implies |b| ≤ 1. The cases where there is an entry that has absolute value 2 are extremely restricted. The following Lemma describes the complete list. Lemma 4. Up to strong equivalence, the only indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5) and containing an entry of modulus greater than 1 are: (−2), (2),

⎞ ⎛ ⎞ ⎛ 2100 210 ⎜1 0 1 0⎟ 21 ⎝ 2 1 ⎟ , 1 0 1⎠,⎜ , ⎝0 1 0 1⎠ . 10 1 −1 010 0010

(The ﬁrst two matrices listed in Lemma 4 are equivalent, but not strongly equivalent.) Proof. Each of the ﬁve 1-by-1 matrices (−2), (−1), (0), (1), (2) was grown in all possible ways to larger indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5), allowing entries from {−2, −1, 0, 1, 2} in accordance with Lemma 3. After producing a provisional list of 2-by-2 matrices, this list was weeded by reduction, as described in §2.4. Repeating this growing process three more times revealed that there are no 5-by-5 examples containing an entry having modulus greater than 1, and by interlacing the same must be true for all larger indecomposable integer symmetric matrices. The output of this computation also established the advertised list. Having reduced to the problem of considering matrices that have absolute value at most 1, we now further restrict the possible entries in each row. Lemma 5. Let A be an indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). Then each row of A has at most 4 non-zero entries. Proof. After Lemma 4, we can suppose that all entries in A are from the set {−1, 0, 1}. If Lemma 5 were false, then by interlacing (and making use of strong equivalence) there would be a small-span integer symmetric matrix M with all eigenvalues in the interval [−2, 2.5) and with M being one of

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⎞ 0111 11 11111 −1 1 1 1 1 ⎟ ⎜ ⎜ 1 a b c d⎟ ⎜1 a b c d⎟ ⎜1 a b c d e⎟ ⎟ ⎜1 b f g h i⎟ ⎟ ⎜ ⎜ ⎟, ⎜ 1 b e f g⎟,⎜1 b e f g⎟,⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ 1 c f h i⎠ ⎝1 c f h i⎠ ⎜1 c g j k l⎟ ⎝1 d h k m n⎠ 1d g ij 1d g ij 1e i l no ⎛

⎞ ⎛

⎞

⎛

where the unspeciﬁed entries are all from {−1, 0, 1}. A computer search showed that no such matrix M exists. 2.6

The Algorithm

Lemma 1 and Corollary 1 suggest a means of ‘growing’ indecomposable smallspan integer symmetric matrices with all eigenvalues in [−2, 2.5) from smaller matrices. This idea has been used before for computing integer symmetric matrices with small spectral radius or small Mahler measure ([13] and [14]). Having established Lemmas 4 and 5, we grow indecomposable matrices with all entries coming from the set {−1, 0, 1}, and with the extra restriction that each row can contain no more than four non-zero entries. After producing a provisional list of r-by-r matrices, this list is weeded by reduction, as described in §2.4, before growing to produce a list of (r + 1)-by-(r + 1) matrices. The complete search up to 13-by-13 matrices was completed in under ﬁve hours on a single processor. This was enough to provide the computational element of the proof of Theorem 3 below. The computation was pushed up to 20-by-20 matrices in under six days; perfect agreement of the results with Theorems 2 and 3 for larger matrices provided conﬁdence in the correctness of the output for smaller matrices. The PARI code for all of this is freely available from the author on request. After each growing of a list of (n − 1)-by-(n − 1) matrices to a list of n-by-n matrices, any examples from the ﬁrst list that had not been grown to one or more examples in the second were recorded in a list of maximal examples. Some of these maximal examples ﬁtted into inﬁnite families, described in Theorem 2; others did not, and these we call sporadic.

3

Results

We shall call an indecomposable small-span integer symmetric matrix that has all eigenvalues in the interval [−2, 2.5) maximal if it cannot be obtained by deleting rows (and corresponding columns) from any larger indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5). It turns out that every indecomposable small-span integer symmetric matrix with all eigenvalues in the interval [−2, 2.5) can be grown to a maximal one (part of Theorem 3). In view of Corollary 1, it is enough to describe all the maximal matrices. Up to strong equivalence there are 197 sporadic examples and 10 inﬁnite families. In this section we tabulate the number of sporadic examples of

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each size, found by computation as outlined above. The inﬁnite families and the proof of completeness of the classiﬁcation will follow in §4 (Theorems 2 and 3). Members of the inﬁnite families all in fact have eigenvalues in the smaller interval [−2, 2]. The following table includes the three maximal examples from Lemma 4. Maximal examples that are members of the inﬁnite families of Theorem 2 are excluded: only the sporadic cases are counted. The computations had been done up to size 20-by-20, but the only maximal cases that were not covered by the inﬁnite families of Theorem 2 were 12-by-12 or smaller. That no further sporadic maximal examples arise is the point of Theorem 3. Sporadic maximal indecomposable small-span integer symmetric matrices with all eigenvalues in [−2, 2.5), up to strong equivalence n n-by-n cosine examples n-by-n non-cosine examples total 1 1 0 1 2 0 1 1 3 0 1 1 4 10 9 19 5 0 19 19 6 0 43 43 7 0 28 28 8 11 39 50 9 0 15 15 10 0 15 15 11 0 2 2 12 0 3 3 total 22 175 197 For degrees up to 8, most small-span irreducible polynomials arise as characteristic polynomials of integer symmetric matrices: it is simpler to record which of Robinson’s polynomials from [18] do not arise. It is interesting to note that all examples of degrees 4 and 5 appear. The missing examples for degrees 2 and 3 are those mentioned in §1.3 above, namely x2 − 3 and x3 − 4x − 1. The other missing polynomials are numbers 6g, 6i, 6k, 7j, 7k, 7l, 8a, 8c, 8l, 8m, 8t, 8u, 8y in Robinson’s list [18]. For degree 9, both of the inequivalent cosine polynomials arise as characteristic polynomials, and three other irreducibles: x9 − x8 − 9x7 + 7x6 + 28x5 − 15x4 − 34x3 + 10x2 + 12x − 1, x9 − 4x8 − 2x7 + 21x6 − 5x5 − 37x4 + 12x3 + 24x2 − 5x − 4, x9 − 3x8 − 5x7 + 18x6 + 7x5 − 34x4 − x3 + 20x2 − 3x − 1. For degree 10, the only irreducible small-span characteristic polynomial is the non-cosine example x10 − 5x9 + x8 + 26x7 − 21x6 − 49x5 + 40x4 + 42x3 − 20x2 − 15x − 1. For degree 11, the only one (up to equivalence) is the cosine case. For degree 13 and above, Theorem 3 (below) gives a complete description of which characteristic polynomials arise. All degree-13 examples that have span below 4 and all eigenvalues in the interval [−2, 2.5) in fact have all eigenvalues in the subinterval [−2, 2] (this is the content of Theorem 3), and hence are described in Theorem 2.

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The following table compares the complete lists of [18] and [2] with the results of the computations for characteristic polynomials, restricting to irreducible polynomials. Degree

1 2 3 4 5 6 7 8 9 10 11 12 13

4

Number of irreducible Number that arise as small-span polynomials characteristic polynomials up to equivalence: of integer symmetric matrices: cosine + non-cosine = total cosine + non-cosine = total 1+0=1 1+0=1 3+1=4 2+1=3 2+3=5 2+2=4 4 + 10 = 14 4 + 10 = 14 1 + 14 = 15 1 + 14 = 15 4 + 13 = 17 1 + 13 = 14 0 + 15 = 15 0 + 12 = 12 5 + 21 = 26 5 + 14 = 19 2 + 19 = 21 2+3=5 3 + 15 = 18 0+1=1 1 + 10 = 11 1+0=1 7 + 9 = 16 0+0=0 0+4=4 0+0=0

Classification of Small-Span Integer Symmetric Matrices

One result of our computations is that any indecomposable small-span 13-by13 integer symmetric matrix with all its eigenvalues in [−2, 2.5) in fact has all its eigenvalues in [−2, 2]. We shall now prove that this holds for all larger indecomposable matrices too. As a ﬁrst step, we classify those indecomposable small-span integer symmetric matrices that have all their eigenvalues in the interval [−2, 2]. After Lemma 4, we are reduced to considering matrices that have entries 0, 1 or −1. These are conveniently represented by charged signed graphs. Vertices are labelled with their charges (corresponding to diagonal entries of the matrix); oﬀ-diagonal entries 1 and −1 are represented respectively by solid and dotted edges. Zero charges can be omitted to reduce clutter. For example, the matrix ⎞ ⎛ t t t 1 1 0 0 @ 1 qq 1 ⎜1 0 1 1⎟ ⎟ ⎜ @ qq . ⎝ 0 1 1 −1 ⎠ is drawn as @qt 0 1 −1 0 In the graphs below, the symbol k t t

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denotes a path with k solid edges (and all vertices uncharged) between the displayed end vertices (if k = 0 then these end vertices are identiﬁed as a single vertex). Deﬁne + − + − graphs O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , Yn+ , Yn− , Zk,l , Zk,l as shown. 2k − 3 t t

2k − 1 t t J

J

Jt

tq q q q q q q t O2k (k ≥ 2)

+ − (k ≥ 1) O2k+1 (k ≥ 1) O2k+1

k t t ± Pk+3 (k ≥ 0)

1t

2k − 1 t t q J q J qq Jt

t−1

t @

@ tq q q [email protected] t1

1t

t @ @ k tq q q [email protected] t qt + Xk+5 (k ≥ 0)

1t

X4+ t @

@ tq q q [email protected] t−1 X4−

−1 t

t @ @ k tq q q [email protected] q t t − Xk+5 (k ≥ 0)

−1 t

1t

1t

@ 1 tq q q [email protected] q t1

1 tq q q [email protected] qt

@

@

@

Y3+ −1 t

−1 t

@

−1 t

@

@

@t−1

−1 t

Y3−

t1

−1 t

+ Yk+4

k t

k t

t @

t + Zk,l

k

t (k ≥ 0)

k @t t − Yk+4 (k ≥ 0)

@

@ l tq q q [email protected] q t t (k ≥ 0, l ≥ 0)

t1

t−1

t1

t−1

t1

t @

t − Zk,l

@ l tq q q [email protected] t qt (k ≥ 0, l ≥ 0)

t−1

Theorem 2. Every indecomposable small-span integer symmetric matrix M1 that has all its eigenvalues in the interval [−2, 2] is a submatrix of an indecomposable

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281

small-span integer symmetric matrix M2 that is maximal subject to being smallspan and having all its eigenvalues in [−2, 2.5). Up to strong equivalence, the possibilities for M2 are the sporadic maximal examples tabulated in Section 3 and the + adjacency matrices of the charged signed graphs O2k (k ≥ 4), O2k+1 (k ≥ 3), − ± + − + O2k+1 (k ≥ 2), Pn (n ≥ 6), Xn (n ≥ 7), Xn (n ≥ 4), Yn (n ≥ 6), Yn− (n ≥ 3), + − Zk,l (k ≥ l ≥ 0, except for (k, l) ∈ {(0, 0), (1, 0), (1, 1), (2, 1)}), Zk,l (k ≥ l ≥ 0) pictured above. Proof. This is a tedious but easy extension of the work in [13, §12] where all examples with eigenvalues in the open interval (−2, 2) were described; here we relax this to consider the intervals (−2, 2] and [−2, 2). A convenient technique is that of Gram vectors. If an integer symmetric matrix A has all its eigenvalues in [−2, 2], then both B = A + 2I and C = −A + 2I have all eigenvalues at least 0. Thus there are lists of Gram vectors v1 , . . . , vn and w1 , . . . , wn contained in Rn such that the (i, j)-entry of B (respectively C) is given by vi · vj (respectively wi · wj ). Now −2 is an eigenvalue of A if and only if v1 , . . . , vn are linearly dependent, and 2 is an eigenvalue of A if and only if w1 , . . . , wn are linearly dependent. We start by noting that the following charged signed graphs have span 4: in each case one readily writes down linearly dependent sets of Gram vectors as above, showing that both −2 and 2 are eigenvalues, following the ideas in [13]. t t t t t qt qt qt qq q q qq @ q @ q q q t t q qq @ q [email protected] t t tq q q [email protected] qt t t t @t t t k k k 1t −1 t t t @ @ @ @ @ @ @ @ t tq q q [email protected] qt qt t @t t tq q q [email protected] t 1 tq q q [email protected] −1 k k 1t 1 1 1 −1 t t t t−1 q qt @ q q qq @ @ [email protected] qq @ @ 1 tqq q q [email protected] 1 tq q q [email protected] q t1 q t tq q q q q t1 −1 t t−1 @t t k k −1 t 1t t t t t−1 1 t t t−1 @ −1 @ @ k @ @ @ −1 t −1 1 @t tq q q [email protected] t t t @t−1 1 tq q q q q t −1 k t t @ @ @ @ 1t 1t tq q q [email protected] t t t−1 t t tq q q [email protected] t−1 k k 1t t1 t t −1 @ @ @ @ @ @ 1 tq q q [email protected] 1t q t−1 @t−1 t t tq q q [email protected] q t t t−1 k k

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+ − Next we note that the charged signed graphs O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , + − + − Yn , Yn , Zk,l , Zk,l have all eigenvalues between −2 and 2 (they are equivalent to subgraphs of those listed in [13, §4]), and have span less than 4 (writing down Gram vector representations for each graph and its negative, one ﬁnds that in every case exactly one of the sets of Gram vectors is linearly independent). Finally we check readily that any connected subgraph of one of those in [13, §4] that does not contain any subgraph equivalent to one of the span-4 examples + − listed above must be a subgraph of one of O2k , O2k+1 , O2k+1 , Pn± , Xn+ , Xn− , + − + − Yn , Yn , Zk,l , Zk,l . The restrictions on n, k and l require a trawl through the sporadic examples to see which of them contain any of the members of these 10 inﬁnite families as subgraphs. For example, P5± is a subgraph of the maximal sporadic example t−1 t1 t t t t−1 .

Theorem 3. Up to strong equivalence, the indecomposable small-span integer symmetric matrices with all eigenvalues in the interval [−2, 2.5) are precisely the indecomposable submatrices of the 197 sporadic cases accounted for in §3 and the 10 inﬁnite families of Theorem 2. In particular, every such matrix with more than 12 rows has all its eigenvalues in the interval [−2, 2]. Proof. In view of Theorem 2 and the computational results of §3, it is enough to show that every indecomposable integer symmetric matrix with more than 12 rows and all its eigenvalues in the interval [−2, 2.5) in fact has all its eigenvalues in the interval [−2, 2]. Suppose for a contradiction that this is not the case. Let A be a counterexample that has as few rows as possible. We know from our computations that A has at least 14 rows, and this minimal counterexample would then have the property that any proper submatrix has all its eigenvalues in the interval [−2, 2]. The result now follows from the classiﬁcation of all integer symmetric matrices minimal subject to not all eigenvalues being in the interval [−2, 2]: there are no such matrices with more than 10 rows [14]. But the current case is much easier, so we outline a direct proof. The key idea in the proof is that the property of having all eigenvalues in the interval [−2, 2] is essentially described by local structure. In the general case treated in [14] this local structure is much more complicated than in the small-span case treated here. Let G be the charged signed graph with adjacency matrix A (using Lemma 4). Pick vertices u and v as far apart as possible in G. Deleting either u or v leaves a connected (Lemma 2) charged signed graph with all eigenvalues in [−2, 2] and with at least 13 vertices, and hence a connected subgraph of one of the inﬁnite families of Theorem 2. Deleting u leaves an underlying graph that is either a cycle or not. Suppose ﬁrst that the underlying graph of G with u deleted is a cycle. Since u and v are maximally distant in G, we deduce that u is joined to vertices as far (or almost as far) as possible from v on this cycle, and since deleting v from G must give a connected subgraph of one of the inﬁnite families of Theorem 2, the only possibility for G (up to strong equivalence) is a charged signed graph of the

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283

+ shape formed by identifying the end vertices of Zk,l , with the charges removed. But then A has all eigenvalues in the interval [−2, 2] (see [13]) and in fact also has span 4, giving two contradictions. Now suppose that deleting u does not leave a cycle. Then it leaves a structure that is up to strong equivalence either an uncharged path (perhaps with one + − negative edge) or is one of Pn± , Xn+ , Xn− , Yn+ , Yn− , Zk,l , Zk,l , perhaps with one or more vertices removed in a way that does not disconnect the graph. Then either v is near the middle and u is adjacent to vertices at or near both ends of this structure, or v is at one end and u is adjacent to vertices at or near the other end. Again one sees (on considering deleting v, and using the classiﬁcation in [13]) that A must have all eigenvalues in [−2, 2], giving a contradiction.

5

Low-Degree Counterexamples to a Conjecture of Estes and Guralnick

Let f (x) be a monic, irreducible, totally real, small-span polynomial of degree n > 6 that has all its eigenvalues in the interval [−2, 2.5) but is not the characteristic polynomial of an integer symmetric matrix. Suppose further that f (x) is not a cosine polynomial. Then f (x) cannot be the minimal polynomial of any integer symmetric matrix. For if it were, then the smallest such matrix would be indecomposable and have characteristic polynomial f (x)r for some r > 1. But Theorem 3 precludes the existence of such characteristic polynomials, since the degree rn would be greater than 12. In particular, none of the polynomials x7 − x6 − 7x5 + 5x4 + 15x3 − 5x2 − 10x − 1, x7 − 8x5 + 19x3 − 12x − 1 or x7 − 2x6 − 6x5 + 11x4 + 11x3 − 17x2 − 6x + 7 is the minimal polynomial of an integer symmetric matrix. These provide degree-7 counterexamples to the conjecture of Estes and Guralnick [6]. Finally we remark that none of the three degree-6 cosine polynomials x6 − 5 x − 6x4 + 6x3 + 8x2 − 8x + 1, x6 − 7x4 + 14x2 − 7 and x6 − 6x4 + 9x2 − 3 is the minimal polynomial of any integer symmetric matrix. Our computations revealed that these three do not arise as characteristic polynomials, nor as minimal polynomials for any 12-by-12 or 18-by-18 matrix. Moreover the smallest span of an indecomposable 19-by-19 matrix is already larger than the spans of all three of these degree-6 polynomials, so by interlacing they cannot appear as the minimal polynomial of any larger matrix. It remains an open problem as to whether or not there exists a degree-5, monic, separable, totally real polynomial that does not arise as the minimal polynomial of an integer symmetric matrix. All the small-span cases are covered, so the techniques of this paper cannot be applied.

Acknowledgments This work was prompted by conversations with Georges Rhin and Chris Smyth at a workshop on Discovery and Experimentation in Number Theory, at the

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Fields Institute, Toronto in September 2009: I am grateful to the organisers of that workshop. I have also beneﬁted from conversations with Gary Greaves. Finally, I thank the referees for their numerous helpful suggestions.

References 1. Bass, H., Guralnick, R., Estes, D.: Eigenvalues of symmetric matrices and graphs. J. Algebra 168, 536–567 (1994) 2. Capparelli, S., Del Fra, A., Sci` o, C.: On the span of polynomials with integer coeﬃcients. Math. Comp. 79, 967–981 (2010) 3. Cauchy, A.: Sur l’´equation a l’aide de laquelle on determine les in´egalit´es s´eculaires des mouvements des plan`etes. In: Oeuvres Compl`etes d’ Augustin Cauchy Seconde S´erie IX, pp. 174–195. Gauthier-Villars, Berkeley (1891) 4. Dobrowolski, E.: A note on integer symmetric matrices and Mahler’s measure. Canadian Mathematical Bulletin 51(1), 57–59 (2008) 5. Estes, D.: Eigenvalues of symmetric integer matrices. J. Number Theory 42, 292– 296 (1992) 6. Estes, D.R., Guralnick, R.M.: Minimal polynomials of integral symmetric matrices. Linear Algebra and its Applications 192, 83–99 (1993) 7. Fisk, S.: A very short proof of Cauchy’s interlace theorem. Amer. Math. Monthly 112, 118 (2005) 8. Godsil, C., Royle, G.: Algebraic Graph Theory. In: Graduate Texts in Mathematics, vol. 207. Springer, New York (2000) 9. Hartley, B., Hawkes, T.O.: Rings, modules and linear algebra. Chapman and Hall, Boca Raton (1970) 10. Hwang, S.-G.: Cauchy’s interlace theorem for eigenvalues of Hermitian matrices. Amer. Math. Monthly 112, 157–159 (2004) 11. Kronecker, L.: Zwei s¨ atse u ¨ber gleichungen mit ganzzahligen coeﬃcienten. J. Reine Angew. Math. 53, 173–175 (1857) 12. Lehmer, E.: A numerical function applied to cyclotomy. Bull. Amer. Math, Soc. 36, 291–298 (1930) 13. McKee, J.F., Smyth, C.J.: Integer symmetric matrices having all their eigenvalues in the interval [−2, 2]. J. Algebra 317, 260–290 (2007) 14. McKee, J.F., Smyth, C.J.: Integer symmetric matrices of small spectral radius and small Mahler measure, arXiv:0907.0371v1 15. Batut, C., Belebas, K., Bernardi, D., Cohen, H., Olivier, M.: PARI/GP version 2.3.4, http://pari.math.u-bordeaux.fr/ 16. Petrovi´c, M.M.: On graphs whose spectral spread does not exceed 4. Publ. Inst. Math. Beograd 34(48), 169–174 (1983) 17. Robinson, R.M.: Intervals containing inﬁnitely many sets of conjugate algebraic integers. In: Mathematical Analysis and Related Topics: Essays in Honor of George P´ olya, Stanford, pp. 305–315 (1962) 18. Robinson, R.M.: Algebraic equations with span less than 4. Math. Comp. 18(88), 547–559 (1964) ¨ 19. Schur, I.: Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeﬃzienten. Math. Z. 1, 377–402 (1918)

Decomposition Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field Koh-ichi Nagao Dept. of Engineering, Kanto Gakuin Univ., 1-50-1 Mutsuura Higashi Kanazawa-ku Yokohama 236-8501, Japan [email protected]

Abstract. We propose some kind of new attack which gives the solution of the discrete logarithm problem for the Jacobian of a curve defined over an extension field Fqn , considering the set of the union of factor basis and large primes B0 given by points of the curve whose x-coordinates lie in Fq . In this attack, an element of the divisor group which is written by a sum of some elements of factor basis and large primes is called (potentially) decomposed and the set of the factors that appear in the sum, is called decomposed factors. So, it will be called decomposition attack. In order to analyze the running of the decomposition attack, a test for the (potential) decomposedness and the computation of the decomposed factors are needed. Here, we show that the test to determine if an element of the Jacobian (i.e., reduced divisor) is written by an ng sum of the elements of the decomposed factors and the computation of decomposed factors are reduced to the problem of solving some multivariable polynomial system of equations by using the Riemann-Roch theorem. In particular, in the case of hyperelliptic curves of genus g, we construct a concrete system of equations, which satisfies these properties and consists of (n2 − n)g quadratic equations. Moreover, in the case of (g, n) = (1, 3), (2, 2) and (3, 2), we give examples of the concrete computation of the decomposed factors by using the computer algebra system Magma. Keywords: Decomposition Attack, Hyperelliptic curve, Discrete logarithm problem, Weil descent attack.

1

Introduction

In this work, we treat the solution of the discrete logarithm problem of the Jacobian of a curve C of genus g deﬁned over an extension ﬁeld Fqn (n ≥ 2) by decomposition attack. In particular, when C is a hyperelliptic curve and ng(≥ 3) is a small integer, we give the concrete algorithm for computing what is called decomposed factors. In [6], Gaudry proposes the decomposition attack for the Jacobian of a hyperelliptic curve deﬁned over a general ﬁnite ﬁeld Fq considering a set of factor basis given by the Fq -rational points of the curve. This attack is usually called ’Index Calculus’ and such variations are widely used [3], [11]. However, the behavior of this attack, when it is used for solving the discrete G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 285–300, 2010. c Springer-Verlag Berlin Heidelberg 2010

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logarithm of algebraic curve, is quite diﬀerent to the original index calculus, which is a method to compute indices, that is, discrete logarithms in multiplicative groups of ﬁnite prime ﬁelds. Because of this, we use the name decomposition attack to refer to the attack. By recent works on the decomposition attack, which are the improvements of [6], it is known that the techniques of 1) using rebalancing [5] and 2) using large primes [15], [13], [7] are available. On the contrary, the techniques of large prime variations of normal index calculus associated to number ﬁeld sieve are known as no contribution and do not lead to a decrease of the complexity. In [8](also c.f. [4]), Gaudry also presents the decomposition attack for an elliptic curve deﬁned over an extension ﬁeld Fqn considering the set of factor basis given by points of the curve whose x-coordinates lie in Fq . Actually, Gaudry proposes also the rebalancing and the large prime variations. In these variations, the set of factor basis B is taken by some subset of B0 which is given by points of the curve whose x-coordinates lie in Fq and an element in B0 \B is called large prime. In these methods, the test for the potential decomposedness of P ∈ E(Fqn ) (i.e., for being a sum of n elements of the B0 ) and the computation of the decomposed factors (i.e., n elements of B0 whose summation equals to P ) are reduced to the problem of solving some system of multivariable polynomial equations of degree 2n−1 , n variables, and n equations, using Semaev’s summation polynomials [14]. Moreover, Gaudry generalizes this decomposition attack to the case of the abelian varieties deﬁned over an extension ﬁeld, including the case of Jacobians of curves. However, in the case of non-elliptic curves, Semaev’s summation polynomials are not available. It is, in principle, possible to derive a similar system of equations using the group law. Unfortunately, such is cumbersome. In fact, in the case of the Jacobian of a hyperelliptic curve of genus g, the sum of ng generic points is needed. Assuming that an element of Jacobian is written by the Mumford representation and that the group law is done by the Cantor algorithm [2], since the Cantor algorithm needs g − 1 times reduction steps, explosions of the degree and terms occur in this computation. In this work, we show that instead of using the group law, another system of equations is obtained from the theory of Riemann-Roch spaces (only in the case of Jacobians of curves). With this tool, the system of the equations is now simple to compute, and its parameters are easily controlled. In particular, in the case of Jacobians of hyperelliptic curves, this system of the equations consists of (n2 − n)g quadratic equations in (n2 − n)g indeterminates. So, under the heuristic assumption that this system of the equations is (essentially) projectively 0-dimensional, the computational amount for solving this 2 system of equations is estimated by O(2(n −n)g·C ) where C is some constant less than 3. In the case of an elliptic curve (i.e., g = 1), this computational amount heuristically equals to that of Gaudry’s original equations system using Semaev’s summation polynomials.

Decomposition Attack for the Jacobian of a Hyperelliptic Curve

2

287

Decomposition Attack for the Jacobian of a General Plane Curve

This section adapts the idea of [8] to the setting of a smooth plane curve with a single missing point at inﬁnity, and presents an overview of the decomposition attack for the Jacobian of a general plane curve using the Riemann-Roch theorem. Let Ca be the aﬃne curve of genus g deﬁned over an extension ﬁeld Fqn (i.e., n ≥ 2) given by the equation f (x, y) = 0, and let C be the corresponding non-singular complete curve. Assume that Ca is non-singular. From this, we have a canonical embedding ι : Ca → C. It is also assumed that C\ι(Ca ) only consists of a single Fqn -valued point, which is denoted by ∞ and is called the point at inﬁnity. These assumptions are true for hyperelliptic curves so there is no problem for the main results of this work. Let D0 be a divisor of the form D0 = Q1 + .. + Qg − (g)∞

(1)

where Q1 , .., Qg ∈ C(Fqn ) and the multiset {Q1 , .., Qg } is stable under the action of galois group Gal(Fqn /Fqn ). Put φ1 (x) :=

g (x − x(Qi ))

(2)

i=1

and note that it is in Fqn [x]. Also put B0 := {P ∈ C |P = (x, y) ∈ C(Fqn ), x ∈ Fq }, as a set of factor basis and large primes. (Strictly saying, B0 must be a subset of JacC (Fqn ), and it is the set of the elements of the divisors P − ∞ where P has the above properties. Here, the term “−∞” is omitted for simplicity.) Assumption 1. Let n be a ﬁxed positive integer. Then the number of the mulng tisets P = {P , .., P } with P ∈ B , which satisfy the relation 1 ng i 0 i=1 Pi ∼ ng P for some diﬀerent (P = P ) multiset P = {P , .., P } with P 1 ng i ∈ B0 , i=1 i ng−ε is less than q , where ε is some positive constant. Here, we shortly state the validity of this assumption in the case of hyperelliptic curve. Let C : y 2 = f (x) be the equation of hyperelliptic curve. For any P = (x, y) ∈ C, put P¯ = (x, −y) ∈ C. So, there are series of trivial relations P + P¯ ∼ P + P¯ for any P, P ∈ B0 . The number of the multisets satisfying the condition of Assumption 1 and coming from these trivial relations is only O(q ng−1 ) and it seems to be no series including many trivial relations. So, Assumption 1 seems to be valid. Assumption 2. |B0 | ≈ q. Here, we also state the validity of this assumption in the case of hyperelliptic curve. Let C : y 2 = f (x) be the equation of hyperelliptic curve. If f (x) is chosen

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randomly, the probability that f (x) (x ∈ Fq ) is square in Fqn is around 1/2 and this assumption seems to hold. In the following, we assume Assumption 1 and Assumption 2. From these assumptions, we see easily that since “the number of the divisors of the form(1)”≈ q gn , the probability that there are some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed) such that + P2 + ... + Png − (ng)∞ D0 + P1 g = i=1 Qi + P1 + P2 + ... + Png − (ng + g)∞ ∼ 0,

(3)

is approximately 1/(gn)!, when q ng. Definition 1. If a divisor D0 is written by the form (3) for some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed), D0 is called potentially decomposed and in this case, the elements P1 , P2 , .., Png are called decomposed factors and the multiset {Pi }ng i=1 is called decomposed divisor. We now ﬁx D0 and discuss how it can be tested that D0 is potentially decomposed and the decomposed factors can be computed. So, Q1 , ..., Qg and φ1 (x), which are dependent on D0 , are also ﬁxed. Let D = P ∈C(Fqn ) np P , np ∈ Z be a divisor of C/Fqn . Assume that D is sta ble under the action of galois group Gal(Fqn /Fqn ). Put deg(D) := P ∈C(Fqn ) np , and L(D) := {f ∈ Fqn (C) | (f )+D ≥ 0}∪{0}. From the Riemann-Roch theorem (cf [10] Corollary A.4.2.3), we have the following lemma. Lemma 1. (Riemann-Roch) 1) L(D) is an Fqn vector space. 2) If deg(D) ≥ 2g − 1, dim L(D) = deg(D) − g + 1. g From this Lemma, dim L((ng)∞ − D0 ) = dim L((ng + g)∞ − i=1 Qi ) = ng − g + 1. Let {f0 (x, y), f1 (x, y), ..., fng−g (x, y)} be a base of L((ng)∞ − D0 )) and an element h ∈ L((ng)∞ − D0 ) is written by a0 f0 (x, y) + a1 f1 (x, y) + ... + ang−g fng−g (x, y)

(4)

where ai are values in Fqn . From Hess [9], we have the following lemma. Lemma 2. A base of L((ng)∞ − D0 ) is computable within Poly(ng log q) time. Let h(x, y) := A0 f0 (x, y) + A1 f1 (x, y) + ... + Ang−g fng−g (x, y)

(5)

be a multivariable polynomial in Fqn [A0 , ..., Ang−g , x, y]. For aaﬀ = (a0 , a1 , ..., ang−g ) ∈ Ang−g+1 (Fqn ) and some polynomial p(x) ∈ Fqn [A0 , ..., Ang−g , x], let paaff (x) be the polynomial obtained from p(x) by substituting ai for Ai .

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Definition 2. A multivariable polynomial p(x) in Fqn [A0 , ..., Ang−g , x] is called A-homogenous, when paaﬀ (x) = Const × pkaaﬀ (x) holds for all aaﬀ = (a0 , a1 , ..., ang−g ) ∈ Ang−g+1 (Fqn ) and k ∈ F∗qn . For

apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn )

and some A-homogenous polynomial p(x) ∈ Fqn [A0 , ..., Ang−g , x], let monic(paaff (x)) be the polynomial obtained from p(x) by substituting ai for Ai and dividing by the leading coeﬃcient. Now, we compute the intersections of hapro (x, y) = 0 on C. Remember that the equation of Ca is f (x, y) = 0. Put S(x) := Resultanty (f (x, y), h(x, y)). From this construction, we then have the following lemma. Lemma 3. 1) S(x) is a multivariable A-homogeneous polynomial in Fqn [A0 , .., Ang−g , x]. 2) degx S(x) = ng + g. 3) φ1 (x) | S(x). Proof. 1) is trivial. For any apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn ), since hapro (x, y) has only poles (ng + g)∞ on points at inﬁnity, we have 2) and since hapro (x, y) have zeros at each Qi ’s, we have 3). Put g(x) := S(x)/φ1 (x). Since φ1 (x) ∈ Fqn [x], g(x) is also a multivariable Ahomogeneous polynomial in Fqn [A0 , .., Ang−g , x]. Thus, g(x) is written in the form g(x) = Cng xng + Cng−1 xng−1 + ... + C0 where each Ci ∈ Fqn [A0 , .., Ang−g ] has the same multi degree of Ai . Note that if the indeterminates Ai s are replaced by values ai and the obtained polynomial is divided by the leading coeﬃcient, then one obtains a polynomial monic(gapro (x)) in Fqn [x]. The solutions of monic(gapro (x)) = 0 mean the x-coordinates of the intersections hapro (x, y) = 0 on C except Q1 , ..., Qg . So, we have the following lemma. Lemma 4. The condition that D0 is potentially decomposed is equivalent to the following: There is some apro = (a0 , a1 , ..., ang−g ) ∈ Png−g (Fqn ) such that monic(gapro (x)) ∈ Fq [x] and monic(gapro (x)) ∈ Fq [x] factors completely in Fq [x]. Now, we ﬁnd such ai ’s. Let [α0 (= 1), α1 , .., αn−1 ] be a base of Fqn /Fq . We ﬁx this base. Let Ai,j (1 ≤ i ≤ ng, 0 ≤ j ≤ n − 1) be new indeterminates over Fq , and let us consider the polynomials obtained by substituting A0 by 1 and Ai by n−1 j=0 Ai,j αj (1 ≤ i ≤ ng − g) in g(x). Let us denote the coeﬃcients obtained in this way again by Ci . Then the coeﬃcients can be written in the form Ci =

n−1

Ci,j αj ,

Ci,j ∈ Fq [∪1≤i≤ng, 0≤j≤n−1 {Ai,j }].

j=0

Then, the condition that there is some apro ∈ Png−g (Fqn ) satisfying 1) monic(gapro (x)) ∈ Fq [x] and

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2) First coordinate of apro is non-zero, is equivalent to the condition that the system of the equations Ci,j = Ti Cng,j

(0 ≤ i ≤ ng − 1, 0 ≤ j ≤ n − 1)

(6)

of (n2 + n)g indeterminates ∪{Ai,j } and T0 , ..., Tng−1 deﬁned over Fq has some solutions Ai,j = ai,j , Ti = ti in Fq . In this case, monic(gapro (x)) is written by xng−g + tng−g−1 xng−g−1 + ... + t1 x + t0 .

(7)

Thus, the test of the decomposedness of D0 and the computation of the decomposed factors are reduced to ﬁnd the solutions of the system of the equations (6) and factorizations of the polynomials (7). In the next section, we will investigate the case of the hyperelliptic curve. In this case, there is a concrete representation of the Riemann-Roch space, and so we have a more concrete system of equations.

3

Decomposition Attack for the Jacobian of a Hyperelliptic Curve

Now, we discuss the special case of Jacobians of hyperelliptic curves. In this case, there are concrete representations of the Riemann-Roch space and some techniques that g(x) can be taken as a monic polynomial, and from this, a simple system of equations is derived. Let C be a hyperelliptic curve (including an elliptic curve) of genus g of the form C : y 2 = f (x), where f (x) = x2g+1 + a2g x2g + ... + a0 over Fqn where the characteristic of Fq is not 2 and n ≥ 2. Put ∞ by the unique point at inﬁnity on C. Let D0 be a reduced divisor (i.e.,Fqn -rational point of the Jacobian) of C. To represent D0 , we use the so-called Mumford representation: D0 = (φ1 (x), φ2 (x)), where φ1 (x) ∈ Fqn [x] is a monic polynomial with deg(φ1 (x)) ≤ g and φ2 (x) ∈ Fqn [x] satisﬁes deg(φ2 (x)) < deg(φ1 (x)) and f (x) − φ2 (x)2 ≡ 0 mod φ1 (x). In the following, we will assume deg(φ1 (x)) = g. This assumption holds for all but a negligible fraction of divisor classes D0 . Note that there are Q1 , .., Qg ∈ C(Fqn )\{∞} satisfying the equation (1) and the multiset {Q1 , .., Qg } is stable under the action of galois group Gal(Fqn /Fqn ). Similarly, put B0 := {P ∈ C |P = (x, y) ∈ C(Fqn ), x ∈ Fq } as a set of factor basis and large primes. Then, from the Assumption 1 and Assumption 2, we can see easily that the probability, that there are some P1 , P2 , .., Png ∈ B0 (exactly ng elements, Pi = Pj for some i = j being allowed) satisfying the equation (3), is approximately 1/(gn)!, when q ng. In the following, we ﬁx a reduced divisor D0 . So, φ1 (x), φ2 (x), and Q1 , ..., Qg , which are dependent on D0 , are also ﬁxed. In this work, we show the following theorem.

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Theorem 1. Let V1 , V2 , ..., V(n2 −n)g be indeterminates and let D0 be a reduced divisor of C/Fqn . Then there are some computable degree 2 polynomials Ci,j ∈ Fq [V1 , V2 , ..., V(n2 −n)g ]

(0 ≤ i ≤ ng − 1, 0 ≤ j ≤ n − 1)

satisfying the following: The condition that D0 is potentially decomposed is equivalent to the following 1) and 2): 1) The system of equations {Ci,j = 0 | 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1} has some 2 solution v = (v1 , .., v(n2 −n)g ) ∈ A(n −n)g (Fq ). 2) Put ci = Ci,0 (v1 , .., v(n2 −n)g ) for 0 ≤ i ≤ ng − 1. Then G(x) = xng + cng−1 xng−1 + ... + c0 ∈ Fq [x] factors completely. Moreover, if D0 is potentially decomposed, the x-coordinates of the decomposed factors are the solutions of G(x) = 0. From this theorem, the test, whether D0 is potentially decomposed and the computation of the decomposed factors (if possible), is reduced to solving the system of the equations {Ci,j = 0 | 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1} and factorizing the polynomials G(x) obtained form the solutions of the system of these equations. In the following, we construct such multivariable polynomials {Ci,j } and show Theorem 1. From the equation of C, we see ord∞ x = 2, and ord∞ y = 2g + 1. Put N1 := and N2 := ng−g−1 .

(n+1)g 2 2 Lemma 5. 1) N1 + N2 = ng − 1. 2) N2 + g − 1 < N1 . Proof. Trivial. Lemma 6. {1, x, x2 , .., xN1 , y, xy, ...xN2 y} is a base of L((ng + g)∞). Proof. From ord∞ x = 2, ord∞ y = 2g + 1, each element in the above list is in L( (ng +g)∞). The independence is from the deﬁnition of the hyperelliptic curve. Thus, since the number of the elements of the list N1 + N2 + 2 = ng + 1 is the same as the dim L((ng + g)∞) (from Lemma 1), we ﬁnish the proof. Lemma 7 {φ1 (x), φ1 (x)x, ..., φ1 (x)xN1 −g , (y − φ2 (x)), (y − φ2 (x))x, ..., (y − φ2 (x))xN2 } is a base of L((ng)∞ − D0 ) = L((ng + g)∞ − gi=1 Qi ). Proof. From the deﬁnition of φ1 (x) and φ2 (x), each element in the list has a zero at each Qi . Since deg(φ1 (x)) = g, deg(φ2 (x)) ≤ g − 1, and N2 + g − 1 < N1 (from Lemma 5), each element in the list has at most (ng + g) poles at ∞. Then they are in L((ng)∞ − D0 ). Now, we show the independence. Assume they are not independent, and there are some non zero f1 (x), f2 (x) ∈ Fqn [x] such that φ1 (x)f1 (x) + (y − φ2 (x))f2 (x) = 0. However, the relation φ1 (x)f1 (x) + (y − φ2 (x))f2 (x) = 0 induces yf2 (x) ∈ Fqn [x] and f1 (x) = f2 (x) = 0. As this is a contradiction, they are independent. On the other hand, the number of the elements in the list is N1 + N2 + 2 − g = ng − g + 1 from Lemma 5, which is the same as the dim L((ng)∞ − D0 ). So we ﬁnish the proof.

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From Lemma 7, an element h ∈ L((ng)∞ − D0 ) is written by h(x, y) = φ1 (x)(a0 + a1 x+ ...+ aN1 −g xN1 −g )+ (y − φ2 (x))(b0 + b1 x+ ...+ bN2 xN2 ) (8) where ai ,bi are values in Fqn . Lemma 8. Let h(x, y) ∈ L((ng)∞ − D0 ). Assume div(h(x,y)) is written in the form P1 + P2 + ... + Png + gi=1 Qi − (ng + g)∞ for Pi ∈ C(Fqn )\{∞}. Then we have the following: 1) aN1 −g = 0 when ng + g is even. 2) bN2 = 0 when ng + g is odd. Proof. When ng + g is even, assume aN1 −g = 0, thus we have the order of the pole of h(x, y) at ∞ being truly less than ng + g and div(h(x, y)) is not written by the form of (3). Similarly, when ng + g is odd, assume bN2 = 0. Thus we have the order of the pole of h(x, y) at ∞ being truly less than ng + g and div(h(x, y)) is not written by the form of (3). So, we can assume that aN1 −g = 0, if ng + g is even, and bN2 = 0, if ng + g is odd. Now, we compute the intersections of h(x, y) = 0 on C. For this purpose, y must be eliminated. Note that the point (x, y) fulﬁlls h(x, y) = 0, if and only if the equation −φ1 (x)(a0 + a1 x + ... + aN1 −g xN1 −g ) + φ2 (x)(b0 + b1 x + ... + bN2 xN2 ) . b0 + b1 x + ... + bN2 xN2 (9) holds. By this y’s representation, the number of the parameters must be decreased. So, put aN1 −g = 1 when ng + g is even and put bN2 = 1 when ng + g is odd (this can be done from the above lemma). Also put M1 = when ng + g is even N2 N1 − g − 1 when ng + g is even , and M2 = . N1 − g N2 − 1 when ng + g is odd when ng + g is odd Note that M1 + M2 = ng − g − 2 from Lemma 5. Put −(denominator of (9))2 f (x) + (numerator of (9))2 , if ng + g is even s(x) := . (denominator of (9))2 f (x) − (numerator of (9))2 , if ng + g is odd y=

and let S(x) be the multivariable polynomial obtained from the deﬁnition of s(x) replacing the values ai and bi by the indeterminates Ai and Bi . From the construction, S(x) is a monic polynomial of the degree ng + g, whose coeﬃcients are degree 2 polynomials in Fqn [A0 , .., AM1 , B0 , .., BM2 ], and φ1 (x)|S(x). Put g(x) := S(x)/φ1 (x). Since φ1 (x) is a monic polynomial in Fqn [x], g(x) is also a monic polynomial of degree ng, whose coeﬃcients are degree 2 polynomials in Fqn [A0 , .., AM1 , B0 , .., BM2 ]. Put Ci ∈ Fqn [A0 , .., AM1 , B0 , .., BM2 ] by i-th coeﬃcient of g(x), i.e., g(x) = xng + Cng−1 xng−1 + ... + C0 .

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Similarly, for v = (a0 , ...aM1 , b0 , ..., bM2 ) ∈ AM1 +M2 +2 (Fqn ) and some polynomial p(x) in Fqn [A0 , ..., AM1 , BM0 , ..., BM2 , x], let pv (x) be the polynomial obtained from p(x) by substituting ai and bi for Ai and Bi . Then, the zeros of gv (x) = 0 are the x-coordinate of the intersections of h(x, y) = 0 on C except Q1 , ..., Qg . Thus, we have the following lemma. Lemma 9. The condition that D0 is a potentially decomposed reduced divisor is equivalent to the following: There is some v = (a0 , .., aM1 , b0 , ...bM2 ) ∈ AM1 +M2 +2 (Fqn ) such that gv (x) ∈ Fq [x] and gv (x) ∈ Fq [x] factors completely in Fq [x]. We now show how to ﬁnd ai in Fqn (0 ≤ i ≤ M1 ) and bi in Fqn (0 ≤ i ≤ M2 ) such that gv (x) in Fq [x]. Let [α0 (= 1), α1 , .., αn−1 ] be a base of Fqn /Fq and ﬁx this base. Let Ai,j (0 ≤ i ≤ M1 , 0 ≤ j ≤ n − 1) and Bi,j (0 ≤ i ≤ M2 , 0 ≤ j ≤ n − 1) be new indeterminates over Fq . Note that the number of the indeterminates {Ai,j } ∪ {Bi,j } is (M1 + M2 + 2)n = (N1 + N2 − g + 1)n = (n2 − n)g. For simplicity, substitute the variables Ai,j (0 ≤ i ≤ M1 , 0 ≤ j ≤ n − 1) and Bi,j (0 ≤ i ≤ M2 , 0 ≤ j ≤ n − 1) by {V1 , V2 , ..., V(n2 −n)g }. Let us consider the n−1 n−1 polynomials obtained by substituting Ai by j=0 Ai,j αj and Bi by j=0 Bi,j αj in g(x). Also let us denote the coeﬃcients obtained in this way again by Ci . Then the coeﬃcients can be written in the form Ci =

n−1

Ci,j αj ,

Ci,j ∈ Fq [V1 , V2 , ..., V(n2 −n)g ].

j=0

Thus from Lemma 9, the condition gv (x) ∈ Fq [x] is equivalent to the condition that there are some v1 , v2 , ..., v(n2 −n)g ∈ Fq such that Ci,j (v1 , v2 , ..., v(n2 −n)g ) = 0 for 0 ≤ i ≤ ng − 1, 1 ≤ j ≤ n − 1. Moreover, when gv (x) ∈ Fq [x], g(x) = xng + Cng−1,0 xng−1 + ... + C0,0 . The condition that gv (x) factors completely in Fq [x] is equivalent to the above condition, and G(x) := xng + cng−1 xng−1 + ... + c0 factors completely in Fq [x] where ci = Ci,0 (v1 , v2 , ..., v(n2 −n)g ). In this case, the solutions of G(x) = 0 are the xcoordinates of the decomposed factor. Then, we ﬁnish the proof of proposition 1 and construct the equation system {Ci,j = 0}.

4

Example

In this section, we examine three computational experiments of the decomposed factors of Jacobian. The computations are done by using the computer algebra

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system magma on a Windows XP preinstalled PC (CPU:Pentium M 2GHz, RAM:1GB). (In order to solve equation system, the function “variety” prepared in magma is used.) We compute three cases 1) (g, n) = (1, 3), 2) (g, n) = (2, 2), and 3) (g, n) = (3, 2) where g and n are the genus and the extension degree of the deﬁnition ﬁeld of the chosen hyperelliptic/elliptic curve, respectively. In all cases, one trial, which means the judge as to whether a given element of Jacobian is decomposed or not and compute its decomposed factor, if it is decomposed, is done within 1 second. Since the probability that an element of Jacobian is decomposed is approximately 1/(gn)!, the amount of the time for obtaining one potentially decomposed reduced divisor is within 6 sec, 24 sec, and 720 sec, respectively. Further, we will give the following three examples. Case 1. Let q = 1073741789(prime number), Fq3 := Fq [t]/(t3 + 456725524t2 + 251245663t + 746495860), and let E/Fq3 be an elliptic curve deﬁned by y 2 = x3 + (1073741788t2 + t)x + (126t + 3969) and P0 := (t, t + 63) ∈ E. We investigate whether nP0 : n = 1, 2, ..30 are decomposed and ﬁnd the following 7 decompositions. (24P0 is written by 2 forms.) 2P0 = (1050861583, 6509843t2 + 387051565t + 920296030) + (742900894, 362262801t2 + 6480079t + 886701711) + (571975376, 938916909t2 + 910769097t + 139897863) 5P0 = (806296922, 113931706t2 + 863383473t + 133427995) + (797256157, 360646567t2 + 663390692t + 1012046566) + (389333914, 986077188t2 + 829314065t + 687783827) 8P0 = (1063441336, 113661172t2 + 942865616t + 744283566) + (894045278, 863335768t2 + 637284565t + 937810737) + (694935460, 740353309t2 + 505910431t + 597402219) 20P0 = (996570058, 341336613t2 + 450680674t + 72874200) + (141768271, 589122734t2 + 930205049t + 713557032) + (73505168, 432994198t2 + 405986289t + 233154172) 24P0 = (529735815, 20343700t2 + 780030904t + 490121669) + (515960254, 269821984t2 + 561547517t + 348990487) + (207183771, 712543643t2 + 356522343t + 895634732) = (818683055, 1034251164t2 + 705927333t + 1062879754) + (754504105, 23461217t2 + 961620879t + 1015889110) + (489159707, 271295793t2 + 600348670t + 1022482426) 26P0 = (628174301, 138296704t2 + 104824480t + 858118320) + (371888603, 417445284t2 + 850151153t + 126970733) + (55411433, 560274594t2 + 609956706t + 821692494) Case 2. Let q = 1073741789(prime number), Fq2 := Fq [t]/(t2 + 746495860t + 206240189), and let C/Fq2 be a hyperelliptic curve deﬁned by y 2 = x5 + (673573223t + 771820244)x + 6t + 9 and let D0 := (x2 + 1073741787tx + 327245929t + 867501600, (1023168391t + 350252228)x + 658555356t + 446913597)

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be a reduced divisor of C. We investigate whether nD0 : n = 1, 2, ..100 are decomposed and ﬁnd the following 9 decompositions. (71D0 is written by 2 forms.) 6D0 ∼ (1025731975, 776505688t + 911495013) + (728060789, 648475468t + 1067025179) + (341799975, 145077925t + 187604034) + (61964999, 227570631t + 639782700) − 4∞ 19D0 ∼ (1039361498, 15180988t + 396695374) + (828360115, 179412594t + 719919461) + (483171045, 677645208t + 604714840) + (34566209, 753841024t + 14375633) − 4∞ 33D0 ∼ (970690833, 608141084t + 889165804) + (260086243, 894605411t + 261264640) + (208957980, 43330622t + 581461318) + (190782894, 124873649t + 510328990) − 4∞ 35D0 ∼ (699447787, 267523741t + 562899544) + (559470007, 197827114t + 99971197) +(472594781, 579187919t+266558458) +(453661772, 449424806t+977318920)− 4∞ 48D0 ∼ (1009979214, 959734525t + 990871450) + (995813251, 44186049t + 288496638) +(521299995, 556594200t+468424666) +(17946008, 977064852t+1071618742)− 4∞ 71D0 ∼ (1019155056, 573896856t + 103042116) + (944470217, 829781939t + 184620624) + (727156004, 462612591t + 582877732) + (281900623, 553507533t + 42660552) − 4∞ ∼ (502979299, 412632304t + 1036827718) + (74527656, 927651409t + 452588110) + (50078888, 801072540t + 888737005) + (2986754, 556402789t + 236723678) − 4∞ 73D0 ∼ (843747137, 682161676t + 600252618) + (829302257, 145878028t + 853397395) +(290487906, 645896278t+279001181) +(184873704, 567002729t+620354511)− 4∞ 80D0 ∼ (907811987, 216534804t + 936839244) + (808513243, 873487475t + 273845273) +(520893378, 757248670t+381150138) +(486203744, 494475019t+791571132)− 4∞

Case 3. Let q = 1073741789(prime number), Fq2 := Fq [t]/(t2 + 746495860t + 206240189), and let C/Fq2 be a hyperelliptic curve deﬁned by y 2 = x7 + (111912375t + 1046743132)x + 6t + 9 and let D0 := (x2 + 1073741787tx + 327245929t + 867501600, (473621736t + 256126568)x + 145989647t + 687383736) be a reduced divisor of C. We investigate whether nD0 : n = 1, 2, ..3000 are decomposed and find the following 6 decompositions. 414D0 ∼ (1001437837, 752632260t+700158497)+(747112084, 656073918t+400137619) + (620249588, 127943213t + 635474623) + (614180498, 206297635t + 445250468) +(515769009, 607297126t+554290493) +(488549466, 627952783t+854182612)− 6∞ 657D0 ∼ (939617127, 695261735t + 239531611) + (933351280, 935312661t + 961494096) + (799612924, 341923983t + 677495100) + (294787599, 279723229t + 760003067) + (273118782053704103t + 577497766) + (153381525, 983211238t + 517037777) − 6∞ 921D0 ∼ (1034634787, 400751409t+829801342)+(763888873, 757155774t+829936954) + (619620874, 800641683t + 200272230) + (603032615, 115219564t + 655011145) +(436423191, 285214454t+450812747) +(125198811, 884750621t+123305741)− 6∞ 1026D0 ∼ (1024020017, 267457905t+41452942)+(794174628, 615676821t+723336407) + (738567269, 433647609t + 128304659) + (629287731, 465842490t + 789390318) + (435082408, 878213106t + 603353206) + (79621979, 479459622t + 672937516) − 6∞ 1121D0 ∼ (764081031, 812350603t+347878564)+(673426715, 687737442t+381588704) + (6102522082007139t + 99219637) + (467560104, 619342780t + 228756808) + (179787786, 333322906t + 75482151) + (59221667, 860686653t + 625301206) − 6∞ 2289D0 ∼ (729358563, 482925408t + 170057124) + (529840657, 42328987t + 857983002)

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+ (514618236, 436901100t + 416530686) + (350106356, 183495333t + 950710579) + (175898979, 411808870t + 427518366) + (96240558, 703780413t + 461022225) − 6∞

5

Conclusion

In this manuscript, we have proposed an algorithm which checks whether a reduced divisor is potentially decomposed or not, and we have computed the decomposed factors, if it is potentially decomposed. From this algorithm, concrete computations of decomposed factors are done by computer experiments when the pairs of the genus of the hyperelliptic curve and the degree of extension ﬁeld are (1, 3), (2, 2), and (3, 2).

Acknowledgment The author would like to thank Professor Kazuto Matsuo in the Institute of Information Security for useful comments and fruitful discussions and Professor Lisa Bond in Kanto Gakuin University for English writing. Also, the author would like to thank the anonymous reviewers who pointed out many mistakes and suggested a revisal plan.

References 1. Adleman, M., DeMarrais, J., Huang, M.-D.: A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 28–40. Springer, Heidelberg (1994) 2. Cantor, D.G.: Computing in the Jacobian of hyperelliptic curve. Math. Comp. 48, 95–101 (1987) 3. Diem, C.: An Index Calculus Algorithm for Plane Curves of Small Degree. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 543–557. Springer, Heidelberg (2006) 4. Diem, C.: On the discrete logarithm problem in class groups (2009) (preprint), http://www.math.uni-leipzig.de/~ diem/preprints/small-genus.pdf 5. Enge, A., Gaudry, P.: A general framework for subexponential discrete logarithm algorithms. Acta Arith. 102(1), 83–103 (2002) 6. Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000) 7. Gaudry, P., Thom´e, E., Th´eriault, N., Diem, C.: A double large prime variation for small genus hyperelliptic decomposed attack. Math. Comp. 76, 475–492 (2007) Preprint Version, http://eprint.iacr.org/2004/153/ 8. Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. Journal of Symbolic Computation 44(12), 1690– 1702 (2009), Preprint version http://eprint.iacr.org/2004/073 9. Hess, F.: Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symb. Comp. 11, 1–22 (2001) 10. Hindry, M., Silverman, J.H.: Diophantine Geometry An introduction. In: Graduate Texts in Math., vol. 201. Springer, Heidelberg (2000)

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11. Granger, R., Vercauteren, F.: On the Discrete Logarithm Problem on Algebraic Tori. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 66–85. Springer, Heidelberg (2005) 12. LaMacchia, B.A., Odlyzko, A.M.: Solving large sparse linear systems over finite fields. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 109–133. Springer, Heidelberg (1991) 13. Nagao, K.: Index calculus for Jacobian of hyperelliptic curve of small genus using two large primes. Japan Journal of Industrial and Applied Mathematics 24(3) (2007); Preprint version entitled by Improvement of Th´eriault Algorithm of decomposed attack for Jacobian of Hyperelliptic Curves of Small Genus, http://eprint.iacr.org/2004/161 14. Semaev, I.: Summation polynomials and the discrete logarithm problem on elliptic curves (2004) (preprint) 15. Th´eriault, N.: Index calculus for hyperelliptic curves of small genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003) 16. Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory IT-32(1), 54–62 (1986)

6

Appendix

In the appendix, we estimate the complexity of the decomposition attack, as a function of q, for ﬁxed g, n (i.e., g, n are considered as constants) under the Assumption 1 and Assumption 2. Here, we apply the ideas of the “Rebalancing method” [5],“One large prime method” [15], and “Two large prime method” [13] [7], which are the techniques of solving discrete logarithm of the Jacobian of a hyperelliptic curve over a general ﬁnite ﬁeld, to our cost estimation for the case of an extension ﬁeld. Note that as g and n are ﬁxed, the input length is linear in log q. These techniques are very complicated, and we only give the outline of the algorithm and estimation of the complexity. In this estimation, since n, g are ﬁxed, the cost for solving the system of the equations is considered as Poly(log q). For simplicity, the terms of Poly(log q)˜ part of the complexity is omitted. For this purpose, we denote the symbol O ˜ where the complexity O(N (q)) is estimated by ˜ (q)) < x(log q)y N (q) for some constants x, y ∈ R>0 , O(N and the symbol ≈ that the relation N1 (q) ≈ N2 (q) is deﬁned by N2 (q) < N1 (q) < x2 (log q)y2 N2 (q) for some constants x1 , x2 , y1 , y2 ∈ R>0 , x1 (log q)y1 where N (q), N1 (q) and N2 (q) are functions of input size q. Now, let G be a general ﬁnite abelian group whose group law is written additively and we consider the general decomposition attack over G. In the following, we also assume that i) The group order is known, and ii) G has a prime order. The assumption ii) is not an essential assumption, but make here for simplicity. Let us now ﬁx a set B0 subset of G.

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Definition 3. Let N be a ﬁxed positive integer(ﬁxed constant). 1) An element of g ∈ G written by g = g1 + .. + gN for g1 , ...gN ∈ B0 is called potentially decomposed. 2) g1 , ..., gN are then called decomposed factors and the multiset {g1 , ..., gN } is called decomposed divisor. Further, we also assume the following iii), iv), v), and vi): iii) The probability that g ∈ G is potentially decomposed is O(1). iv) For a g ∈ G, the cost for checking whether g is potentially decomposed or ˜ not is O(1). v) For the potentially decomposed g ∈ G the cost of computing decomposed ˜ divisor {g1 , ..., gN } from g is O(1). (If there are several decomposed divisors, the computation of all decomposed divisors is needed.) vi) |B0 |2 |G|. ˜ from iv). (Otherwise, the Note that o(|G|) < |B0 |N from ii) and |B0 |N < O(|G|) ˜ ε ) for some ε > 0 expected number of decomposed divisors is bigger than O(q and iv) does not hold.) In the normal index calculus, the number of B0 which are used for the decomposition is basically large (i.e.,N 1). So, the randomly chosen element is basically written by some linear sum of B0 in many ways. However, it is diﬃcult to compute such linear sums, so, by the use of the lifting to integer or number ﬁeld ring and by the use of the sieving method, one can ﬁnd some decomposition of randomly chosen element. So, remark carefully that the prerequisite condition of the normal index calculus for number ﬁeld sieve and that of the decomposition attack for the Jacobian of algebraic curve is quite diﬀerent. In our case (i.e., G being the Jacobian of a hyperelliptic curve of genus g over extension ﬁeld Fqn , B0 being the set of Fqn -rational point of the curve whose x-coordinate lie in Fq , N = ng), iii) is from Assumption 1 and Assumption 2, iv) and v) are from Theorem 1, and vi) is from the notations. Let us now ﬁx a set B subset of B0 . The set B is called the factor base and an element in B0 \B is called a large prime. Definition 4. 1) An element of g ∈ G written by g = g1 + .. + gN for g1 , ...gN ∈ B is called decomposed. 2) An element of g ∈ G written by g = g1 + .. + gN for one gi ∈ B0 \B, and the other gj ∈ B (1 ≤ j ≤ N, j = i) is called almost decomposed. 3) An element of g ∈ G written by g = g1 + .. + gN for two gi1 , gi2 ∈ B0 \B, and the other gj ∈ B (1 ≤ j ≤ N, j = i1 , i2 )is called 2-almost decomposed. 4)In every case, g1 , ..., gN are also called decomposed factors and the multiset {g1 , ..., gN } is called decomposed divisor. Now, we give the outlines of the algorithms named ’rebalancing method’, ’one large prime method’ and ’two large prime method’, which are the variants of the decomposition attack [5], [15], [13], and [7], by Algorithm 1 and Algorithm 2. Note that Algorithm 1 and Algorithm 2 are probabilistic, since they need random numbers. Also note that the probability that r1 a + r2 b is potentially

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Algorithm 1. The outline of the Rebalancing method Input: a, b ∈ G s.t. a = nb for some unknown n ∈ Z/|G|Z. Output: find n. 1: Initializing the list of the relations L = {} 2: while |L| < suitable number N0 do 3: For a pair of random numbers (r1 , r2 ), computing r1 a + r2 b. 4: if r1 a + r2 b being decomposed then 5: adding the informations of (r1 , r2 ) and the decomposed factor to L. 6: (If there are several decomposed factors, choosing one decomposed factor randomly.) 7: Solving the linear algebraic computation of roughly |B| × |B| size, modulo |G| 8: Computing n

Algorithm 2. The outlines of the One (resp. Two)large prime method Input: a, b ∈ G s.t. a = nb for some unknown n ∈ Z/|G|Z. Output: find n. 1: Initializing the list of the relations L = {} 2: while |L| < suitable numberN1 (resp. N2 ) do 3: For a pair of random numbers (r1 , r2 ), computing r1 a + r2 b. 4: if r1 a + r2 b being almost-decomposed (resp. 2-almost decomposed) then 5: adding the informations of (r1 , r2 ) and the decomposed factor to L. 6: (If there are several decomposed factors, choosing one decomposed factor randomly.) 7: Updating L by the elimination of the terms of external elements. 8: Solving the linear algebraic computation of roughly |B| × |B| size, modulo |G| 9: Computing n

decomposed is O(1), since |G| is a prime number and r1 a+ r2 b can be considered as a random element of G. In Algorithm 1 and Algorithm 2, N0 (resp. N1 , resp N2 ) be the number of decomposed (resp. almost decomposed , resp. 2-almost decomposed ) elements of G which are required in the rebalancing method (resp. one large prime method, resp. two large prime method). From the ideas of [5], [15], [13], and [7], the estimations of the following conjecture is expected. Conjecture . 1) N0 is estimated by Const × |B|, i.e., N0 = O(|B|). 2) N12 /|B0 | is estimated by Const × |B|, i.e., N1 = O(|B|1/2 |B0 |1/2 ). 3) N2 is estimated by Const × |B0 |, i.e., N2 = O(|B0 |). Further, we have the following estimations of the complexity. Lemma 10. Under the assumptions of i),ii),iii), iv) v), vi), and Conjecture, we have the following: 1) The complexity of the general decomposition attack taking B as a set of factor basis by the rebalancing method is minimized at |B| ≈ |B0 |N/(N −1) , and it is ˜ 0 |(2N )/(N +1) ). estimated by O(|B 2) The complexity of the general decomposition attack taking B as a set of factor basis and taking B0 \B as a set of large primes by the one large prime method is min˜ 0 |(4N −2)/(2N +1) ). imized at |B| ≈ |B0 |(2N −1)/(2N +1) , and it is estimated by O(|B

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3) The complexity of the general decomposition attack taking B as a set of factor basis and taking B0 \B as a set of large primes by the two large prime method is ˜ 0 |(2N −2)/N ). minimized at |B| ≈ |B0 |(N −1)/N , and it is estimated by O(|B Proof. (Sketch of the proof) In every case, the cost of the part of linear algebra 2 ˜ is O(|B| ), and for the rebalance, which is needed for minimizing the complexity, it is the same as the cost of the collecting divisors. So, we only need to estimate the optimized size |B|. 1)In the case of rebalancing method: The probability that the randomly chosen g ∈ G is a decomposed is O(|B/B0 |N ). So, the cost to obtain one decomposed g ˜ 0 /B|N ). From Conjecture , we must have O(|B|) number of such g. So is O(|B |B0 /B|N · |B| ≈ |B|2 where the left hand side is the cost for collecting enough decomposed group elements, and the right hand side is the cost for the linear algebra. Thus we have |B| ≈ |B0 |N/(N +1) . 2) In the case of one large prime method: The probability that the randomly chosen g ∈ G is an almost decomposed is O(|B/B0 |N −1 ). From Conjecture , we must have O(|B|1/2 |B0 |1/2 ) number of such g. Similarly, we have |B0 /B|N −1 · |B|1/2 |B0 |1/2 ≈ |B|2 and |B| ≈ |B0 |(2N −1)/(2N +1) is obtained. 3) In the case of two large prime method: The probability that the randomly chosen g ∈ G is a 2-almost is O(|B/B0 |N −2 ). From Conjecture , we must have O(|B0 |) number of such g. Similarly, we have |B0 /B|N −2 · |B0 | ≈ |B|2 and |B| ≈ |B0 |(N −1)/N is obtained. Now, we apply this lemma for the decomposition attack for the Jacobian of a curve over an extension ﬁeld. Note that B0 = {P − ∞ |x(P ) ∈ Fq }, |B0 | ≈ q, N = ng and thus, we have the following claim, which is based on the assumptions i),ii),iii),iv),v),vi),and Conjecture. Claim . 1) The complexity of the decomposition attack with the rebalancing ˜ (2ng)/(ng+1) ). method is estimated by O(q 2) The complexity of the decomposition attack with the one large prime method ˜ (4ng−2)/(2ng+1) ). is estimated by O(q 3) The complexity of the decomposition attack with the two large prime method ˜ (2ng−1)/(ng) ). is estimated by O(q

Factoring Polynomials over Local Fields II Sebastian Pauli Department of Mathematics and Statistics University of North Carolina at Greensboro, Greensboro, NC 27412, USA s [email protected]

Abstract. We present an algorithm for factoring polynomials over local fields, in which the Montes algorithm is combined with elements from Zassenhaus Round Four algorithm. This algorithm avoids the computation of characteristic polynomials and the resulting precision problems that occur in the Round Four algorithm.

1

Introduction

Polynomial factorization is fundamental in working with local ﬁelds. In addition to the irreducible factors of a given polynomial, computer algebra systems that support extensions of local ﬁelds (e.g., Magma [1], Sage [16]) require explicit representations of the unramiﬁed and totally ramiﬁed parts of the extensions generated by arbitrary irreducible polynomials, as these systems represent such extensions as a tower of unramiﬁed and totally ramiﬁed extensions. Moreover, there are many applications of global ﬁelds that include the construction of integral bases, decomposition of ideals, and the computation of completions. The algorithms [2,4,7,14] for factoring a polynomial Φ(x) over a local ﬁeld ﬁnd successively better approximations to the irreducible factors of Φ(x) until gaining suﬃcient precision to apply Hensel lifting. The algorithms diﬀer in how the approximations are computed. Algorithms based on the Zassenhaus Round Four algorithm (e.g. [3,4,14]) suﬀer from loss of precision in computing characteristic polynomials and approximating greatest common divisors. The Montes algorithm [10,11,7,8] avoids the computation of characteristic polynomials by exploiting Newton polygons of higher order. Here the most expensive operations are division with remainder and polynomial factorization over ﬁnite ﬁelds. We present the algorithm of Montes in the terminology of [14] and use the techniques of the Round Four algorithm to derive a factorization when a breaking element is found. We also give a complexity analysis. Notation Let K be a ﬁeld complete with respect to a non-archimedian exponential valuation ν with ﬁnite residue class ﬁeld K ∼ = Fq of characteristic p; we call K a local ﬁeld. Assume ν is normalized with ν(π) = 1 for the uniformizing element G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 301–315, 2010. c Springer-Verlag Berlin Heidelberg 2010

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π in the valuation ring OK of K. For γ ∈ OK denote by γ the class γ + (π) in K. The unique extension of ν to an algebraic closure K of K (or to any intermediate ﬁeld) is also denoted ν. In our algorithm we will be concerned with the ﬁrst non-zero coeﬃcient of the expansion of an element in a ﬁnite subextension of K/K. We introduce an equivalence relation on the elements of K which reﬂects this (also see [9]). ∗

∗

Definition 1. For γ ∈ K and δ ∈ K we write γ ∼ δ if ν(γ − δ) > ν(γ) and make nthe supplementary assumption 0 ∼ 0. For ϕ(x) = ϑ(x) = i=0 ϑi xi in K[x] we write ϕ(x) ∼ ϑ(x) if

n

i=0

ϕi xi and

min 0≤i≤n ν(ϕi − ϑi ) > min 0≤i≤n ν(ϕi ). Let L be a ﬁnite extension of K with uniformizing element πL . Two elements γ = γ0 πLv ∈ L and δ = δ0 πLw ∈ L with ν(γ0 ) = ν(δ0 ) = 0 are equivalent with respect to ∼ if and only if v = w and γ0 ≡ δ0 mod (πL ). It follows immediately that the relation ∼ is symmetric, transitive, and reﬂexive.

2

Reducibility

Assume we want to factor a polynomial Φ ∈ OK [x] of degree N . If Φ(x) splits into the product of two co-prime factors over the residue class ﬁeld K of K, say Φ(x) = Φ1 (x) · Φ2 (x), then Hensel lifting yields a factorization of Φ(x) to any given precision. In addition to this classic situation we give two further situations that we can exploit to obtain a factorization of Φ(x). We consider a polynomial ϑ(x) ∈ OK [x] as a representative of an element in the algebra K[x]/(Φ(x)) and determine a polynomial χϑ (x) ∈ K[x] from ϑ(x) such that χϑ (ϑ(ξ)) = 0 for all roots ξ of Φ(x). Definition 2. Let Φ(x) = N j=1 (x − ξj ) ∈ OK [x], where ξj ∈ K for 1 ≤ j ≤ N and ϑ(x) ∈ K[x]. Then we set χϑ (y) :=

N (y − ϑ(ξi )) = resx (Φ(y), y − ϑ(x)). i=1

Assume we ﬁnd ϑ ∈ K[x] such that χϑ (y) = χ1 (y)χ2 (y) with gcd(χ1 , χ2 ) = 1. Reordering the roots ξi (1 ≤ i ≤ N ) of Φ(x) if necessary, we may write χ1 (y) = (y − ϑ(ξ1 )) · · · (y − ϑ(ξr )) and χ2 (y) = (y − ϑ(ξr+1 )) · · · (y − ϑ(ξN )), where 1 ≤ r < N and obtain a proper factorization of Φ(x): Φ(x) = gcd(Φ(x), χ1 (ϑ(x))) · gcd(Φ(x), χ2 (ϑ(x))).

(1)

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Definition 3. We say a polynomial ϑ(x) ∈ K[x] with χϑ (t) ∈ OK [t] passes the Hensel test if χϑ (t) = ρ(t)g for some irreducible polynomial ρ(t) ∈ K[t]. If ϑ(x) ∈ K[x] fails the Hensel test, that is, χϑ (y) splits into two co-prime factors over K, say χϑ (y) = χ1 (y)χ2 (y), then Hensel lifting yields a factorization χϑ (y) = χ1 (y)χ2 (y) and equation (1) gives a proper factorization of Φ(x). ∗ Definition 4. For ϑ ∈ K[x] we set vΦ (ϑ) := minΦ(ξ)=0 ν(ϑ(ξ)) and say the polynomial ϑ(x) passes the Newton test if ν(ϑ(ξ)) = ν(ϑ(ξ )) for all roots ξ and ξ of Φ(x).

If ϕ(x) ∈ K[x] fails the Newton test, the Newton polygon of χϕ (y) consists of at ∗ least two segments. Let h/e = vΦ (ϕ) be the minimum of the valuations ν(ϕ(ξi )) (1 ≤ i ≤ N ) in lowest terms. Then −h/e is the gentlest slope of the segments of the Newton polygon of χϕ (y). We set ϑ(x) := ϕ(x)e /π h and obtain ν(ϑ(ξ)) = 0 for all roots ξ of Φ(x) with ν(ϕ(ξ)) = h/e and ν(ϑ(ξ)) > 0 for all roots ξ of Φ(x) with ν(ϕ(ξ)) > h/e. Thus χϑ (t) splits into two co-prime factors and the considerations above yield a proper factorization of Φ(x).

3

Irreducibility and the Sequence ϕt(x) t

In the polynomial factorization algorithm we construct a sequence of polynomials ϕt (x) ∈ OK [x] such that ν(ϕt+1 (ξ)) > ν(ϕt (ξ)) for all roots ξ of Φ(x) until we either ﬁnd a polynomial that fails the Newton test, which leads to a factorization of Φ(x) or we have established the irreducibility of Φ(x). If we assure that the degrees of the polynomials ϕt (x) are less than or equal to the degree of all irreducible factors of Φ(x), we either obtain a factorization of Φ(x) or we establish the irreducibility of Φ(x) in ﬁnitely many steps [14]: Theorem 5. Let ξ1 , . . . , ξN be elements of an algebraic closure of a local ﬁeld K and assume the following hypotheses hold. N – Φ(x) = j=1 (x − ξj ) is a square-free polynomial in OK [x]. – ϕ(x) ∈ K[x]. – N ν(ϕ(ξj )) > 2ν(disc Φ) for 1 ≤ j ≤ N . – The degree of any irreducible factor of Φ(x) is greater than or equal to deg ϕ. Then N = deg ϕ and Φ(x) is irreducible over K. While we construct the sequence of polynomials ϕt (x) we gather information about the extensions generated by the irreducible factors of Φ(x). In particular we will at all times know divisors Et and Ft of the ramiﬁcation index and inertia degree of these extensions respectively. If we ﬁnd that not all of these extensions have the same inertia degree and ramiﬁcation index, we will have encountered a polynomial that fails the Hensel or the Newton test. On the other hand if Et · Ft = deg Φ we know that Φ(x) is irreducible.

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Definition 6. Let Φ(x) ∈ OK [x] be irreducible and let ξ be a root of Φ(x). We call a pair of Π(x) ∈ K[x] and Γ (x) ∈ K[x] with ν(Π(ξ)) = 1/E polynomials and F = K Γ (ξ) : K such that E ·F = deg Φ a two element certiﬁcate for the irreducibility of Φ(x). Remark 7. If a two element certiﬁcate exists then Φ(x) is irreducible and an integral basis of the extension of K(ξ)/K generated by a root ξ of Φ(x) is given by the elements Γ (ξ)i Π(ξ)j with 0 ≤ i ≤ F − 1 and 0 ≤ j ≤ E − 1. In the polynomial factorization algorithm we construct a sequence of polynomials (ϕt (x))t∈N where ϕt ∈ OK [x] such that 1. ν(ϕt+1 (ξ)) > ν(ϕt (ξ)) for all roots ξ of Φ(x), 2. ν(ϕt (ξ)) = ν(ϕt (ξ )) for all roots ξ and ξ of Φ(x), and 3. the degree of ϕt (x) is less than or equal to the degree of any irreducible factor of Φ(x). In the following we assume that all polynomials that occur in our constructions pass the Hensel and Newton tests, as we can otherwise derive a factorization of Φ(x). For convenience of notation we deﬁne: ∗ ∗ Definition 8. If vΦ (ϕ−ϑ) > vΦ (ϕ) for polynomials ϕ(x) ∈ K[x] and ϑ(x) ∈ K[x] n we write ϕ ∼ ϑ. For polynomials χ(y) = i=0 ai (x)y i ∈ K[x][y] and τ (y) = Φ n i i=0 bi (x)y ∈ K[x][y] we write χ(y) ∼ τ (y) if Φ

∗ (ai min 0≤i≤n vΦ

4

∗ − bi ) > min 0≤i≤n vΦ (ai ).

The First Iteration

N Let Φ(x) = i=0 ci xi and ϕ1 (x) := x ∈ OK [x]. Assume the Newton polygon of Φ(x) consists of one segment and let −h1 /E1 be its slope in lowest terms. Then ν(ϕ1 (ξ)) = ν(ξ) = h1 /E1 for all roots ξ of Φ(x). This implies that the ramiﬁcation index of all extension generated by irreducible factors of Φ(x) is divisible by E1 . Let β ∈ K with β E1 = π h1 where π is the uniformizing element of K. We ﬂatten the Newton polygon of Φ(x) so that it lies on the x-axis: Φ (y) :=

N

Φ(βy) = ci β i−N y i . βN i=0

Because we can only have ν(ci β i−N ) = 0 when E1 | i, we have

Φ (y) ∼

N/E1

cj·E1 π h1 (j−N/E1 ) y j·E1 .

j=0

Replacing y

E1

by z yields N/E1

A1 (z) :=

j=0

cj·E1 π h1 (j−N/E1 ) z j .

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The polynomial A1 (z) ∈ K[z] is called the associated polynomial [11,10] or residual polynomial [7,8] of Φ(x) with respect to ϕ1 (x). Assume that A1 (z) = ρ1 (z)r for some irreducible polynomial ρ1 ∈ K. Otherwise ϕ1 (x)E1 /π h1 = xE1 /π h1 would fail the Hensel test and (1) would yield a factorization of Φ(x). All ﬁelds K(ξ), where ξ is a root of Φ(x), contain an element ξ E1 /π h1 , whose minimal polynomial is a power of ρ1 (z) over K[z]; therefore their ramiﬁcation indices are divisible by F1 := deg ρ1 . Let γ1 ∈ K be a root of a lift ρ1 (z) ∈ OK [z] of ρ1 (z). In the unramiﬁed extension K1 := K(γ1 ) we have the relation xE1 ∼ π h1 · γ1 . Since Φ ν ρ1 (ϕ1 (ξ)E1 /π h1 ) > 0 for all roots ξ of Φ(x), we get

1 ϕ1 (ξ)E1 h1 F1 > ν(π h1 ) = ν ϕE ρ1 ν π 1 (ξ) > ν ϕ1 (ξ) = ν(ξ). h 1 π We set ϕ2 (x) := π h1 F1 ρ1 (ϕ1 (x)E1 /π h1 ) and continue the construction of our sequence of polynomials (ϕt )t . Obviously deg ϕ2 = E1 F1 , which divides the degree of every irreducible factor of Φ(x). Remark 9. Because the Newton polygon of ϕ2 (x) consists of one segment of slope −h1 /E1 with gcd(h1 , E1 ) = 1 and its associated polynomial with respect to x is ρ1 (z) of degree F1 , the extensions K(α), where α is a root of ϕ2 (x), have inertia degree F1 and ramiﬁcation index E1 . Hence ϕ2 (x) with deg ϕ2 = E1 F1 is irreducible.

5

The Second Iteration

Definition 10. Let Φ(x) ∈ OK [x] of degree N and ϕ(x) ∈ OK [x] of degree n be monic polynomials and assume n | N . We call N/n

Φ(x) =

ai (x)ϕi (x)

i=0

with deg(ai ) < deg(ϕ) the ϕ-expansion of Φ(x). We use the ϕ2 -expansion of Φ(x) to ﬁnd the valuations ν(ϕ2 (ξ)). Set n2 := deg ϕ2 N/n and let Φ(x) = i=0 2 ai (x)ϕi2 (x) be the ϕ2 -expansion of Φ(x). For each root ξ of Φ(x) we have N/n2 0 = Φ(ξ) = ai (ξ)ϕi2 (ξ). i=0

Hence χ2,ξ (y) =

m

ai (ξ)y i ∈ OK(ξ) [y]

i=0

with m = N/n2 = deg(Φ)/ deg(ϕ2 ) is a polynomial with root ϕ2 (ξ). Assume n2 −1 ai,j xj . As the valuations that ai (x) = j=0 ∗ (ϕ1 ) = vΦ

h1 (E1 − 1)h1 ∗ , . . . , vΦ (ϕ1E1 −1 ) = E1 E1

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are distinct (and not in Z) and ϕ1 (x)E1 1, ∼ γ1 , . . . , Φ π h1

ϕ1 (x)E1 π h1

F1 −1

∼ γ1F1 −1 Φ

are linearly independent over K, we have ∗ vΦ (ai ) =

min

0≤j≤n2 −1

ν(ai,j )(h1 /E1 )j.

If the Newton polygon of χ2,ξ (y) consists of more than one segment then ϕ2 (x) fails the Newton test and we can derive a factorization of Φ(x). Otherwise let −h2 /e2 be the slope of the Newton polygon of χ2,ξ (y) in lowest terms. Then ν(ϕ2 (ξ)) = h2 /e2 for all roots ξ of Φ(x). We set E2+ := e2 / gcd(E1 , e2 ). For all roots ξ of Φ(x) the ramiﬁcation index of K(ξ) is divisible by E2 := E1 · E2+ . Because the denominator of E2+ h2 /e2 is a divisor of E1 there is ψ2 (x) := π sπ ϕ1 (x)s1 = π sπ xs1 ∈ K[x] ∗ with s1 ∈ {0, . . . , E1 − 1} and sπ ∈ Z such that vΦ (ψ2 ) = E2+ h2 /e2 . + We ﬂatten the Newton polygon of χ2,ξ (y). Let β ∈ K with β E2 = ψ2 (x) and consider the polynomial χ2,ξ (y) := χ2,ξ (βy)/β m . As only the valuations of the +

coeﬃcients of y i·E2 (0 ≤ i ≤ m/E2+ ) can be zero we get χ2,ξ (y)

m/E2+

=

i=0 m/E2+

=

i=0 E1

+

2

+

+

ai·E + (ξ)ψ2 (ξ)i−m/E2 y i·E2 ∈ K2 [y]. 2

∼ π h1 · γ1 , which is independent of ξ, we ﬁnd coeﬃcients

Using the relation x

Φ

i−m/E2+

ai ∼ ai·E + (x)ψ2 ai ∈ K1 with Φ

+

ai·E + (ξ)β i·E2 −m y i·E2

2

m/E2+

A2 (z) :=

(x). We set

ai z i ∼ Φ

i=0

m/E2+

i=0

i−m/E2+

ai·E + (x)ψ2 2

(x)z i

and obtain the associated polynomial A2 (z) ∈ K1 [z] of Φ(x) with respect to ϕ2 (x). If A2 (y) splits into two or more co-prime factors over K1 = K(γ1 ), we can derive a factorization of Φ(x): Since deg ψ2 (x) is less than the degree of any irreducible factor of Φ(x) we have gcd(ψ2 (x), Φ(x)) = 1 and the extended Euclidean algorithm yields ψ2−1 (x) ∈ OK1 [x] such that ψ2 (x) · ψ2−1 (x) ≡ 1 mod Φ(x). The E+

polynomial ϕ2 2 (x) · ψ2−1 (x) fails the Hensel test. Otherwise A2 (z) = ρ2 (z)r2 for some irreducible polynomial ρ2 (z) ∈ K1 [z]. We set K2 := K(γ2 ) where γ2 is a root of a lift ρ2 (z) ∈ OK1 [z] of ρ2 (z) ∈ K1 [z], let +

F2+ := deg ρ2 , and obtain ϕ2 (x)E2 ∼ γ2 ψ2 (x). Φ

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307

∗ ∗ Next we construct ϕ3 (x) ∈ OK [x] with vΦ (ϕ3 ) > vΦ (ϕ2 ) and deg ϕ3 = E2 F2 . The coeﬃcients of ρ2 (z) ∈ OK1 can be written as polynomials in γ1 ∼ xE1 /π h1 , Φ say +

F2 F1 −1

ρ2 (z) =

ri,j γ1j z i

i=0 j=0

where ri,j ∈ OK . We are looking for

F2+

ϕ3 (x) ∼ ψ2 (x) Φ

ρ2

+

ϕ2 (x)E2 ψ2 (x)

+

=

F2 F1 −1

ri,j

i=0 j=0

xE1 π h1

j

+

ψ2 (x)F2

−i

+

ϕ2 (x)iE2

∗ ρ1 (xE1 /π h1 ) > 0. If we write with deg ϕ3 = E2 F2 = E2+ F2+ E1 F1 . We have vΦ ρ1 (z) = z F1 + ρ∗1 (z) with deg(ρ∗1 ) < F1 this implies 1 F1 ϕE ∼ −(π h1 )F1 ρ∗1 1

Φ

xE1 π h1

.

It follows that we can ﬁnd a polynomial Ri,j (x) with deg Ri,j < E1 F1 such that

Ri,j (x) ∼ ri,j Φ

xE1 π h1

j

F2+ −i

ψ2 (x)

= ri,j

xE1 π h1

j

+

(π sπ xs1 )F2

−i

.

Thus the polynomial E2+ F2+

ϕ3 (x) = ϕ2 (x)

F2+ −1 F1 −1

+

i=0

+

Ri,j (x)ϕ2 (x)iE2

j=0

∗ ∗ has the desired properties vΦ (ϕ3 ) > vΦ (ϕ2 ) and deg ϕ3 = E2 F2 .

Remark 11. ϕ3 (x) ∈ OK [x] is irreducible.

6

Data and Relations

In the algorithm we continue the construction of the sequence of polynomials (ϕt )t from the previous two sections. In the following steps the computation of ψt (x), the valuation of the coeﬃcients ai (x) of the ϕt -expansion of Φ(x), the coeﬃcients of the associated polynomial, and ϕt+1 becomes more involved and relies on the data computed in the previous iteration. We initially set K0 := K,

ϕ1 := x,

E0 := 1,

and compute the following data in every iteration:

F0 := 1

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S. Pauli

ϕt (x) ∈ OK [x]

∗ ∗ with vΦ (ϕt ) > vΦ (ϕt−1 ) and nt = deg(ϕt ) = Et−1 Ft−1 ; an approximation to an irreducible factor of Φ(x)

∗ ht /et = vΦ (ϕt ) et Et+ = gcd(Et−1 , et ) Et = Et+ ·Et−1 t−1 ψt (x) = π sπ i=1 ϕsi i

with gcd(ht , et ) = 1 the increase of the maximum known ramiﬁcation index the maximum known ramiﬁcation index E+

∗ ∗ with sπ ∈ Z and 0 ≤ si < Ei+ such that vΦ (ψt ) = vΦ (ϕt t )

At (y) ∈ Kt−1 [y]

the associated polynomial of Φ(x) with respect to ϕt (x)

ρt (y) ∈ Kt−1 [y]

irreducible with ρtrt (y) = At (y)

γt ∈ K t

such that ϕt t ∼ γt ψt

Kt = Kt−1 (γt )

the maximum known unramiﬁed subﬁeld

Ft+ = [Kt : Kt−1 ]

the increase of the maximum known inertia degree

Ft =

7

E+

Φ

Ft+ ·Ft−1

the maximum known inertia degree

The u-th Iteration

Assume we have computed the data and relations given above for t up to u − 1 and that ϕu (x) of degree nu = Eu Fu is the best approximation to an irreducible factor of Φ(x) found so far. We compute the ϕu -expansion Φ(x) = N/nu N/nu i i i=0 ai (x)ϕu (x) of Φ(x) and set χu (y) := i=0 ai (x)y . Definition 12. Let a(x) ∈ OK [x] with deg a < Et−1 Ft−1 . We call + + Et−1 Ft−1 −1

a(x) =

jt−1 ϕt−1 (x)

jt−1 =0

E2+ F2+ −1

···

ϕj22 (x)

j2 =0

E1 F1 −1

xj1 · aj1 ,...,jt−1 ,

j1 =0

where aj1 ,...,jt−1 ∈ OK (0 ≤ ji ≤ Ei , 0 ≤ i ≤ t), the (ϕ1 , . . . , ϕt−1 )-expansion of a(x). From the (ϕ1 , . . . , ϕu−1 )-expansion of ai (x) we obtain the valuations of ai (ξ) and see that they are independent of the choice of the root ξ of Φ(x). Since, by construction, the values E + −1

∗ ∗ ∗ ∗ vΦ (ϕ1 ), . . . , vΦ (ϕ1E1 −1 ), vΦ (ϕ2 ), . . . , vΦ (ϕ2 2

E+

u−1 ∗ ∗ ), vΦ (ϕ3 ), . . . . . . , vΦ (ϕu−1

−1

)

are distinct (and not in Z) and for 0 ≤ t ≤ u − 1 the elements +

F + −1

1, γt ∼ ϕt (x)Et /ψt (x), . . . , γt t Φ

F + −1 + ∼ ϕt (x)Et /ψt (x) t Φ

are linearly independent over Kt−1 = K(γ1 , . . . , γt−1 ) we have (see [7, Lemma 4.21]):

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309

Lemma 13. Let a(x) ∈ OK [x] with deg a < Et−1 Ft−1 and let aj1 ,...,jt−1 , with 0 ≤ ji < Ei+ Fi+ − 1, be the coeﬃcients of the (ϕ1 , . . . , ϕt−1 )-expansion of a(x). Then jt−1 ∗ ∗ vΦ ϕt−1 (x) · · · ϕj22 (x) · xj1 · aj1 ,...,jt−1 . (a) = min vΦ 1≤i≤t−1 + 1≤ji <E i

If the Newton polygon of χt (y) consists of one segment, say of slope −hu /eu , with u gcd(hu , eu ) = 1, then ϕt (x) passes the Newton test. We set Eu+ := gcd(Eeu−1 ,eu ) and construct u−1 sπ ψu (x) = π ϕt (x)st t=1 ∗ with sπ ∈ Z and 0 ≤ st < (1 ≤ t < u) such that vΦ (ψu ) = Eu+ hu /eu using the following algorithm. For q ∈ Q we denote by den(q) the denominator of q in lowest terms.

Et+

Algorithm 14 (Psi) ∗ Input: vΦ (ϕi ) and Ei+ for 0 ≤ i ≤ t, E = E0+ · · · Et+ , v ∈ Q with E |den(v). ∗ Output: sπ ∈ Z, 0 ≤ si ≤ Ei+ (1 ≤ i ≤ t) such that vΦ (π sπ ϕs00 · · · ϕst t ) = v. – d ← E, i ← t – for i from t to 1 by −1: ∗ • d ← d/Ei+ , v ← v · d, e ← vΦ (ϕi ) · d • Find si such that e · si ≡ v mod den(d · e) ∗ • v ← v − si vΦ (ϕi ) – sπ ← v – return sπ , s1 , . . . , st Next we determine the associated polynomial Au (y) of Φ(x) with respect to ϕu (x). Because we have representations of ai (x) (0 ≤ i ≤ N/ni ) and ψu (x) by + power products of π, ϕ1 , . . . , ϕu−1 we can use the relations ϕt (x)Et ∼ γt ψt (x) Φ

+

ai ∼ ai·Eu+ (x)ψu (x)i−m/Eu . We get to ﬁnd the coeﬃcients ai ∈ Ku−1 such that Φ

the associated polynomial + m/Eu

Au (z) =

ai z i

i=0

where m = N/nu . Assume that Au (z) = ρu (z)r for some irreducible polynomial +

ρu (z) ∈ Ku−1 (z). Otherwise we can ﬁnd ϑ(x) ∈ K[x] with ϑ(x) ∼ ϕu (x)Eu /ψu (x) Φ

that fails the Hensel test, which yields a factorization of Φ(x). Let ρu (z) ∈ Ku−1 be a lift of ρu (z), and set Fu+ := deg ρu . Finally we construct ϕu+1 (x) ∈ OK [x] of degree Eu Fu = Eu+ Fu+ Eu−1 Fu−1 such that +

ϕu+1 (x) ∼ Φ

Fu i=0

+

+

E+

ϑi (x)ϕu (x)iEu = ψu (x)Fu ρu (ϕu u (x)/ψu (x)),

(2)

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S. Pauli

where the ϑi (x) are sums of power products of π, ϕ1 , . . . , ϕu−1 . For t = u − 1, u − 2, . . . , 0 we recursively apply

E + ϕt t ∗ ρt >0 vΦ ψt to reduce the maximum exponent of ϕt (x) to Et+ Ft+ − 1, such that the de+ + gree of the ϕt (x) term is at most deg(ϕt (x)Et Ft −1 ) = (Et−1 Ft−1 )(Et+ Ft+ − 1). Thus we can ﬁnd a ϕu+1 (x) that fulﬁlls the degree condition deg ϕu+1 = Eu Fu . Furthermore

+ + ϕu (x)Eu Fu+ F ∗ ∗ ∗ ∗ vΦ (ϕu+1 ) = vΦ ψu ρu > vΦ ψu u ≥ vΦ (ϕu ). ψu (x) As a preparation for the next iteration we set Ku := Ku−1 (γu ) with γu a root of + ρu (z) and obtain the relation ϕEu (x) ∼ γu ψu (x). Φ

Remark 15. ϕu+1 (x) ∈ OK [x] is irreducible.

8

The Algorithm

We summarize the steps for the construction of the sequence (ϕt (x))t in an algorithm. Although we use the unramiﬁed extensions Kt /K above and in the algorithm, in practice the γi are represented as elements in the residue class ﬁeld Kt . Furthermore, many of the manipulations in the algorithm can be conducted on the representations of ψt (x) as power products of π, ϕ1 (x), . . . , ϕt−1 (x) and of ai (x) as sums of power products of π, ϕ1 (x), . . . , ϕt−1 (x) thus reducing these operations to operations of vectors of integers. Algorithm 16 (Polynomial Factorization) Input: a monic, separable, squarefree polynomial Φ(x) over a local ﬁeld K. Output: a proper factorization of Φ(x) if one exists, a two-element certiﬁcate for Φ(x) otherwise. (1) Initialize t ← 1, ϕ1 (x) ← x, E0 = 1, F0 = 1, K0 = K. (2) Repeat: deg ϕt (a) Find the ϕt expansion Φ(x) = N/ ai (x)ϕ(x)i of Φ(x). i=1 ∗ (b) Find vΦ (ai ) for 0 ≤ i ≤ N/ deg ϕt . (c) If ϕt (x) fails the Newton test: return a proper factorization of Φ(x). et ∗ (d) ht /et ← vΦ (ϕ) with gcd(ht , et ) = 1; Et+ ← gcd(e ; Et ← Et+ · Et−1 . t ,E) t−1 ∗ ∗ (ψt ) = Et+ vΦ (ϕt ), sπ ∈ N, (e) Construct ψt (x) = π sπ i=1 ϕi (x)si with vΦ + 0 ≤ si < Ei (1 ≤ i ≤ t − 1), deg ψt < Ei Fi . (f ) Compute the associate polynomial At (z). (g) Find a factorization of At (z) ∈ Kt (z). (h) If At (z) has two co-prime factors: return a proper factorization of Φ(x).

Factoring Polynomials over Local Fields II

311

(i) Ft+ ← deg ρ where ρt (z)r = At (z), ρt (z) ∈ Kt−1 [z] irreducible; Ft ← Ft+ · Ft−1 , Kt ← K[x]/(ρt (x)). (j) If Et Ft = deg Φ: return a two-element certiﬁcate for Φ(x). + (k) Find ϕt+1 (x) ∼ ρt ϕt (x)Et ψt (x)deg(ρ) of degree nt+1 = Et Ft in OK [x]. (l) t ← t + 1.

Φ

Certificates for Irreducibility If Φ(x) is irreducible we will have Et Ft = N for some t. We obtain the two element certiﬁcate (Deﬁnition 6) for the irreducibility of Φ(x) as follows. A poly∗ nomial Π(x) ∈ K[x] with vΦ (Π) = 1/Et can be found using Algorithm 14. If Ft = 1 we can choose Γ (x) = x. If Ft = 1, let i be maximal with Fi+ = 0. We + ﬁnd Γ (x) ∈ K[x] with Γ (x) ∼ ϕi (x)Ei /ψi (x). Φ

9

Complexity

We restrict our analysis of the complexity of the algorithm to the main loop. The ﬁrst complexity estimate for the Montes algorithm, restricted to irreducibility testing, was given by Veres [17] and improved by Ford and Veres [5]. The complexity estimate for determining the irreducibility of a polynomial Φ(x) ∈ Zp [x] of degree N using this algorithms is O(N 3+ε ν(disc Φ) + N 2+ε ν(disc Φ)2+ε ). The running time of the Round Four algorithm is analyzed in [14], but without taking into account the precision loss in the computation of greatest common divisors. Both estimates rely on Theorem 5 to bound the number of iterations and the required precision and only diﬀer slightly in the exponent of the discriminant of Φ(x). Lemma 17. Let Φ(x) ∈ OK [x] be of degree N and let ϕ(x) ∈ OK [x] be monic of degree n. Then the ϕ-expansion of Φ(x) can be computed in O(N 2 ) operations in OK . N/n Proof. In order to determine the ϕ-expansion Φ(x) = i=1 ai (x)ϕ(x)i we ﬁrst compute q0 (x), a0 (x) ∈ OK [x] with Φ(x) = ϕ(x)q0 (x) + a0 (x), which can be done in O((N − n)n) operations in OK [x]. Next we determine q1 (x), a1 (x) ∈ OK [x] with q0 (x) = ϕ(x)q1 (x) + a1 (x) (O((N − 2n)n) operations in OK [x]), and so on. Therefore the ϕ-expansion of Φ(x) can be computed in ⎛ ⎛ ⎞⎞ N/n 2 N −n O((N −n)n)+O((N −2n)n)+· · ·+O((2n)n) = O⎝n ⎝ i⎠⎠ = O(N 2 ) n i=0 operations in OK . The computation of the (ϕ1 , . . . , ϕt−1 )-expansion of a polynomial a(x) ∈ OK [x] of degree m ≤ deg ϕt −1 consists of the recursive computation of ϕt−1 , ϕt−2 , . . . ,

312

S. Pauli

ϕ2 , and ϕ1 -expansions. Let ni = deg ϕi (1 ≤ i ≤ t). The ϕt−1 -expansion of a(x) yields up to m/nt−1 polynomials of degree less than nt . The ϕt−2 -expansions of these polynomials yield up to m/nt−1 · nt−1 /nt−2 = m/nt−2 of degree less than nt−2 . Thus the (ϕ1 , . . . , ϕt−1 )-expansion of a(x) can be computed in m 2 2 + · · · + O + O(m) O m2 + O nmt nt 2 + O nm n n t−1 1 n1 t−1 operations in OK . Because ni+1 /ni ≥ 2 this is less than 2 2 log m + O(m) = O m2 i=0 2 2−i = O(m2 ). O m2 + O m2 + · · · + O 2m t−1 Lemma 18. The (ϕ0 , . . . , ϕt−1 )-expansion of a(x) ∈ OK [x] with m = deg a ≤ deg ϕt − 1 can be computed in O(m2 ) operations in OK . ∗ (ϕt ) > 2ν(disc Φ) for By Theorem 5 the polynomial Φ(x) is irreducible, if N vΦ ∗ ∗ some t ∈ N. In every iteration the increase from vΦ (ϕt ) to vΦ (ϕt+1 ) is at least 2/N , unless E = N , but that would imply irreducibility. Thus the algorithm terminates after at most ν(disc Φ) iterations. In our analysis of the cost of the steps in the main loop we exclude the cost of ﬁnding a proper factorization to a desired precision using the methods of section 2 in steps (c) and (h). We assume that two polynomials of degree up to n can be multiplied in O(n log n log log n) = O(n1+ε ) operations in their coeﬃcient ring [15].

(a,b,c,d) By Lemma 18 the ϕt -expansion Φ(x) = ϕt (x)N/nt +

N/nt −1

ai (x)ϕt (x)i

i=0

of Φ(x) and the (ϕ1 , . . . , ϕt )-expansion of the ai (x) can be computed in O(N 2 ) operations in OK . (e) The exponents sπ , s1 , . . . , st−1 in ψt (x) = π sπ ϕ1 (x)s0 · · · ϕt−1 (x)st−1 with ∗ vΦ (ψ) = ht /et can be computed with Algorithm 14. The most expensive computation is the extended Euclidean construction, which for integers less than N runs in time O((log N )2 ), at most log2 N times. (f ) We have a representation of ai (x)ψt (x)i−(N/nt ) (1 ≤ i ≤ N/nt ) as nt sums of power products of π, ϕ1 (x), . . . , ϕt−1 (x). In this representation only the exponents of ϕi (x) where Ei+ Fi+ = 1 are non-zero. There are at most log2 N such indices i. Let mt be the number of i < t with Ei+ Fi+ = 1. Reducing the coeﬃcients of the associated polynomial in this representation using the mt + relations ϕi (x)Ei /ψi (x) ∼ γi (1 ≤ i ≤ mt ) takes at most N i=1 i = Φ

O(N (log N )2 ) integer additions and N (t − 1) = O(N log N ) multiplications in the ﬁnite ﬁeld Kt with q F elements. (g,h) The factorization of a polynomial of degree at most N/F over a ﬁnite ﬁeld with at most q F elements can be done in O((N/F )2 log q F ) bit operations [6].

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(j) The cost of ﬁnding the exponents for the representation of Π(x) ∈ K[x] with ∗ vΦ (Π) = 1/E as a power product of π, ϕ1 (x), . . . , ϕt (x) is the same as the cost of ﬁnding ψ(x) in step (f ). The polynomial Γ (x) can be computed in the same way as the coeﬃcients ϑi (x) in step (l). + (k) The polynomial ϕt+1 (x) is constructed as a polynomial in ϕt (x)Et of degree Ft+ with coeﬃcients ϑi (x), 0 ≤ i ≤ Ft+ , (see (2)), obtained from the representations of the elements γu as ϕu (x)Eu /ψu (x) and ∗ vΦ ρu (ϕu (x)Eu /ψu (x)) > 0 for 1 ≤ u ≤ t−1. This is done by manipulating the exponents in the representation of the polynomials as sums of power products of π, ϕ1 (x), . . . , ϕt (x). + The computation of ϕt (x)Et takes log2 Et multiplications of polynomials of + j degree up to Et+ Et−1 Ft− t < N . For 2 ≤ j ≤ Ft+ the polynomial ϕt (x)Et can be computed in Ft+ multiplications of polynomials of degree up to Et Ft < N . For 1 ≤ t − 2 the exponent of ϕi (x) in the representation of ϑi (x) as a power product of ϕ1 (x), . . . , ϕt−1 (x) is less than Ei+ Fi+ . This gives less than log N multiplications of polynomials of degree less than N . As in (e) the exponents of at most log N of the ϕi (x) are nonzero. Therefore in total this step can be conducted in O(N 2+ε ) operations in OK [x]. ∗ By Theorem 5 the maximum of the valuations ν(vΦ (ξ)), where ξ is a root of Φ(x), is less than 2 ν(disc Φ) /N . This is also the maximal (absolute) slope of the Newton polygon of the polynomials under consideration. Therefore a precision of 2ν(disc Φ) is suﬃcient for all operations in the main loop.

Theorem 1. Let p be a ﬁxed prime. We can ﬁnd a breaking element or a two element certiﬁcate for the irreducibility of a polynomial Φ(x) ∈ Zp [x] in at most O(N 2+ε ν(disc Φ)2+ε ) operations of integers less than p.

10

Example

We show that Φ(x) = x32 + 16 ∈ Z2 [x] is irreducible using Algorithm 16. Initially we set ϕ1 (x) = x, E0 = 1, F0 = 1, K0 = Q2 . 32 (a) The ϕ1 -expansion of Φ(x) is Φ(x) = i=0 ai (x)ϕ0 (x)i = x32 + 16. ∗ ∗ (b) The valuations of the coeﬃcients are vΦ (a0 ) = 4, vΦ (ai ) = ∞ for 1 ≤ i ≤ 31, ∗ and vΦ (a32 ) = 0. 4 ∗ (c,d) ϕ1 (x) passes the Newton test; we get vΦ (ϕ1 ) = he11 = 32 = 18 , so E1+ = 8 and E1 = 8. E+ ∗ ∗ (e) We set ψ1 (x) = 2 as vΦ (ϕ1 1 ) = vΦ (x8 ) = 1. 4 (f,g) A1 (z) = z + 1 with A1 (z) = (z − 1)4 in F2 [z]. 8 + (h,i) ϕψ11(x) (x) passes the Hensel test; we get F1 = 1, K1 = Q2 , F1 = 1. (k) We obtain the next approximation of an irreducible factor of Φ(x):

8 x − 1 = x8 − 2. ϕ2 (x) = 2 2

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Second iteration: (a) The ϕ2 -expansion of Φ(x) is Φ(x) = ϕ2 (x)4 + 8ϕ2 (x)3 + 24ϕ2 (x)2 + 32ϕ2 (x) + 32. ∗ ∗ ∗ (32) = 5, vΦ (24) = 3, vΦ (8) = 3, (b) The valuations of the coeﬃcients are vΦ ∗ and vΦ (1) = 0. (c,d) ϕ2 (x) passes the Newton test; we get he22 = 54 , so E2+ = 1, E2 = 8. 2

∗ (ψ2 ) = 54 . (e) We set ψ2 (x) = x2 , so that vΦ (f,g) The associated polynomial with respect to ϕ2 (x) is A2 (z) = z 4 + 1 = (z − 1)4 ∈ F2 [z]. + 2 (x) (h,i) ϕ ψ2 (x) passes the Hensel test, we get F2 = 1, K2 = Q2 , F2 = 1. (l) We set

ϕ2 (x) − 1 = x8 − 2x2 − 2. ϕ3 (x) = ψ2 (x) ψ2 (x)

Third iteration: (a) The ϕ3 -expansion of Φ(x) is Φ(x) = ϕ3 (x)4 + a3 (x)ϕ3 (x)3 + a2 (x)ϕ3 (x)2 + a1 (x)ϕ3 (x) + a0 (x) where a3 (x) = 8x2 + 8, a2 (x) = 24x4 + 48x2 + 24, a1 (x) = 32x6 + 96x4 + 96x2 + 48, a0 (x) = 64x6 + 96x4 + 96x2 + 64. ∗ ∗ ∗ (b) The valuations of the coeﬃcients are vΦ (a0 ) = 21 4 , vΦ (a1 ) = 4, vΦ (a2 ) = 3, ∗ ∗ vΦ (a3 ) = 3, and vΦ (1) = 0. + ∗ (c,d) ϕ3 (x) passes the Newton test; we get vΦ (ϕ3 ) = he33 = 21 16 , E3 = 2, E3 = 16. E+

∗ ∗ (e) We ﬁnd ψ3 (x) = 22 x5 ; so that vΦ (ψ3 ) = vΦ (ϕ3 3 ) = 21 8 . (f,g) The associated polynomial with respect to ϕ3 (x) is A2 (z) = z 2 + 3 = (z − 1)3 ∈ F2 [z]. + 3 (x) (h,i) ϕ ψ3 (x) passes the Hensel test; we get F3 = 1, K3 = Q2 , F3 = 1. (l) We set ϕ4 (x) = x16 − 4x10 − 4x8 − 4x5 + 4x4 + 8x2 + 4.

Fourth iteration: (a) Let Φ(x) = ϕ4 (x)2 + a1 (x)ϕ4 (x) + a0 (x) be the ϕ4 -expansion of Φ(x). ∗ ∗ (b) We have vΦ (a0 ) = 85/16 and vΦ (a1 ) = 3. + (c,d) ϕ4 (x) passes the Newton test; we get he44 = 85 32 , E4 = 2, E4 = 32. (g) Now E4 F4 = 32 = deg Φ which implies the irreducibility of Φ(x) = x32 + 16.

Acknowledgments The author would like to thank the anonymous referees and David Ford for their numerous comments. He apologizes to them for the large number of small mistakes.

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References 1. Cannon, J.J., et al.: The computer algebra system Magma. University of Sydney (2010), http://magma.maths.usyd.edu.au/magma/ 2. Cantor, D.G., Gordon, D.: Factoring polynomials over p-adic fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 185–208. Springer, Heidelberg (2000) 3. Ford, D., Letard, P.: Implementing the Round Four maximal order algorithm. Journal de Th´eorie des Nombres de Bordeaux 6, 39–80 (1994) 4. Ford, D., Pauli, S., Roblot, X.-F.: A Fast Algorithm for Polynomial Factorization over Qp . Journal de Th´eorie des Nombres de Bordeaux 14, 151–169 (2002) 5. Ford, D., Veres, O.: On the Complexity of the Montes Ideal Factorization Algorithm. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX, July 19-23. LNCS, vol. 6197, pp. 174–185. Springer, Heidelberg (2010) 6. Kaltofen, E., Shoup, V.: Subquadratic-time factoring of polynomials over finite fields. Math. Comp. 67 (1998) 7. Guardia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory (2008), arXiv:0807.2620 8. Guardia, J., Montes, J., Nart, E.: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields (2008), arXiv:0807.4065 9. MacLane, S.: A Construction for absolute values in polynomial rings. Trans. Amer. Math. Soc. 40, 363–395 (1936) 10. Montes, J., Nart, E.: On a Theorem of Ore. Journal of Algebra 146, 318–334 (1992) 11. Montes, J.: Pol´ıgonos de Newton de orden superior y aplicaciones aritm´eticas, PhD Thesis, Universitat de Barcelona (1999) ¨ Newtonsche Polygone in der Theorie der algebraischen K¨ 12. Ore, O.: orper. Math. Ann. 99, 84–117 (1928) 13. PARI/GP, version 2.3.4, Bordeaux (2008), http://pari.math.u-bordeaux.fr/ 14. Pauli, S.: Factoring polynomials over local fields. J. Symb. Comp. 32, 533–547 (2001) 15. Sch¨ onhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971) 16. Stein, W., et al.: SAGE: Software for Algebra and Geometry Experimentation (2007), http://www.sagemath.org 17. Veres, O.: On the Complexity of Polynomial Factorization over p-adic Fields, PhD Dissertation, Concordia University, Montreal (2009)

On a Problem of Hajdu and Tengely Samir Siksek1 and Michael Stoll2 1 2

Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK [email protected] Mathematisches Institut, Universit¨ at Bayreuth, 95440 Bayreuth, Germany [email protected]

Abstract. We prove a result that ﬁnishes the study of primitive arithmetic progressions consisting of squares and ﬁfth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a2 , b2 , c2 , d5 ) is (1, 1, 1, 1). For the proof, we ﬁrst reduce the problem to that of determining the sets of rational points on three speciﬁc hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We ﬁnd generators for a subgroup of ﬁnite index of the Mordell-Weil group of the last curve. Applying Chabauty’s method, we prove that the only rational points on this curve are the obvious ones.

1

Introduction

Euler ([9, pages 440 and 635]) proved Fermat’s claim that four distinct squares cannot form an arithmetic progression. Powers in arithmetic progressions are still a subject of current interest. For example, Darmon and Merel [8] proved that the only solutions in coprime integers to the Diophantine equation xn + y n = 2z n with n ≥ 3 satisfy xyz = 0 or ±1. This shows that there are no non-trivial three term arithmetic progressions consisting of n-th powers with n ≥ 3. The result of Darmon and Merel is far from elementary; it needs all the tools used in Wiles’ proof of Fermat’s Last Theorem and more. An arithmetic progression (x1 , x2 , . . . , xk ) of integers is said to be primitive if the terms are coprime, i.e., if gcd(x1 , x2 ) = 1. Let S be a ﬁnite subset of integers ≥ 2. Hajdu [11] showed that if (a11 , . . . , akk )

(1)

is a non-constant primitive arithmetic progression with i ∈ S, then k is bounded by some (inexplicit) constant C(S). Bruin, Gy˝ ory, Hajdu and Tengely [2] showed that for any k ≥ 4 and any S, there are only ﬁnitely many primitive arithmetic progressions of the form (1), with i ∈ S. Moreover, for S = {2, 3} and k ≥ 4, they showed that ai = ±1 for i = 1, . . . , k. A recent paper of Hajdu and Tengely [12] studies primitive arithmetic progressions (1) with exponents belonging to S = {2, n} and {3, n}. In particular, they G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 316–330, 2010. c Springer-Verlag Berlin Heidelberg 2010

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show that any primitive non-constant arithmetic progression (1) with exponents i ∈ {2, 5} has k ≤ 4. Moreover, for k = 4 they show that (1 , 2 , 3 , 4 ) = (2, 2, 2, 5) or (5, 2, 2, 2).

(2)

Note that if (ai i : i = 1, . . . , k) is an arithmetic progression, then so is the reverse progression (ai i : i = k, k − 1, . . . , 1). Thus there is really only one case left open by Hajdu and Tengely, with exponents (1 , 2 , 3 , 4 ) = (2, 2, 2, 5). This is also mentioned as Problem 11 in a list of 22 open problems recently compiled by Evertse and Tijdeman [10]. In this paper we deal with this case. Theorem 1. The only arithmetic progression in coprime integers of the form (a2 , b2 , c2 , d5 ) is (1, 1, 1, 1). This together with the above-mentioned results of Hajdu and Tengely completes the proof of the following theorem. Theorem 2. There are no non-constant primitive arithmetic progressions of the form (1) with i ∈ {2, 5} and k ≥ 4. The primitivity condition is crucial, since otherwise solutions abound. Let for example (a2 , b2 , c2 , d) be any arithmetic progression whose ﬁrst three terms are squares — there are inﬁnitely many of one can take a = r2 − 2rs − s2 , these; 2 2 2 2 2 2 b = r + s , c = r + 2rs − s — then (ad ) , (bd2 )2 , (cd2 )2 , d5 ) is an arithmetic progression whose ﬁrst three terms are squares and whose last term is a ﬁfth power. For the proof of Thm. 1, we ﬁrst reduce the problem to that of determining the sets of rational points on three speciﬁc hyperelliptic curves of genus 4. A 2-cover descent computation (following Bruin and Stoll [3]) shows that there are no rational points on two of these curves. We ﬁnd generators for a subgroup of ﬁnite index of the Mordell-Weil group of the last curve. Applying Chabauty’s method, we prove that the only rational points on this curve are the obvious ones. All our computations are performed using the computer package MAGMA [1]. The result we prove here may perhaps not be of compelling interest in itself. Rather, the purpose of this paper is to demonstrate how we can solve problems of this kind with the available machinery. We review the relevant part of this machinery in Sect. 3, after we have constructed the curves pertaining to our problem in Sect. 2. Then, in Sect. 4, we apply the machinery to these curves. The proofs are mostly computational. We have tried to make it clear what steps need to be done, and to give enough information to make it possible to reproduce the computations (which have been performed independently by both authors as a consistency check).

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Construction of the Curves

Let (a2 , b2 , c2 , d5 ) be an arithmetic progression in coprime integers. Since a square is ≡ 0 or 1 mod 4, it follows that all terms are ≡ 1 mod 4, in particular, a, b, c and d are all odd. Considering the last three terms, we have the relation √ √ (−d)5 = b2 − 2c2 = (b + c 2)(b − c 2) . Since b√and c are odd and coprime, the two factors √ on the right are coprime in R = Z[ 2]. Since R× /(R× )5 is generated by 1 + 2, it follows that √ √ √ √ (3) b + c 2 = (1 + 2)j (u + v 2)5 = gj (u, v) + hj (u, v) 2 with −2 ≤ j ≤ 2 and u, v ∈ Z coprime (with u odd and v ≡ j + 1 mod 2). The polynomials gj and hj are homogeneous of degree 5 and have coeﬃcients in Z. Now the ﬁrst three terms of the progression give the relation a2 = 2b2 − c2 = 2gj (u, v)2 − hj (u, v)2 . Writing y = a/v 5 and x = u/v, this gives the equation of a hyperelliptic curve of genus 4, Cj : y 2 = fj (x) where fj (x) = 2gj (x, 1)2 −hj (x, 1)2 . Every arithmetic progression of the required form therefore induces a rational point on one of the curves Cj . We observe that taking conjugates in (3) leads to √ √ √ (−1)j b + (−1)j+1 c 2 = (1 + 2)−j (u + (−v) 2)5 , which implies that f−j (x) = fj (−x) and therefore that C−j and Cj are isomorphic and their rational points correspond to the same arithmetic progressions. We can therefore restrict attention to C0 , C1 and C2 . Their equations are as follows. C0 : y 2 = f0 (x) = 2x10 + 55x8 + 680x6 + 1160x4 + 640x2 − 16 C1 : y 2 = f1 (x) = x10 + 30x9 + 215x8 + 720x7 + 1840x6 + 3024x5 + 3880x4 + 2880x3 + 1520x2 + 480x + 112 C2 : y 2 = f2 (x) = 14x10 + 180x9 + 1135x8 + 4320x7 + 10760x6 + 18144x5 + 21320x4 + 17280x3 + 9280x2 + 2880x + 368 The trivial solution a = b = c = d = 1 corresponds to j = 1, (u, v) = (1, 0) in the above and therefore gives rise to the point ∞+ on C1 (this is the point at inﬁnity where y/x5 takes the value +1). Changing the signs of a, b or c leads to ∞− ∈ C1 (Q) (the point where y/x5 = −1) or to the two points at inﬁnity on the isomorphic curve C−1 .

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319

Background on Rational Points on Hyperelliptic Curves

Our task will be to determine the set of rational points on each of the curves C0 , C1 and C2 constructed in the previous section. In this section, we will give an overview of the methods we will use, and in the next section, we will apply these methods to the given curves. We will restrict attention to hyperelliptic curves, i.e., curves given by an aﬃne equation of the form C : y 2 = f (x) where f is a squarefree polynomial with integral coeﬃcients. The smooth projective curve birational to this aﬃne curve has either one or two additional points ‘at inﬁnity’. If the degree of f is odd, there is one point at inﬁnity, which is always a rational point. Otherwise there are two points at inﬁnity corresponding to the two square roots of the leading coeﬃcient of f . In particular, these two points are rational if and only if the leading coeﬃcient is a square. For example, C1 above has two rational points at inﬁnity, whereas the points at inﬁnity on C0 and C2 are not rational. We will use C in the following to denote the smooth projective model; C(Q) denotes as usual the set of rational points including those at inﬁnity. 3.1

Two-Cover Descent

It will turn out that C0 and C2 do not have rational points. One way of showing that C(Q) is empty is to verify that C(R) is empty or that C(Qp ) is empty for some prime p. This does not work for C0 or C2 ; both curves have real points and p-adic points for all p. (This can be checked by a ﬁnite computation.) So we need a more sophisticated way of showing that there are no rational points. One such method is known as 2-cover descent. We sketch the method here; for a detailed description, see [3]. An important ingredient of this and other methods is the algebra L := Q[T ] =

Q[x] , Q[x] · f (x)

where T denotes the image of x. If f is irreducible (as in our examples), then L is the number ﬁeld generated by a root of f . In general, L will be a product of number ﬁelds corresponding to the irreducible factors of f . We now assume that f has even degree 2g + 2, where g is the genus of the curve. This is the generic case; the odd degree case is somewhat simpler. We can then set up a map, called the descent map or x − T map: x − T : C(Q) −→ H :=

L× . Q× (L× )2

Here L× denotes the multiplicative group of L, and (L× )2 denotes the subgroup of squares. On points P ∈ C(Q) that are neither at inﬁnity nor Weierstrass points (i.e., points with vanishing y coordinate), the map is deﬁned as

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(x − T )(P ) = x(P ) − T mod Q× (L× )2 . Rational points at inﬁnity map to the trivial element, and if there are rational Weierstrass points, their images can be determined using the fact that the norm of x(P ) − T is y(P )2 divided by the leading coeﬃcient of f . If we can show that x − T has empty image on C(Q), then it follows that C(Q) is empty. We obtain information of the image by considering again C(R) and C(Qp ). We can carry out the same construction over R and over Qp , leading to an algebra Lv (v = p, or v = ∞ when working over R), a group Hv and a map (x − T )v : C(Qv ) −→ Hv

(where Q∞ = R).

We have inclusions C(Q) → C(Qv ) and canonical homomorphisms H → Hv . Everything ﬁts together in a commutative diagram x−T

C(Q) v C(Qv )

v (x−T )v

/

/H v

Hv

where v runs through the primes and ∞. If we can show that the images of the lower horizontal map and of the right vertical map do not meet, then the image of x − T and therefore also C(Q) must be empty. We can verify this by considering a ﬁnite subset of ‘places’ v. In general, we obtain a ﬁnite subset of H that contains the image of x−T ; this ﬁnite subset is known as the fake 2-Selmer set of C/Q. It classiﬁes either pairs of (isomorphism classes of) 2-covering curves of C that have points everywhere locally, i.e., over R and over all Qp , or else it classiﬁes such 2-covering curves, in which case it is the (true) 2-Selmer set. Whether it classiﬁes pairs or individual 2-coverings depends on a certain condition on the polynomial f . This condition is satisﬁed if either f has an irreducible factor√ of odd degree, or if deg f ≡ 2 mod 4 and f factors over a quadratic extension Q( d) as a constant times the product of two conjugate polynomials. A 2-covering of C is a morphism π : D → C that is unramiﬁed and becomes Galois over a suitable ﬁeld extension of ﬁnite degree, with Galois group (Z/2Z)2g . It is known that every rational point on C lifts to a rational point on some 2-covering of C. The actual computation splits into a global and a local part. The global computation uses the ideal class group and the unit group of L (or the constituent number ﬁelds of L) to construct a ﬁnite subgroup of H containing the image of x − T . The local computation determines the image of (x − T )v for ﬁnitely many places v. 3.2

The Jacobian

Most other methods make use of another object associated to the curve C: its Jacobian variety (or just Jacobian). This is an abelian variety J (a higherdimensional analogue of an elliptic curve) of dimension g, the genus of C. It

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reﬂects a large part of the geometry and arithmetic of C; its main advantage is that its points form an abelian group, whereas the set of points on C does not carry a natural algebraic structure. For our purposes, we can more or less forget the structure of J as a projective variety. Instead we use the description of the points on J as the elements of the degree zero part of the Picard group of C. The Picard group is constructed as a quotient of the group of divisors on C. A divisor on C is an element of ¯ of all algebraic points on C. The the free abelian group DivC on the set C(Q) absolute Galois group of Q acts on DivC ; a divisor that is ﬁxed by this action is rational. This does not mean that the points occurring in the divisor must be rational; points with the same multiplicity can be permuted. A nonzero rational ¯ has an associated divisor div(h) that function h on C with coeﬃcients in Q records its zeros and poles (with multiplicities). If h has coeﬃcients in Q, then div(h) is rational. The homomorphism deg : DivC → Z induced by sending each ¯ to 1 gives the degree of a divisor. Divisors of functions have degree point in C(Q) zero. Two divisors D, D ∈ DivC are linearly equivalent if their diﬀerence is the divisor of a function. The equivalence classes are the elements of the Picard group PicC deﬁned by the following exact sequence. × div ¯ × −→ Q(C) ¯ 0 −→ Q −→ DivC −→ PicC −→ 0

Since divisors of functions have degree zero, the degree homomorphism descends ¯ is isomorphic as a to PicC . We denote its kernel by Pic0C . It is a fact that J(Q) group to Pic0C . The rational points J(Q) correspond to the elements of Pic0C left invariant by the Galois group. In general it is not true that a point in J(Q) can be represented by a rational divisor, but this is the case when C has a rational point, or at least points everywhere locally. The most important fact about the group J(Q) is the statement of the Mordell-Weil Theorem: J(Q) is a finitely generated abelian group. For this reason, J(Q) is often called the Mordell-Weil group of J or of C. If P0 ∈ C(Q), then the map C P → [P − P0 ] ∈ J is a Q-deﬁned embedding of C into J. We use [D] to denote the linear equivalence class of the divisor D. The basic idea of the methods described below is to try to recognise the points of C embedded in this way among the rational points on J. We need a way of representing elements of J(Q). Let P → P − denote the hyperelliptic involution on C; this is the morphism C → C that changes the sign of the y coordinate. Then it is easy to see that the divisors P + P − all belong to the same class W ∈ PicC . An eﬀective divisor D (a divisor such that no point occurs with negative multiplicity) is in general position if there is no point P such that D − P − P − is still eﬀective. Divisors in general position not containing points at inﬁnity can be represented in a convenient way by pairs of polynomials (a(x), b(x)). This pair represents the divisor D such that its image on the projective line (under the x-coordinate map) is given by the roots of a; the corresponding points on C are determined by the relation y = b(x). The polynomials have to satisfy the relation f (x) ≡ b(x)2 mod a(x). This is

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the Mumford representation of D. The polynomials a and b can be chosen to have rational coeﬃcients if and only if D is rational. (The representation can be adapted to allow for points at inﬁnity occurring in the divisor.) If the genus g is even, then it is a fact that every point in J(Q) has a unique representation of the form [D] − nW where D is a rational divisor in general position of degree 2n and n ≥ 0 is minimal. The Mumford representation of D is then also called the Mumford representation of the corresponding point on J. It is fairly easy to add points on J using the Mumford representation, see [5]. This addition procedure is implemented in MAGMA, for example. There is a relation between 2-coverings of C and the Jacobian J. Assume C is embedded in J as above. Then if D is any 2-covering of C that has a rational point P , D can be realised as the preimage of C under a map of the form Q → 2Q + Q0 on J, where Q0 is the image of P on C ⊂ J. A consequence of this is that two rational points P1 , P2 ∈ C(Q) lift to the same 2-covering if and only if [P1 − P2 ] ∈ 2J(Q). 3.3

The Mordell-Weil Group

We will need to know generators of a ﬁnite-index subgroup of the Mordell-Weil group J(Q). Since J(Q) is a ﬁnitely generated abelian group, it will be a direct sum of a ﬁnite torsion part and a free abelian group of rank r; r is called the rank of J(Q). So what we need is a set of r independent points in J(Q). The torsion subgroup of J(Q) is usually easy to determine. The main tool used here is the fact that the torsion subgroup injects into J(Fp ) when p is an odd prime not dividing the discriminant of f . If the orders of the ﬁnite groups J(Fp ) are coprime for suitable primes p, then this shows that J(Q) is torsion-free. We can ﬁnd points in J(Q) by search. This can be done by searching for rational points on the variety parameterising Mumford representations of divisors of degree 2, 4, . . . . We can then check if the points found are independent by again mapping into J(Fp ) for one or several primes p. The hard part is to know when we have found enough points. For this we need an upper bound on the rank r. This can be provided by a 2-descent on the Jacobian J. This is described in detail in [16]. The idea is similar to the 2cover descent on C described above in Sect. 3.1. Essentially we extend the x − T map from points to divisors. It can be shown that the value of (x − T )(D) only depends on the linear equivalence class of D. This gives us a homomorphism from J(Q) into H, or more precisely, into the kernel of the norm map NL/Q : H → Q× /(Q× )2 . It can be shown that the kernel of this x − T map on J(Q) is either 2J(Q), or it contains 2J(Q) as a subgroup of index 2. The former is the case when f satisﬁes the same condition as that mentioned in Sect. 3.1. We can then bound (x − T )(J(Q)) in much the same way as we did when doing a 2-cover descent on C. The global part of the computation is identical. The local part is helped by the fact that we now have a group homomorphism (or a homomorphism of F2 -vector spaces), so we can use linear algebra. We obtain a bound for the order of J(Q)/2J(Q), from which we can deduce a bound for

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the rank r. If we are lucky and found that same number of independent points in J(Q), then we know that these points generate a subgroup of ﬁnite index. The group containing (x−T )(J(Q)) we compute is known as the fake 2-Selmer group of J [13]. If the polynomial f satisﬁes the relevant condition, then this fake Selmer group is isomorphic to the true 2-Selmer group of J (that classiﬁes 2coverings of J that have points everywhere locally). 3.4

The Chabauty-Coleman Method

If the rank r is less than the genus g, there is a method available that allows us to get tight bounds on the number of rational points on C. This goes back to Chabauty [6], who used it to prove Mordell’s Conjecture in this case. Coleman [7] reﬁned the method. We give a sketch here; more details can be found for example in [15]. Let p be a prime of good reduction for C (this is the case when p is odd 1 and does not divide the discriminant of f ). We use ΩC (Qp ) and ΩJ1 (Qp ) to denote the spaces of regular 1-forms on C and J that are deﬁned over Qp . If P0 ∈ C(Q) and ι : C → J, P → [P − P0 ] denotes the corresponding embedding 1 of C into J, then the induced map ι∗ : ΩJ1 (Qp ) → ΩC (Qp ) is an isomorphism that is independent of the choice of basepoint P0 . Both spaces have dimension g. There is an integration pairing Q 1 ΩC (Qp ) × J(Qp ) −→ Qp , (ι∗ ω, Q) −→ ω = ω, log Q . 0

In the last expression, log Q denotes the p-adic logarithm on J(Qp ) with values in the tangent space of J(Qp ) at the origin, and ΩJ1 (Qp ) is identiﬁed with the dual of this tangent space. If r < g, then there are (at least) g −r linearly independent 1 diﬀerentials ω ∈ ΩC (Qp ) that annihilate the Mordell-Weil group J(Q). Such a diﬀerential can be scaled so that it reduces to a non-zero diﬀerential ω ¯ mod p. Now the important fact is that if ω ¯ does not vanish at a point P¯ ∈ C(Fp ), then there is at most one rational point on C(Q) whose reduction is P¯ . (There are more general bounds valid when ω ¯ does vanish at P¯ , but we do not need them here.)

4

Determining the Rational Points

In this section, we determine the set of rational points on the three curves C0 , C1 and C2 . To do this, we apply the methods described in Sect. 3. We ﬁrst consider C0 and C2 . We apply the 2-cover-descent procedure described in Sect. 3.1 to the two curves and ﬁnd that in each case, there are no 2-coverings that have points everywhere locally. For C0 , only 2-adic information is needed in addition to the global computation, for C2 , we need 2-adic and 7adic information. Note that the number ﬁelds generated by roots of f0 or f2 are suﬃciently small in terms of degree and discriminant that the necessary class and unit group computations can be done unconditionally. This leads to the following.

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Proposition 3. There are no rational points on the curves C0 and C2 . Proof. The 2-cover descent procedure is available in recent releases of MAGMA. The computations leading to the stated result can be performed by issuing the following MAGMA commands. > SetVerbose("Selmer",2); > TwoCoverDescent(HyperellipticCurve(Polynomial( [-16,0,640,0,1160,0,680,0,55,0,2]))); > TwoCoverDescent(HyperellipticCurve(Polynomial( [368,2880,9280,17280,21320,18144,10760,4320,1135,180,14]))); We explain how the results can be checked independently. We give details for C0 ﬁrst. The procedure for C2 is similar, so we only explain the diﬀerences. The polynomial f0 is irreducible, and it can be checked √ that the number ﬁeld generated by one of its roots is isomorphic to L = Q( 10 288). Using MAGMA or ˜ pari/gp, one checks that this ﬁeld has trivial class group. The ﬁnite subgroup H × × 2 of H containing the Selmer set is then given as OL,S /(Z× (O ) ), where S L,S {2,3,5} is the set of primes in OL above the ‘bad primes’ 2, 3 and 5. The set S contains two primes above 2, of degrees 1 and 4, respectively, and one prime above 3 and 5 each, of degree 2 in both cases. Since L has two real embeddings and four pairs ˜ is of complex embeddings, the unit rank is 5. The rank (or F2 -dimension) of H then 7. (Note that 2 is a square in L.) The descent map takes its values in the ˜ consisting of elements whose norm is twice a square. This subset is subset of H of size 32; elements of OL representing it can easily be obtained. Let δ be such a representative. We let T be a root of f0 in L and check that the system of equations y 2 = f0 (x), x − T = δcz 2 has no solutions with x, y, c ∈ Q2 , z ∈ L ⊗Q Q2 . The second equation leads, after expanding δz 2 as a Q-linear combination of 1, T, T 2, . . . , T 9 , to eight homogeneous quadratic equations in the ten unknown coeﬃcients of z. Any solution to these equations gives a unique x, for which f0 (x) is a square. The latter follows by taking norms on both sides of x − T = δcz 2 . So we only have to check the intersection of eight quadrics in P9 for existence of Q2 -points. Alternatively, we × × 2 evaluate the descent map on C0 (Q2 ), to get its image in H2 = L× 2 /(Q2 (L2 ) ), where L2 = L ⊗Q Q2 . Then we check that none of the representatives δ map into this image. When dealing with C2 , the ﬁeld L is generated by a root of x10 − 6x5 − 9. Since the leading coeﬃcient of f2 is 14, we have to add (the primes above) 7 to the bad primes. As before, the class group is trivial, and we have the same splitting behaviour of 2, 3 and 5. The prime 7 splits into two primes of degree 1 and two primes of degree 4. The group of S-units of L modulo squares has now ˜ has rank 10, and the subset of H consisting of elements rank 14, the group H whose norm is 14 times a square has 128 elements. These elements now have to be tested for compatibility with the 2-adic and the 7-adic information, which can be done using either of the two approaches described above. The 7-adic check is

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only necessary for one of the elements; the 127 others are already ruled out by the 2-adic check. We cannot hope to deal with C1 in the same easy manner, since C1 has two rational points at inﬁnity coming from the trivial solutions. We can still perform a 2-cover-descent computation, though, and ﬁnd that there is only one 2-covering of C1 with points everywhere locally, which is the covering that lifts the points at inﬁnity. Only 2-adic information is necessary to show that the fake 2-Selmer set has at most one element, so we can get this result using the following MAGMA command. > TwoCoverDescent(HyperellipticCurve(Polynomial( [112,480,1520,2880,3880,3024,1840,720,215,30,1])) : PrimeCutoff := 2); (In some versions of MAGMA this returns a two-element set. However, as can be checked by pulling back under the map returned as a second value, these two elements correspond to the images of 1 and −1 in L× /(L× )2 Q× and therefore both represent the trivial element. The error is caused by MAGMA using 1 instead of −1 as a ‘generator’ of Q× /(Q× )2 . This bug is corrected in recent releases.) The computation can be performed in the same way as for C0 and C2 . The relevant ﬁeld L is generated by a root of x10 − 18x5 + 9; it has class number 1, and the primes 2, 3 and 5 split in the same way as before. The subset H (in fact ˜ consisting of elements with square norm has size 32. Of these, a subgroup) of H only the element represented by 1 is compatible with the 2-adic constraints.√ We remark that by the way it is given, the polynomial f1 factors over Q( 2) into two conjugate factors of degree 5. This implies that the ‘fake 2-Selmer set’ computed by the 2-cover descent is the true 2-Selmer set, so that there is really only one 2-covering that corresponds to the only element of the set computed by the procedure. We state the result as a lemma. We ﬁx P0 = ∞− ∈ C1 as our basepoint and write J1 for the Jacobian variety of C1 . Then, as described in Sect. 3.2, ι : C1 −→ J1 , P −→ [P − P0 ] is an embedding deﬁned over Q. Lemma 4. Let P ∈ C1 (Q). Then the divisor class [P − P0 ] is in 2J1 (Q). Proof. Let D be the unique 2-covering of C1 (up to isomorphism) that has points everywhere locally. The fact that D is unique follows from the computation of the 2-Selmer set. Any rational point P ∈ C1 (Q) lifts to a rational point on some 2-covering of C1 . In particular, this 2-covering then has a rational point, so it also satisﬁes the weaker condition that it has points everywhere locally. Since D is the only 2-covering of C1 satisfying this condition, P0 and P must both lift to a rational point on D. This implies by the remark at the end of Sect. 3.2 that [P − P0 ] ∈ 2J1 (Q). To make use of this information, we need to know J1 (Q), or at least a subgroup of ﬁnite index. A computer search reveals two points in J1 (Q), which are given in Mumford representation (see Sect. 3.2) as follows.

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Q1 = x4 + 4x2 + 45 , −16x3 − 96 5 x 3 3 36 2 48 36 Q2 = x4 + 24 − 1712 5 x + 5 x + 5 x+ 5 , 75 x −

976 2 25 x

−

1728 25 x

−

2336 25

We note that 2Q1 = [∞+ − ∞− ]; this makes Lemma 4 explicit for the known two points on C1 . Lemma 5. The Mordell-Weil group J1 (Q) is torsion-free, and Q1 , Q2 are linearly independent. In particular, the rank of J1 (Q) is at least 2. Proof. The only primes of bad reduction for C1 are 2, 3 and 5. It is known that the torsion subgroup of J1 (Q) injects into J1 (Fp ) when p is an odd prime of good reduction. Since #J1 (F7 ) = 2400 and #J1 (F41 ) = 2633441 are coprime, there can be no nontrivial torsion in J1 (Q). We check that the image of Q1 , Q2 in J1 (F7 ) is not cyclic. This shows that Q1 and Q2 must be independent. The next step is to show that the Mordell-Weil rank is indeed 2. For this, we compute the 2-Selmer group of J1 as sketched in Sect. 3.3 and described in detail in [16]. We give some details of the computation, since it is outside the scope of the functionality that is currently provided by MAGMA (or any other software package). √ We ﬁrst remind ourselves that f1 factors over Q( 2). This implies that the kernel of the x − T map on J(Q) is 2J(Q). Therefore the ‘fake 2-Selmer group’ that we compute is in fact the actual 2-Selmer group of J1 . Since J1 (Q) is torsionfree, the order of the 2-Selmer group is an upper bound for 2r , where r is the rank of J1 (Q). The global computation is the same as that we needed to do for the 2-cover descent. In particular, the Selmer group is contained in the group H from above, consisting of the S-units of L with square norm, modulo squares and modulo {2, 3, 5}-units of Q. For the local part of the computation, we have to compute the image of J1 (Qp ) under the local x− T map for the primes p of bad reduction. We check that there is no 2-torsion in J1 (Q3 ) and J1 (Q5 ) (f1 remains irreducible both over Q3 and over Q5 ). This implies that the targets of the local maps (x − T )3 and (x − T )5 are trivial, which means that these two primes need not be considered as bad primes for the descent computation. The real locus C1 (R) is connected, which implies that there is no information coming from the local image at the inﬁnite place. (Recall that C1 denotes the smooth projective model of the curve. The real locus of the aﬃne curve y 2 = f1 (x) has two components, but they are connected to each other through the points at inﬁnity.) Therefore, we only need to use 2-adic information in the computation. We set L2 = L⊗Q Q2 and compute the natural homomorphism μ2 : H −→ H2 =

L× 2 × 2 Q2 (L× 2)

.

Let I2 be the image of J1 (Q2 ) in H2 . Then the 2-Selmer group is μ−1 2 (I2 ).

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It remains to compute I2 , which is the hardest part of the computation. The 2-torsion subgroup J1 (Q2 )[2] has order 2 (f1 splits into factors of degrees 2 and 8 over Q2 ); this implies that J1 (Q2 )/2J1 (Q2 ) has dimension g + 1 = 5 as an F2 -vector space. This quotient is generated by the images of Q1 and Q2 and of three further points of the form [Di ] − deg2Di W , where Di is the sum of points on C1 whose x-coordinates are the roots of D1 : x − 12 x − 14 , D2 : x2 − 2x + 6 , D3 : x4 + 4x3 + 12x2 + 36 , respectively. These points were found by a systematic search, using the fact that the local map (x − T )2 is injective in our situation. We can therefore stop the search procedure as soon as we have found points whose images generate a ﬁvedimensional F2 -vector space. We thus ﬁnd I2 ⊂ H2 and then can compute the 2-Selmer group. In our situation, μ2 is injective, and the intersection of its image with I2 is generated by the images of Q1 and Q2 . Therefore, the F2 -dimension of the 2-Selmer group is 2. Lemma 6. The rank of J1 (Q) is 2, and Q1 , Q2 ⊂ J1 (Q) is a subgroup of finite odd index. Proof. The Selmer group computation shows that the rank is ≤ 2, and Lemma 5 shows that the rank is ≥ 2. Regarding the second statement, it is now clear that we have a subgroup of ﬁnite index. The observation stated just before the lemma shows that the given subgroup surjects onto the 2-Selmer group under the x − T map. Since the kernel of the x − T map is 2J1 (Q), this implies that the index is odd. Now we want to use the Chabauty-Coleman method sketched in Sect. 3.4 to show that ∞+ and ∞− are the only rational points on C1 . To keep the computations reasonably simple, we want to work at p = 7, which is the smallest prime of good reduction. For p a prime of good reduction, we write ρp for the two ‘reduction mod p’ maps J1 (Q) → J1 (Fp ) and C1 (Q) → C1 (Fp ). Lemma 7. Let P ∈ C1 (Q). Then ρ7 (P ) = ρ7 (∞+ ) or ρ7 (P ) = ρ7 (∞− ). Proof. Let G = Q1 , Q2 be the subgroup of J1 (Q) generated by the two points Q1 and Q2 . We ﬁnd that ρ7 (G) has index 2 in J1 (F7 ) ∼ = Z/10Z ⊕ Z/240Z. By Lemma 6, we know that (J1 (Q) : G) is odd, so we can deduce that ρ7 (G) = ρ7 (J1 (Q)). The group J1 (F7 ) surjects onto (Z/5Z)2 . Since ρ7 (J1 (G)) has index 2 in J1 (F7 ), ρ7 (G) = ρ7 (J1 (Q)) also surjects onto (Z/5Z)2 . This implies that the index of G in J1 (Q) is not divisible by 5. We determine the points P ∈ C1 (F7 ) such that ι(P ) ∈ ρ7 (2J1 (Q)) = 2ρ7 (G). We ﬁnd the set X7 = {ρ7 (∞+ ), ρ7 (∞− ), (−2, 2), (−2, −2)} .

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Note that for any P ∈ J1 (Q), we must have ρ7 (P ) ∈ X7 by Lemma 4. Now we look at p = 13. The image of G in J1 (F13 ) ∼ = Z/10Z ⊕ Z/2850Z has index 5. Since we already know that (J1 (Q) : G) is not a multiple of 5, this implies that ρ13 (G) = ρ13 (J1 (Q)). As above for p = 7, we compute the set X13 ⊂ C1 (F13 ) of points mapping into ρ13 (2J1 (Q)). We ﬁnd X13 = {ρ13 (∞+ ), ρ13 (∞− )} . Now suppose that there is P ∈ C1 (Q) with ρ7 (P ) ∈ {(−2, 2), (−2, −2)}. Then ι(P ) is in one of two speciﬁc cosets in J1 (Q)/ ker ρ7 ∼ = G/ ker ρ7 |G . On the other hand, we have ρ13 (P ) = ρ13 (∞± ), so that ι(P ) is in one of two speciﬁc cosets in J1 (Q)/ ker ρ13 ∼ = G/ ker ρ13 |G . If we identify G = Q1 , Q2 with Z2 , then we can ﬁnd the kernels of ρ7 and of ρ13 on G explicitly, and we can also determine the relevant cosets explicitly. It can then be checked that the union of the ﬁrst two cosets does not meet the union of the second two cosets. This implies that such a point P cannot exist. Therefore, the only remaining possibilities are that ρ7 (P ) = ρ7 (∞± ). Remark 8. The use of information at p = 13 to rule out residue classes at p = 7 in the proof above is a very simple instance of a method known as the MordellWeil sieve. For a detailed description of this method, see [4]. Now we need to ﬁnd the space of holomorphic 1-forms on C1 , deﬁned over Q7 , that annihilate the Mordell-Weil group under the integration pairing, compare Sect. 3.4. We follow the procedure described in [14]. We ﬁrst ﬁnd two independent points in the intersection of J1 (Q) and the kernel of reduction mod 7. In our case, we take R1 = 20Q1 and R2 = 5Q1 + 60Q2 . We represent these points in the form Rj = [Dj − 4∞− ] with eﬀective divisors D1 , D2 of degree 4. The coeﬃcients of the primitive polynomial in Z[x] whose roots are the x-coordinates of the points in the support of D1 have more than 100 digits and those of the corresponding polynomial for D2 ﬁll several pages, so we refrain from printing them here. (This indicates that it is a good idea to work with a small prime!) The points in the support of D1 and D2 all reduce to ∞− modulo the prime above 7 in their ﬁelds of deﬁnition (which are degree 4 number ﬁelds totally ramiﬁed at 7). Expressing 1 a basis of ΩC (Q7 ) as power series in the uniformiser t = 1/x at P0 = ∞− 1 times dt, we compute the integrals numerically. More precisely, the diﬀerentials η0 =

dx , 2y

η1 =

x dx , 2y

η2 =

x2 dx 2y

and η3 =

x3 dx 2y

1 form a basis of ΩC (Q7 ). We get 1

ηj = t3−j

1 2

−

15 145385 4 2764899 5 t + 115t2 − 1980t3 + t − t + . . . dt 2 4 4

as power series in the uniformiser. Using these power series up to a precision of t20 , we compute the following 7-adic approximations to the integrals.

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⎛

Rj

0

ηi

0≤i≤3,1≤j≤2

⎞ −20 · 7 + O(74 ) −155 · 7 + O(74 ) ⎜−150 · 7 + O(74 ) −13 · 7 + O(74 ) ⎟ ⎟ =⎜ ⎝−130 · 7 + O(74 ) −83 · 7 + O(74 ) ⎠ −19 · 7 + O(74 ) 163 · 7 + O(74 )

From this, it follows easily that the reductions mod 7 of the (suitably scaled) 1 diﬀerentials that kill J1 (Q) ﬁll the subspace of ΩC (F7 ) spanned by 1 ω1 = (1 + 3x − 2x2 )

dx 2y

and ω2 = (1 − x2 + x3 )

dx . 2y

Since ω2 does not vanish at the points ρ7 (∞± ), this implies that there can be at most one rational point P on C1 with ρ7 (P ) = ρ7 (∞+ ) and at most one point P with ρ7 (P ) = ρ7 (∞− ) (see for example [15, Prop. 6.3]). Proposition 9. The only rational points on C1 are ∞+ and ∞− . Proof. Let P ∈ C1 (Q). By Lemma 7, ρ7 (P ) = ρ7 (∞± ). By the argument above, for each sign s ∈ {+, −}, we have #{P ∈ C1 (Q) : ρ7 (P ) = ρ7 (∞s )} ≤ 1. These two facts together imply that #C1 (Q) ≤ 2. Since we know the two rational points ∞+ and ∞− on C1 , there cannot be any further rational points. We can now prove Thm. 1. Proof (of Thm. 1). The considerations in Sect. 2 imply that if (a2 , b2 , c2 , d5 ) is an arithmetic progression in coprime integers, then there are coprime u and v, related to a, b, c, d by (3), such that (u/v, a/v 5 ) is a rational point on one of the curves Cj with −2 ≤ j ≤ 2. By Prop. 3, there are no rational points on C0 and C2 and therefore also not on the curve C−2 , which is isomorphic to C2 . By Prop. 9, the only rational points on C1 (and C−1 ) are the points at inﬁnity. This translates into a = ±1, u = ±1, v = 0, and we have j = ±1. We deduce a2 = 1, b2 = g1 (±1, 0)2 = 1, whence also c2 = d5 = 1.

References 1. Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System I: The User Language. J. Symb. Comp. 24, 235–265 (1997), http://magma.maths.usyd.edu.au/magma 2. Bruin, N., Gy˝ ory, K., Hajdu, L., Tengely, S.: Arithmetic progressions consisting of unlike powers. Indag. Math. 17, 539–555 (2006) 3. Bruin, N., Stoll, M.: 2-cover descent on hyperelliptic curves. Math. Comp. 78, 2347–2370 (2009) 4. Bruin, N., Stoll, M.: The Mordell-Weil sieve: Proving non-existence of rational points on curves. LMS J. Comput. Math. (to appear), arXiv:0906.1934v2 [math.NT] 5. Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48, 95–101 (1987) 6. Chabauty, C.: Sur les points rationnels des courbes alg´ebriques de genre sup´erieur a l’unit´e. C. R. Acad. Sci. Paris 212, 882–885 (1941) (French) `

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7. Coleman, R.F.: Eﬀective Chabauty. Duke Math. J. 52, 765–770 (1985) 8. Darmon, H., Merel, L.: Winding quotients and some variants of Fermat’s last theorem. J. Reine Angew. Math. 490, 81–100 (1997) 9. Dickson, L.E.: History of the theory of numbers. Vol. II: Diophantine Analysis. Chelsea Publishing Co., New York (1966) 10. Evertse, J.-H., Tijdeman, R.: Some open problems about Diophantine equations from a workshop in Leiden in (May 2007), http://www.math.leidenuniv.nl/~ evertse/07-workshop-problems.pdf 11. Hajdu, L.: Perfect powers in arithmetic progression. A note on the inhomogeneous case. Acta Arith. 113, 343–349 (2004) 12. Hajdu, L., Tengely, S.: Arithmetic progressions of squares, cubes and n-th powers. Funct. Approx. Comment. Math. 41, 129–138 (2009) 13. Poonen, B., Schaefer, E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997) 14. Stoll, M.: Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math. 11, 367–380 (2008) 15. Stoll, M.: Independence of rational points on twists of a given curve. Compositio Math. 142, 1201–1214 (2006) 16. Stoll, M.: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98, 245–277 (2001)

Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures Jonathan P. Sorenson Butler University, Indianapolis IN 46208, USA [email protected] http://www.butler.edu/~ sorenson

Abstract. We extend the known tables of pseudosquares and pseudocubes, discuss the implications of these new data on the conjectured distribution of pseudosquares and pseudocubes, and present the details of the algorithm used to do this work. Our algorithm is based on the spacesaving wheel data structure combined with doubly-focused enumeration, run in parallel on a cluster supercomputer.

1

Introduction

It is well-known that testing for primality can be done in polynomial time [1,3]. However, the fastest known deterministic algorithms are conjectured to be the pseudosquares prime test of Lukes, Patterson, and Williams [6], and its generalizations, the pseudocube prime test of Berrizbeitia, M¨ uller, and Williams [4], and the Eisenstein pseudocube test [13,15], all of which run in roughly cubic time, if a suﬃciently large pseudosquare or pseudocube is available. In particular, the pseudosquares prime test is very useful in the context of ﬁnding all primes in an interval [10], where sieving can be used in place of trial division. This, then, motivates our search for larger and larger peudosquares and pseudocubes, and our attempts to predict their distribution. See, for example, Wooding and Williams [14] and also [7,12,8,2,11]. In this paper, we present extensions to the known tables of pseudosquares and pseudocubes in §2. We discuss the implications of this new data on the conjectured distribution of pseudosquares and pseudocubes in §3, and give a minor reﬁnement of the current conjectures. Then we describe our parallel algorithm, based on Bernstein’s doubly-focused enumeration [2], which is used in a way similar, but not identical to the work of Wooding and Williams [14], combined with the space-saving wheel data structure presented in [10, §4.1]. We then suggest ideas for future work in §5.

Supported by a grant from the Holcomb Awards Committe, and computing resources provided by the Frank Levinson Supercomputing Center at Butler University.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 331–339, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Computational Results

Let (x/y) denote the Legendre symbol [5]. For an odd prime p, let Lp,2 , the pseudosquare for p, be the smallest positive integer such that 1. Lp,2 ≡ 1 (mod 8), 2. (Lp,2 /q) = 1 for every odd prime q ≤ p, and 3. Lp,2 is not a perfect square. In other words, Lp,2 is a square modulo all primes up to p, but is not a square. We found the following new pseudosquares: p Lp,2 367 36553 34429 47705 74600 46489 373 42350 25223 08059 75035 19329 379 > 1025 The two pseudosquares listed were found in 2008 in a computation that went up to 5 × 1024 , taking roughly 3 months wall time. The ﬁnal computation leading to the lower bound of 1025 ran for about 6 months, in two 3-month pieces, the second of which ﬁnished on January 1st, 2010. Wooding and Williams [14] had found a lower bound of L367,2 > 120120 × 264 ≈ 2.216 × 1024 . (Note: a complete table of pseudosquares, current as of this writing, is available at http://cr.yp.to/focus.html care of Dan Bernstein). Note that 1025 may be used as a lower bound for L379,2 in the pseudosquares prime test. Together with trial division to guarantee there are no divisors below, say, 1010 , this means the pseudosquares prime test is practical on integers of 35 decimal digits, especially in the context of a prime sieve [10]. Similarly, for an odd prime p, let Lp,3 , the pseudocube for p, be the smallest positive integer such that 1. 2. 3. 4.

Lp,3 ≡ ±1 (mod 9), (q−1)/3 Lp,3 ≡ 1 (mod q) for every prime q ≤ p, q ≡ 1 (mod 3), gcd(Lp,3 , q) = 1 for every prime q ≤ p, and Lp,3 is not a perfect cube.

We found the following new pseudocubes (only listed for p ≡ 1 (mod 3)): p 499 523,541 547 571,577 1 601,607 2 613 67 619

601 1166 41391 62485 41913 44415

25695 14853 50561 73199 74719 80981

21674 91487 50994 87995 36148 24912

Lp,3 16551 89317 02789 15947 78852 27899 69143 39717 42758 90677 90374 06633 > 1027

These pseudocubes were found in about 6 months of total wall time in 2009. Wooding and Williams [14] had found a lower bound of L499,3 > 1.45152 × 1022. For a complete list of known pseudocubes, see [14,4,11].

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3

333

The Distribution of Pseudosquares and Pseudocubes

Let pi denote the ith prime, and qi denote the ith prime such that qi ≡ 1 (mod 3). In [6] it was conjectured that, for a constant c2 > 0, we have Lpn ,2 ≈ c2 2n log pn .

(1)

Using similar methods, in [4] it was conjectured that, for a constant c3 > 0, we have Lqn ,3 ≈ c3 3n (log qn )2 . (2) In a desire to test the accuracy of these conjectures, for integers n > 0 let us deﬁne Lpn ,2 , 2n log pn Lq ,3 . c3 (n) := n n 3 (log qn )2 c2 (n) :=

(3) (4)

We calculated c2 (n) and c3 (n) from known pseudosquares and pseudocubes. We present these computations in Table 1, for pseudosquares, and in Table 2, for pseudocubes, below. From Table 1, we readily see that c2 (n) appears to be bounded between roughly 5 and 162, with an average value near 45. There is no clear trend toward zero or inﬁnity. Due to the common occurence of values of n where Lpn ,2 = Lpn+1 ,2 (for example, n = 56), it should also be clear c2 (n) does not have a limit. Similarly for the pseudocubes, in Table 2 we see that 0.05 < c3 (n) < 6.5 for 10 ≤ n ≤ 53, with an average value of roughly 1.22. And again, there is no clear trend toward zero or inﬁnity, nor can there be a limit for c3 (n). This leads us to the following reﬁnements, if you will, of the conjectures (1),(2) above. Conjecture. For the pseudosquares, we conjecture that Lpn ,2 > 0, 2n log pn Lp ,2 < ∞. lim sup n n 2 log pn n→∞ lim inf n→∞

(5) (6)

Similarly, for the pseudocubes, we conjecture that Lqn ,3 n n→∞ 3 (log qn )2

lim inf

lim sup n→∞

> 0,

Lqn ,3 < ∞. 3n (log qn )2

(7) (8)

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Table 1. Values of c2 (n) based on known pseudosquares

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

pn 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109

Lpn ,2 c2 (n) 73 16.61 241 18.72 1009 32.41 2641 34.42 8089 49.28 18001 49.64 53881 71.48 87481 54.49 117049 33.95 515761 73.34 1083289 73.24 3206641 105.41 3818929 61.97 9257329 73.38 22000801 84.55 48473881 90.70 48473881 44.98 175244281 79.49 427733329 95.70 427733329 47.54 898716289 49.04 2805544681 75.69 2805544681 37.25 2805544681 18.28 10310263441 33.29 23616331489 37.96 85157610409 67.89 85157610409 33.81

n 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

pn Lpn ,2 c2 (n) 113 196265095009 38.67 127 196265095009 18.87 131 2871842842801 137.15 137 2871842842801 67.95 139 2871842842801 33.88 149 26250887023729 152.68 151 26250887023729 76.14 157 112434732901969 161.79 163 112434732901969 80.30 167 112434732901969 39.96 173 178936222537081 31.58 179 178936222537081 15.69 181 696161110209049 30.45 191 696161110209049 15.07 193 2854909648103881 30.84 197 6450045516630769 34.70 199 6450045516630769 17.32 211 11641399247947921 15.46 223 11641399247947921 7.65 227 190621428905186449 62.42 229 196640148121928601 32.14 233 712624335095093521 58.06 239 1773855791877850321 71.92 241 2327687064124474441 47.12 251 6384991873059836689 64.15 257 8019204661305419761 40.11 263 10198100582046287689 25.40 269 10198100582046287689 12.65 271 10198100582046287689 6.32 277 69848288320900186969 21.54 281 208936365799044975961 32.14 283 533552663339828203681 40.99 293 936664079266714697089 35.76 307 936664079266714697089 17.73 311 2142202860370269916129 20.23 313 2142202860370269916129 10.10 317 2142202860370269916129 5.04 331 13649154491558298803281 15.94 337 34594858801670127778801 20.14 347 99492945930479213334049 28.81 349 99492945930479213334049 14.39 353 295363187400900310880401 21.32 359 295363187400900310880401 10.63 367 3655334429477057460046489 65.54 373 4235025223080597503519329 37.86

Sieving for Pseudosquares and Pseudocubes in Parallel

Table 2. Values of c3 (n) based on known pseudocubes n 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

qn Lqn ,3 79 7235857 97 8721539 103 8721539 109 91246121 127 91246121 139 98018803 151 1612383137 157 1612383137 163 7991083927 181 7991083927 193 7991083927 199 20365764119 211 2515598768717 223 6440555721601 229 29135874901141 241 29135874901141 271 29135874901141 277 406540676672677 283 406540676672677 307 406540676672677 313 406540676672677 331 75017625272879381 337 75017625272879381 349 75017625272879381 367 996438651365898469 373 2152984914389968651 379 12403284862819956587 397 37605274105479228611 409 37605274105479228611 421 37605274105479228611 433 205830039006337114403 439 1845193818928603436441 457 7854338425385225902393 463 12904554928068268848739 487 13384809548521227517303 499 60125695216741655189317 523 116614853914870278915947 541 116614853914870278915947 547 4139150561509947885227899 571 16248573199879956914339717 577 16248573199879956914339717 601 24191374719361484275890677 607 24191374719361484275890677 613 674441580981249129037406633

c3 (n) 6.42 2.35 0.764 2.6 0.813 0.281 1.49 0.488 0.795 0.254 0.0827 0.0695 2.8 2.34 3.49 1.14 0.365 1.69 0.558 0.181 0.0598 3.61 1.2 0.394 1.71 1.23 2.34 2.33 0.77 0.254 0.459 1.37 1.91 1.04 0.355 0.527 0.336 0.111 1.31 1.69 0.56 0.274 0.0912 0.845

335

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It has been pointed out, both by one of the referees and by Rich Schroeppel [9], that a value for k > 0 such that Lpn ,2 = Lpn+1 ,2 = · · · = Lpn+k ,2 likely is not bounded. This applies to pseudocubes as well. It implies that we, most likely, cannot simultaneously have both (5) and (6), nor both of (7) and (8). This might be avoided if we, say, multiply our upper bounds by n and divide our lower bounds by n in our conjectures. Our data also has implications on the relative eﬃciently of primality testing. In particular, several researchers have pointed out that if conjectures (1),(2) are true, then the running time of the pseudocube prime test, which depends on 2/3 the value of Lqn ,3 , should eventually outperform the pseudosquare prime test, whose running time depends on Lpn ,2 . In particular, one infers from conjectures (1) and (2) that 2/3 n 2/3 Lqn ,3 3 > 1 (9) Lpn ,2 2 for suﬃciently large n (see [14, §9.1]). This inference follows from our reﬁned conjectures as well. We have our ﬁrst speciﬁc value of n to support (9), namely with n = 48, where 2/3 Lqn ,3 ≈ 2.214 · Lpn ,2 . However, given that c2 (n) averages about 45, and c3 (n) averages just over 1.2, we would reasonably expect (9) to largely be true only for n larger than about 75, under the assumption these averages are maintained. To test this, more pseudosquares and, in particular, more pseudocubes are needed.

4

Algorithm Details

We begin with a review of doubly-focused enumeration, explain how we employ parallelism, and how the space-saving wheel datastructure is utilized. We also discuss the details of our implementation, including the hardware platform and software used. 4.1

Doubly-Focused Enumeration

The main idea is that every integer x, with 0 ≤ x ≤ H, can be written in the form x = t p M n − tn M p (10) where gcd(Mp , Mn ) = 1,

0 ≤ tp ≤

H + Mn Mp , Mn

and 0 ≤ tn < Mn .

(11)

(See [2] or [14, Lemma 1].) This is an explicit version of the Chinese Remainder Theorem.

Sieving for Pseudosquares and Pseudocubes in Parallel

337

To ﬁnd pseudosquares, we set Mn and Mp to be products of small odd primes and 8, choose tp to be square modulo Mp , and −tn to be square modulo Mn . To be precise, in our implementation we set Mp = 7 · 11 · 13 · 17 · 19 · 23 · 29 · 31 · 37 · 41 · 43 · 53 · 89 = 2057 04617 33829 17717 and Mn = 8 · 3 · 5 · 47 · 59 · 61 · 67 · 71 · 73 · 79 · 83 · 97 = 4483 25952 77215 26840. Note that both Mp , Mn < 264 , allowing us to work in 64-bit machine arithmetic. To ﬁnd pseudocubes, the same idea applies, only note that if −tn is a cube modulo Mn , so is tn . We used only 2, 9 and primes congruent to 1 (mod 3) for better ﬁlter rates: Mp = 2 · 7 · 13 · 31 · 43 · 73 · 79 · 127 · 139 · 157 · 181 = 701 85635 61110 39402 and Mn = 9 · 19 · 37 · 61 · 67 · 97 · 103 · 109 · 151 · 163 = 693 11050 43291 92503 4.2

Parallelism and Main Loop

Each processor core was assigned an interval of tp values to process by giving it values of H − and H + . For ﬁnding pseudosquares, H + − H − ≈ Mn · 4.76 × 1011 . For ﬁnding pseudocubes, H + − H − ≈ Mn · 4.99 × 1012 . Parallelism was achieved by having diﬀerent processors working on diﬀerent intervals simultaneously. Once all processors had ﬁnished their current intervals, the work was saved to disk (allowing restarts as needed) and new intervals were assigned. To process an interval, each processor core did the following: 1. Using the wheel datastructure, generate all square or cube values of tp with H − ≤ tp Mn ≤ H + , and store these in an array A[]. 2. The wheel datastructure does not generate the tp values in order, so sort A[] in memory using quicksort. Note that H − and H + are chosen close enough together so that this array held no more than 40 million integers, using at most 320 megabytes of RAM per processor core. 3. Using the ﬁrst and last entries in A[], compute a range of valid tn values to process, and then use a wheel datastructure to generate all tn values in that range such that −tn is square modulo Mn for pseudosquares, or tn is a cube modulo Mn for pseudocubes. We use an outer loop over tn values in the order enumerated by the wheel data structure for Mn , and an inner loop over consecutive tp values drawn from A[].

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4. For each tn generated, we normalize sieve tables for the next 4 primes (101, 103, 107, 109 for pseudosquares, and 193, 199, 211, 223 for pseudocubes) to allow for constant-time table lookup to see if an x-value (see below) is a square/cube modulo these primes, indexed by tp value. The number of primes to use for this depends on how many tp values will be processed for each tn – in our case, it was several hundred on average, so this step improves performance. If it were fewer, say 50, then normalizing the sieve tables would require more work than is saved by having constant-time lookup. 5. For each tn generated, using binary search on A[] to ﬁnd all the tp values it can match with, generate an x = tp Mn − tn Mp within our global search range. (For example, in our last run for pseudosquares, we searched for x values between 7.5 × 1024 and 1025 .) Note: at this point we do not actually compute the value of x. 6. Lookup each tp value in the normalized tables mentioned above. If it fails any of the 4 sieve tests, move on to the next tp value. For pseudosquares, a tp values passes these tests with probability roughly (1/2)4 = 1/16, and for pseudocubes, roughly (1/3)4 = 1/81. Note that this step is the running time bottleneck of the algorithm. 7. The next batch of primes q have precomputed sieve tables that are not normalized, but we precompute Mp and Mn modulo each q so the we can compute x mod q without exceeding 64-bit arithmetic. Continue only if our tp value passes all these sieve tests as well. The expected number of primes q used in this step is constant. 8. Finally, compute x using 128-bit hardware arithmetic, and see if it is a perfect square or perfect cube. If it passes this test, append x to the output ﬁle for this processor core. We had two wheel datastructures, one each for Mp and Mn . For details on how this datastructure works, see [10]. We leave the details for how to modify the datastructure to handle cubes in place of squares to the reader. 4.3

Implementation Details

To compute the tables presented in §2, we used Butler University’s cluster supercomputer, BigDawg, which has 24 compute nodes, each of which has four AMD Opteron 8354 quad-core CPUs at 2.2GHz with 512KB cache, for a total of 384 compute cores. As might be expected, we did not have sole access to this machine for over a year, so the code was designed, and ran, using anywhere from 10 to 24 nodes, or from 160 to 384 cores, depending on the needs of other users. This ﬂexibility is one advantage of our parallelization method – by tp intervals. In [14], they parallelized over residue classes, which restricts the CPU count to a ﬁxed number (180 in their case). BigDawg runs a Linux kernel on its head node and compute nodes, and the code was written in C++ using the gnu compiler (version 4.1.2) with MPI. It has both 10GB ethernet and Inﬁniband interconnect, but inter-processor communication was not a bottleneck for our programs.

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We tested our code by ﬁrst ﬁnding known pseudosquares (all but the highest few) and known pseudocubes, in the process verifying previous results.

5

Future Work

We plan to port our code to work with 8 NVidia GPUs recently added to Butler’s supercomputer, giving it roughly 2-3 times the raw computing power. This will require a major restructuring of the code, and the removal of recursion in the wheel datastructure.

References 1. Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. of Math. 160(2), 781– 793 (2004), http://dx.doi.org/10.4007/annals.2004.160.781 2. Bernstein, D.J.: Doubly focused enumeration of locally square polynomial values. In: High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, pp. 69–76. Amer. Math. Soc., Providence (2004) 3. Bernstein, D.J.: Proving primality in essentially quartic random time. Math. Comp. 76(257), 389–403 (2007), http://dx.doi.org/10.1090/S0025-5718-06-01786-8 (electronic) 4. Berrizbeitia, P., M¨ uller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004) 5. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979) 6. Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Math. Comp. 65(213), 361–372, S25–S27 (1996) 7. Pomerance, C., Shparlinski, I.E.: On pseudosquares and pseudopowers. In: Combinatorial Number Theory, pp. 171–184. Walter de Gruyter, Berlin (2009) 8. Schinzel, A.: On pseudosquares. New Trends in Prob. and Stat. 4, 213–220 (1997) 9. Schroeppel, R.: Private communication (February 2010) 10. Sorenson, J.P.: The pseudosquares prime sieve. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 193–207. Springer, Heidelberg (2006) 11. Stephens, A.J., Williams, H.C.: An open architecture number sieve. In: Number theory and cryptography (Sydney, 1989). London Math. Soc. Lecture Note Ser., vol. 154, pp. 38–75. Cambridge Univ. Press, Cambridge (1990) ´ 12. Williams, H.C.: Edouard Lucas and primality testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. John Wiley & Sons Inc, New York (1998), A Wiley-Interscience Publication 13. Wooding, K.: The Sieve Problem in One- and Two-Dimensions. Ph.D. thesis, The University of Calgary, Calgary, AB (April 2010) http://math.ucalgary.ca/~ hwilliam/files/wooding10thesis.pdf 14. Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006) 15. Wooding, K., Williams, H.C.: Improved primality proving with Eisenstein pseudocubes. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 372–384. Springer, Heidelberg (2010)

On the Extremality of an 80-Dimensional Lattice Damien Stehl´e1,2 and Mark Watkins2 2

1 CNRS and Macquarie University Magma Computer Algebra Group, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia [email protected], [email protected]

Abstract. We show that a speciﬁc even unimodular lattice of dimension 80, ﬁrst investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal lattice in this dimension. The known part of its automorphism group is isomorphic to SL2 (F79 ), which is smaller (in cardinality) than the two previous examples. The technique to show extremality involves using the positivity of the Θ-series, along with fast vector enumeration techniques including pruning, while also using the automorphisms of the lattice.

1

Introduction

We show that a speciﬁc 80-dimensional even unimodular lattice is extremal, that is, that it has no (nonzero) vectors of norm less than 8. It follows that the kissing number of this lattice is 1 250 172 000.1 Although two other even unimodular extremal lattices in dimension 80 are known [3], the one we describe has a construction related to coding theory, and has an automorphism group that contains SL2 (F79 ). In Section 2 we recall some facts and results about extremal lattices. In Section 3 we follow the method of Schulze-Pillot [40] to construct our lattice N80 as a 2-neighbour of a lattice derived from a length 80 extended quadratic residue code over F19 . The prime 19 here is not overly signiﬁcant; the construction√produces ﬁve unimodular lattices in correspondence with the class group of Q( −79), and the ideal class that yields N80 (the only extremal one among the ﬁve) has an ideal of norm 19 in it.2 Alternatively, a variation (see [1]) on a method of Gross [18, §11] can be used to construct N80 , and deals more directly with the ideals of this imaginary quadratic ﬁeld. Via either method, it is fairly immediate that N80 has an automorphism group that contains SL2 (F79 ). In Section 4 we note that various choices of bases make the group action nice (doubly transitive as signed permutations on the coordinates), and then make a speciﬁc basis choice that relates directly to the construction in [1]. 1

2

We do not describe herein any features of these minimal vectors. In fact, the 2 555 orbits of these vectors under the known automorphisms were ﬁrst found (without proof of completeness) by the authors of [1], with whom we started this project. We could also have chosen l = 5 (as indicated in [40, Example 3]), but for technical reasons (in lattice generation) wanted l not to be too small.

G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 340–356, 2010. c Springer-Verlag Berlin Heidelberg 2010

On the Extremality of an 80-Dimensional Lattice

341

In Section 5 we ﬁrst brieﬂy outline our method of proof that the lattice N80 is extremal. We need to show that N80 has no nonzero vectors of norm 6 or smaller. We can almost immediately eliminate vectors of norm 2, while a slightly more involved argument is necessary to show there are no vectors of norm 4. We then use the nonnegativity of the coeﬃcients of the Θ-series of the lattice to reduce the problem of showing that there is no vector of norm 6 to the problem of ﬁnding (almost) all the vectors of norm 10. The latter is feasible due to the fact that we need only ﬁnd one representative in each orbit class under the known automorphisms, whereas the more direct method of an exhaustive search for norm 6 vectors would be signiﬁcantly more time-consuming. After ﬁrst cataloguing the norm 10 orbits that have a nontrivial stabiliser, all the other vectors will have a full orbit under the known automorphisms, and so we can reduce the problem by a factor of approximately #SL2 (F79 ) = 492 960. This leaves us with only 15.3 million orbits of norm 10 to ﬁnd. In Section 6 we describe our method to ﬁnd all the norm 10 orbits. One principal idea is to prune the tree corresponding to the Kannan-Fincke-Pohst enumeration algorithm that ﬁnds all short lattice vectors [21,12]. Our tree pruning strategy, which generalizes that of [38, §7] and improves the one from [39], considers a truncated search domain that is much smaller but still ﬁnds a signiﬁcant proportion of the desired vectors. Note that the pruning strategy we describe and its analysis have been independently discovered by Gama, Nguyen, and Regev [15, §4]. In our case, we need only ﬁnd one vector in each orbit class, so the fact we miss some vectors when searching is unimportant. Another idea to speed the search is to periodically apply a random perturbation to the basis and re-apply lattice reduction (namely LLL with deep insertions [38]), before again searching with tree pruning. As our lattices are of quite high dimension, the new basis is very likely to be diﬀerent than the previous ones. This can help in two ways: ﬁrstly, searching with a given lattice basis for short vectors, even with pruning available, tends to become less cost-eﬀective over time, in terms of the number of vectors found per second; and secondly, and rather surprisingly to us, a “good basis” for searching can sometimes have many orbit classes which will not show up until quite deep in the search. We still do not understand this latter phenomenon, but it is easily overcome via the random perturbations. Section 7 gives our results and veriﬁcation methods, plus related questions. Computations. All timings are given for 2.3Ghz Opteron 8356 processors. If otherwise unspeciﬁed, only one processor is used.

2

Extremal Lattices

The extremality of a lattice is typically deﬁned using Θ-series, as for instance in [7, §7.4].3 In particular, an extremal unimodular even lattice in dimension d with 8|d has a minimum nonzero vector norm of 2(1+d/24), as this is twice the 3

The precise notion of “extremal” seems to vary over time; for instance [6] is more demanding, asking that the minimum be at least 1 + d/8.

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dimension of the associated space of modular forms. For odd lattices, shadow theory is typically used to obtain satisfactory bounds [8]. A relatively recent survey on extremality appears in [14]. In particular, there were already two extremal even unimodular lattices known in dimension 80, both due to Bachoc and Nebe [3] via a coding theory construction. The ﬁrst lattice L80 has an automorphism group 2.A7 ⊗√−7 2.M22 .2 of size 212 34 52 72 11 = 4 470 681 600, and this group is known to be a maximal ﬁnite subgroup of GL80 (Z) (see [3, Theorem 3.2]). The second extremal lattice M80 has known automorphisms [3, Lemma 4.11] of order 212 34 52 = 8 294 400. For comparison, the number of known automorphisms of our lattice is 492 960. Our lattice N80 is isometric neither to L80 nor M80 . The argument for L80 is immediate, as its automorphism group is known to be maximal but 79 does not divide the order. For M80 we can compute the minimal vectors in a few days, and perhaps argue via some property of them versus those for N80 . We can also argue via Aschbacher’s theorem on maximal subgroups of ﬁnite classical groups, and in an appendix, we sketch a proof along these lines, showing that Aut(N80 ) is a maximal ﬁnite subgroup of GL80 (Z) up to a possible index of 4. The idea of extremality can also be extended to include other lattices which are isomorphic to their dual(s). In this case, the full space of modular forms is typically replaced by the subspace that is ﬁxed under the Atkin-Lehner involutions [36]. This then relates the question to a simultaneous maximisation of the minimum of a lattice and that of its shadow; see [13] and [32] for instance. Finally, we note that [28] shows that there are only ﬁnitely many extremal lattices, though the most easily computed bound on maximal dimension still seems to be quite high.4 In the other direction, King [22] classiﬁes all (even) unimodular lattices in dimension 32 with no roots, and ﬁnds there to be at least 107 such; as the lack of roots implies that the lattices have no vectors of norm 2, it follows that each is extremal. Similarly, Peters [33] shows there are at least 1051 extremal lattices in dimension 40.

3

Construction of the Lattice N80

We follow the paper [40] of Schulze-Pillot on quadratic residue codes and cyclotomic lattices, which builds on works from Thompson, Feit [9], and Quebbemann [35, §3] about unimodular lattices with an automorphism of prime order. 4

The proof therein is similar in ﬂavour to the idea we exploit, that is, for suﬃciently large dimension, the ﬁrst form in a triangular basis will have coeﬃcients that are negative, and thus positivity precludes the existence of an extremal lattice. See the recent [42, p. 36] for a brief sketch. Our computations give that the q n+2 term in the expansion is negative for n ≥ 6 775, 6 789, 6 803 for the respective 0, 8, 16 mod 24 classes, which gives an upper bound of 163 264 = (6802· 24) + 16 for the dimension of an even unimodular extremal lattice. Finally, Rains [37] has followed upon the work of Krasikov and Litsyn [27] to obtain that the minimal norm of a unimodular lattice is (asymptotically with dimension d → ∞) smaller than the Siegel bound ∼ d/12 by at least a constant factor (see N = 1 in the Remark after Theorem 4.2 in [37]).

On the Extremality of an 80-Dimensional Lattice

343

The construction gives a unimodular lattice as a sublattice of index p in a (rescaled) direct sum of two lattices of dimensions 2 and (p − 1). In this, the 2-dimensional lattice T2 can be taken as any integral lattice of determinant p. The lattice Up−1 of dimension (p − 1) comes about from an (unpublished) construction of Thompson (see [9, §9]). We let E = Q(ζp ) be cyclotomic, and take ¯ = (d) with d ∈ E + totally positive. This ideal an ideal A ⊆ OE such that AA induces a (positive deﬁnite) lattice of dimension (p − 1) via a basis for the ring of integers Z[ζp ], with the quadratic form given by Q1 (u) = trE ud−1 ). Via a Q (u¯ computation (with the diﬀerent as in [9, Theorem 9.3], or with a Vandermonde determinant) one can show that the lattice Up−1 has determinant pp−2 . To obtain a unimodular lattice of dimension (p + 1), we start with the direct sum T2 ⊕ Up−1 , and take the sublattice of this consisting of all vectors whose norm is a multiple of p. Upon dividing the whole lattice by p, the result will be integral and unimodular, the latter since (p·pp−2 )·p2 /pp+1 = 1. We need to show that this actually yields a sublattice, that is, the resulting subset of the original lattice satisﬁes the group law, and this is most easily done via homomorphic projection maps. We take the lattice N(T2 , Up−1 ) = {(m, u) ∈ T2 ⊕ Up−1 | π(m) = ρ(u)} under the quadratic form Q (m, u) = Q0 (m) + Q1 (u) /p, with the projection maps being π : T2 → R/radQ0 (R) where R = T2 /pT2 , and ρ˜ : A → A/(1 − ζp )A (here ρ˜ is on A, with ρ on Up−1 ). Since (1 − ζp ) has norm p, both images will be vector spaces over Fp of dimension 1, and we can identify them (arbitrarily) by taking m0 ∈ T2 and u0 ∈ A with Q0 (m0 ) ≡ 1 (mod p) and u0 u¯0 d−1 ≡ 1 (mod (1 − ζp )OE ). The lattice N(T2 , Up−1 ) will be even if and only if T2 is even. 3.1

An Odd Lattice

Rather than derive our desired even unimodular lattice directly, we again follow Schulze-Pillot, who ﬁrst constructs an odd lattice for which the automorphism group can be determined via a relation to coding theory, and then passes to an even lattice via Kneser’s neighbouring construction. √ We let K be the imaginary quadratic ﬁeld Q( −79), and d = l = 19 an auxiliary prime that splits. Writing (l)OK = l¯l, the location of l in the class group of K will have a determining factor on the lattice we derive in the end, and so the choice of l is not completely arbitrary. Welet a be the ideal of K generated by l and the twisted Gauss sum 12 1 − 33 a χp (a)ζpa where χp is the quadratic character modulo p. Using the notation of Schulze-Pillot, we have p = −j 2 + 8ml with p = 79, j = 15, m = 2, and l = 19, so that yj ≡ 1 (mod l) together with y ≡ 1 (mod 4) yields y = 33.5 Noting that a¯a = (l) and ¯ = (19) in OE . Letting T2 be taking E = Q(ζ79 ), we write A = aOE so that AA the 2-dimensional lattice (in a basis {w1 , w 2 }) of determinant 79 given by the 5

The import of this numerology only becomes clear when proofs are included, as this choice of y for the scaling factor of the Gauss sum allows one to show that the cyclotomic and coding theory constructions agree.

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19 15 l j , we ﬁx the gluing via π(w 1 ) = ρ( lζp ), = Gram matrix Q0 = 15 16 j 8m where here · gives the map from A to Up−1 . We let No = N(T2 , Up−1 ) with these choices, noting that No is odd.

3.2

Relation to Coding Theory

We can obtain correspondence with coding theory by taking p coordinates the as ei = w1 ⊕ lζpi for 0 ≤ i ≤ p − 1 and an additional one e∞ = jw1 − lw2 , from which a computation shows that these ei form a scaled root system of type 80A1 in No , that is, each ei has the same norm, and they are all mutually orthogonal. Indeed, for all 0 ≤ i ≤ p − 1 we have ei = Q0 (w 1 ) + (p − 1) · (l2 /l) /p = l since Q0 (w 1 ) = l, while e∞ = Q0 (jw 1 − lw2 )/p = l(8ml − j 2 )/p = l. For the inner products, we have

ei , ek = ei + ek − ei − ek i

1 k ¯i ¯k Q0 (2w1 ) + (l2 /l) · trE − 2l = Q (ζ + ζ )(ζ + ζ ) p

1 i−k 4l + l · trE + ζ i+k − 2l = Q 2+ζ p 1 = 4l + l · [2(p − 1) − 1 − 1] − 2l = 0 p when i = k and i, k = ∞, while for i = ∞ we have

ei , e∞ = ei + e∞ − 2l 1 = Q0 (j + 1)w1 − lw2 + (p − 1) · (l2 /l) − 2l p 1 = l(1 + 8ml − j 2 ) + l(p − 1) − 2l = 0. p Using this root system, it follows that the extended quadratic residue code C ⊆ F80 l (or indeed, any self-dual code) gives an integral unimodular lattice via

1 ai ) ∈ C ai ei (¯ (1) NC = l i where the sum is over all 80 coordinates, and a ¯i is reduction mod l of ai . The proof that NC is the same lattice as our lattice No is given in [40, Proposition 1], using the generator matrix and idempotent of the code.6 The appearance of the value y = 33 with the Gauss sum is of relevance therein. 6

We have taken a sublattice of index lp+1 via the scaled root system, and then taken a superlattice of the same index via the construction from coding theory, and so just have to check that these operations are compatible.

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One nicety of this re-visioning is that the code automorphism (of order 4) given by a∞ → a0 , a0 → −a∞ , ai → −χp (i)aj , where ij ≡ −1 (mod p), can be seen to lift to the lattice. Combined with the order p automorphism induced via ζp , which ﬁxes a∞ and cycles a0 → a1 → · · · → ap−1 → a0 , this gives SL2 (Fp ) as a subgroup of the automorphism group Aut(No ) of the lattice. In an appendix, we use the classiﬁcation of ﬁnite simple groups to show that this realisation of SL2 (F79 ) is within a factor of 4 of being a maximal ﬁnite subgroup of GL80 (Z), so that [Aut(No ) : SL2 (F79 )] ≤ 4. 3.3

The Even 2-Neighbours

The above lattice No is odd, while we wish to get an even unimodular lattice. The method of passing to this is given by the neighbouring method of Kneser [26]. Again following Schulze-Pillot, we want to ﬁnd v ∈ No with Q(v) ∈ 4Z, and then take the lattice spanned by v/2 and the sublattice of No whose inner product with v is even. Via linear algebra over F2 , we ﬁnd that there is a 2-dimensional space of such v satisfying the conditions (Schulze-Pillot notes this in general via genus theory). Obviously v = 0 does not help us, while we also need Q(v) ∈ 8Z if the resulting neighbouring lattice is to be even, and this eliminates another of the initial 4 possibilities. This leaves but 2 choices for v, one of which gives a lattice with many vectors of norm 4 (note that v itself must have norm at least 32 if the new lattice is to have minimum 8) and the other of which is our desired lattice N80 . As in [40, Proposition 2], we could construct N80 directly using a diﬀerent choice with T2 in the cyclotomic construction, though the relation to coding theorythen becomes less clear. For instance, [40, Example 3] takes l = 5 and 8 1 to get the same N80 . Finally, the last Remark of [40] notes the Q0 = 1 10 automorphisms of No given by SL2 (Fp ) all transfer to N80 . As noted above, we show in an appendix that [Aut(N80 ) : SL2 (F79 )] ≤ 4 so that in particular N80 and M80 are not isometric, but our proof of extremality does not use this.

4

Nice Bases for N80

We next link N80 to the construction given in [1] that modiﬁes the method of Gross. The authors of [1] construct the lattice from a representation that is irreducible away from 2. In particular, in the basis they obtain, all the coordinates are of the same parity. Furthermore, the automorphisms are given by a doubly transitive signed permutation action on the coordinates. From our construction, we have a lattice N80 with automorphisms generated by two matrices O79 and O4 . We wish to transform this so that the automorphisms are generated by signed permutations σ79 and σ4 (as in the end of Section 3.2), thus giving a doubly transitive coordinate action. One way to achieve this is just to solve the 802 -dimensional linear algebra problem given by

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equating the automorphisms, that is, solve O79 X = Xσ79 and O4 X = Xσ4 for the unknown matrix X (we try solving this with both σ4 and σ43 ). It turns out that the resulting solution space is 2-dimensional, and if we write X1 and X2 for generators of it, then the determinant of the matrix X1 t + X2 u is given by 240 f (t, u)40 where f is a binary quadratic form of discriminant −79 corresponding to the ideal of above. To obtain the representation of [1] we choose the pair (t, u) so that f (t, u) = 8, so that the transform maps vectors of norm 10 in N80 to vectors of norm 16 · 10 in the resulting sublattice of Z80 . The resulting basis has the property that every vector has coordinates all of the same parity. We denote this transform matrix from N80 to Z80 by T16 , and the resulting lattice basis by B80 . 4.1

Identifying Orbits

As noted above, the action of σ79 and σ4 is doubly transitive, and we can exploit this to expedite the ﬁnding of a canonical representative for a given orbit. We ﬁrst ﬁnd the largest coordinate in absolute value, and move it to the front, and then cycle the latter 79 coordinates until the second largest is in the second position. This movement uses 80 · 79 elements of the group, and after modding out by the centre {±1}, we only have 39 possibilities left to check for their 78 latter coordinates (we use a lexicographic ordering). Of course, we could have many ties amongst the two largest coordinates (this is basis-dependent, and we can map to another choice of (t, u) if desired), but this method will still be much faster than looping over all 492 960 possibilities.

5

Method of Proof

We now describe how we shall show that N80 is indeed extremal. Since the lattice N80 is even and unimodular, its Θ-series Θ80 lies in the vector space of modular forms of level 1 and weight 40 (see [30]). This space has dimension 4, and a triangular integral basis is: f0 = 1 + 1 250 172 000 q 4 + 7 541 401 190 400 q 5 + O(q 6 ), f1 = q + 19 291 168 q 4 + 37 956 369 150 q 5 + O(q 6 ), f2 = q 2 + 156 024 q 4 + 57 085 952 q 5 + O(q 6 ), f3 = q 3 + 168 q 4 − 12 636 q 5 + O(q 6 ). We thus know that Θ80 = f0 + a1 f1 + a2 f2 + a3 f3 for some integers ai . We shall derive that a1 = a2 = 0 by showing that there are no vectors of norm 2 or 4 in the lattice. We will then have Θ80 = 1 + a3 q 3 + (· · · )q 4 + (7 541 401 190 400 − 12 636 a3) q 5 + O(q 6 ). By positivity we have a3 ≥ 0, and so by ﬁnding 7 541 401 190 400 vectors of norm 10 in the lattice, we deduce that a3 = 0 so that N80 is extremal as claimed.

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The reader might wonder why we do not simply search for norm 6 vectors, but instead aim to ﬁnd all those of norm 10, as the latter (at ﬁrst glance) seems much harder. However, the search in norm 6 has to be exhaustive, while with norm 10 it need not be: we ﬁnd one vector in each orbit, and apply automorphisms to get the whole set. We estimate an exhaustive search for norm 6 vectors would take more than 1 000 times as much work as our method using norm 10 vectors. 5.1

The Lattice N80 Has No Vectors of Norm 2 or 4

As we noted above in Section 4, we can change the basis by a transform T16 so that each vector has its norm multiplied by 16, with the resulting basis having the property that all the coordinates of any vector will have the same parity. In particular, a vector of norm 2 or 4 will have the square-sum of its coordinates as 32 or 64, with necessarily all coordinates being even. Also, the inner product of any two vectors in this basis will need to be a multiple of 16, a fact we exploit below. Finally, the lattice automorphisms in this new basis are given by signed permutations, with the action doubly transitive. No vectors of norm 2 (roots). One proof (from Elkies) ﬁrst notes that the only root systems with compatible automorphisms are A80 1 and D80 . With the former, any automorphism of order 79 would necessarily ﬁx at least one of the 160 roots, but the 2-dimensional sublattice of N80 ﬁxed by a 79-cycle has no roots. The latter is similarly impossible; a 39-cycle must ﬁx a root since gcd(39, 12 640) = 1, but the 4-dimensional sublattice therein lacks roots. Another way (similar to a comment in [40, Example 3]) would be to use l = 5 and note that we must have i a2i = 2l = 10 in (1), while the minimal distance7 of the extended quadratic residue code of length 80 over F5 is > 10, though care needs to be made here when working with both N80 and the odd lattice L. A direct computation also easily shows that N80 has no roots. After applying suitable reduction, the veriﬁcation typically takes less than 30 minutes. We did not try a similar computation with norm 4, as we estimate that it would likely take a few months. e No vectors of norm 2 or 4. We let B80 be the sublattice of B80 given by e e vectors with even coordinates in the T16 basis, and map B80 → B80 /2 → F80 2 via the additive coordinate map generated by ±2 → ±1 → 1. The image in F80 2 is a binary code C2 , and this inherits the automorphisms from the lattice. e We have 16| v, w for any v, w ∈ B80 , which implies that C2 is doubly-even, that is, each codeword has weight divisible by 4. Similarly, we see that C2 ⊆ C2⊥ , as the inner product between any two codewords is 0 (in F2 ). We then show e equality here by ﬁnding enough vectors in B80 to show that dim(C2 ) ≥ 40. As C2 is self-dual and has automorphism group PSL2 (F79 ), it follows from either [25, Theorem 6.2] or [24, Satz 3.4] that C2 is equivalent to the extended 7

It seems that showing the minimal distance exceeds 20 would take about 58 days, though the computation should parallelise.

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binary quadratic residue code,8 and thus has minimal weight of 16 with 97 565 minimal codewords which lie in 3 orbits under the automorphisms.9 We now check that the preimages of codewords of weight 0 and 16 in C2 do not yield vectors of norm 2 or 4 in N80 .10 This is done using the explicit form −1 −1 of T16 . For weight 0, we need to check that T16 w is non-integral for w = 8, 0, . . . , 0, 4, ±4, 0, . . . , 0, 4, ±4, (. . .) where in this third expression exactly two of the latter 78 coordinates have size 4. By the doubly transitive nature of the automorphism action, this suﬃces. There are thus 3 + 23 78 = 24 027 possibilities to check here. 2 For weight 16, we have 3 orbits of codewords. For each orbit we take a representative, and lift its nonzero coordinates in 216 ways to every choice of sign −1 for ±2. We then apply T16 to each, and note that none are integral. This completes the proof that there are no vectors of norm 2 or 4 in the lattice N80 . e Presumably we could similarly show that B80 has no vectors of norm 96, but extending our observations to odd-coordinate vectors in B80 looks more diﬃcult. 5.2

Vectors with a Nontrivial Stabiliser

We now describe how to use the known automorphisms to reduce our vectorﬁnding quota from 7.5 trillion vectors down to about 15.3 million. We make a separate computation of the norm 10 vectors that have nontrivial stabiliser. If a vector v has a nontrivial stabiliser under the above action of G = SL2 (F79 ), there is some nontrivial element g ∈ G such that the kernel of g − id contains v. So we loop over nontrivial elements (or conjugacy classes) of G, compute this kernel (which is a sublattice), and then search for short vectors in it. The elements of order 3 give a kernel sublattice of dimension 28, for which it takes a few seconds to ﬁnd the vectors of norm ≤ 10. These yield 465 orbit classes under the action. The elements of order 5, 39, and 79 give lattices of dimensions 16, 4, and 2, and yield 15, 2, and 1 orbits respectively. Upon computing the stabilisers, we obtain – – – – 8 9

10

1 orbit with stabiliser size 79 · 39 = 3081 (order 79), 2 orbits with stabiliser size 39 (order 39), 15 orbits with stabiliser size 5 (order 5), 465 orbits with stabiliser size 3 (order 3).

We thank Elkies for recalling this fact, and J. Cannon for the Klemm reference. Here is an alternative method. Assume ﬁrst that there is a codeword w of weight 4 or 8. Take a 79-cycle σ and note that since (8−1)2 < 79 there is some iterate of σ such that w and σw intersect only in the ﬁxed coordinate. This implies that w, σw = 1, which contradicts that C2 is self-dual. Since there are no codewords of weight 4 or 8, we can then apply Gleason’s theorem [16] and get that the weight enumerator is of the form q 0 + (a + 15 200) q 12 + (127 965 + 2a) q 16 + (11 347 488 − 101a) q 20 + . . . for some a ∈ Z, and in an echo of our proof of lattice extermality, show code extremality (no codewords of weight 12) via ﬁnding 12 882 688 codewords of weight 20; for this, we ﬁnd short vectors in the lattice, map to the code, and apply automorphisms. We do not explicitly need the fact that the code is extremal for this step, but only that we have all codewords of length 16 or less.

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None of the other 78 nontrivial conjugacy classes of SL2 (F79 ) yields an orbit with vectors of norm 10. We can also note that there no vectors of norm 6 with a nontrivial stabiliser (though this is not strictly necessary for our proof). An accounting then tells us that there are presumably 7 541 323 277 280 vectors of norm 10 yet unfound, and dividing by #SL2 (F79 ) = 492 960 predicts 15 298 043 orbits with trivial stabiliser. Via a standard coupon-collecting analysis [11, p. 213] we expect that about 250 million suitably random vectors of norm 10 should suﬃce to hit each orbit at least once. In fact, for the purposes of proving the lattice extremal, we need only ﬁnd (15 298 043 − 12 635) orbits (see the q 5 coeﬃcient of f3 , and use the fact that 492 960|a3 as we ﬁnd no vectors of norm 6 with nontrivial stabiliser), and due to the lengthy ﬁnal part of coupon-collecting,11 this reduces the expected running time by about 55%. However, for completeness, we still chose to ﬁnd all orbits.

6

General Search for Vectors of Norm 10

The general method to enumerate short vectors in a lattice is due to Kannan [21] and Fincke and Pohst [12]. This corresponds to the computation of the leaves of a huge tree. As noted by Schnorr and Euchner [38], this tree can be pruned to some extent. This can be thought of as searching ﬁrst in the areas of the search region which are more likely to contain short vectors, or, equivalently, removing the tree nodes that are less likely to produce useful leaves. The initial pruning strategy was later improved in [39]. We describe below a further improvement. 6.1

The Full KFP Tree Search

The basic method iteratively looks at the projections to the span of the ﬁrst i coordinates for decreasing i. We have a basis given by {bi } and wish to solve the inequality i xi bi 2 ≤ 10. Borrowing the common notation for lattice reduction, we take the Gram-Schmidt orthogonalisation, and translate the xi ’s by the μj,i ’s: bi = bi −

μi,j bj so that μi,j =

j

d

bi , bj for i > j, and y = x + μj,i xj . i i bj 2 j=i+1

Here d is the dimension. By substituing yi for xi , we get by positivity leads to the series of inequalities:

i

yi2 bi 2 ≤ 10, which

yd2 bd 2 ≤ 10, 2 bd−1 2 ≤ 10 − yd2 bd 2 , yd−1 ... y12 b1 2

≤ 10 −

d

yi2 bi 2 .

i=2 11

The comparison is between

N

N n=1 n

and

N

N n=12636 n

for N = 15 298 043.

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Note that for all i, the variable xi is an integer, while yi is a shift of xi by a ﬁxed amount (once xi+1 , . . . , xd have been chosen). The KFP method proceeds by look ing at all yd ’s satisfying the ﬁrst inequality, then all pairs yd−1 , yd satisfying the second, etc. In particular, the vectors with yi ≈ 0 for all i up to a given point will be found most easily (and these often correspond to small xi ’s). Also, to ﬁnd more short vectors earlier in the search procedure, it is useful to run over the diﬀerent possible the centre of the interval implied by the inequality yi2 b∗i 2 ≤ xi ’s2 from ∗ 2 10 − j>i yj bj : the variable xi will run across the integers by decreasing proximity to − j>i μj,i xj . This “zig-zag” strategy, introduced by Schnorr and Euchner [38], allows one to split the search of the tree in diﬀerent stages: in the ﬁrst stage, we have xj = 0 for all j > 1; then in the second stage we have xj = 0 for all j > 2 but x2 = 0; etc. We call stage i the period of time during which xj = 0 for all j > i but xi = 0. Stage i means that we have already reached level i in the KFP tree but not yet been in level i + 1 (level 1 corresponding to the leaves). The arithmetic operations corresponding to Gram-Schmidt orthogonalisation computations can be quite slow. The Magma [5] implementation of the KFP tree search replaces them by double precision ﬂoating-point arithmetic operations, in a fully reliable way (using [34]). 6.2

Tree Pruning

Our pruning strategy consists in restricting the above inequalities by a “pruning factor” that depends on the level. So the above inequalities become d

yi2 bi 2 ≤ 10 · Pj , ∀j

i=j

where Pj is the jth pruning factor. A version of this with a speciﬁc choice of Pj appears in [38, §7], and the general description as well as its analysis below have been independently obtained in [15, §4]. In the latter, the authors also introduce the concept of “extreme pruning”, which resembles but diﬀers from our bases switching strategy (see subsection below). The “best” choice for the pruning factors appears to be something like Pj = (d − j + 1)/d. We happened to choose Pj = 1 − (j − 1)/100 in practise. The idea here can be phrased as follows: we have a given quantity of “norm” (here 10) to spend on a vector; if we spend a lot on the coordinates xj to xd , there will then be a lesser chance that we can form an integral vector via some possible choice of the other coordinates, due to positivity and the fact that most coordinates will have at least some nonzero contribution. Eﬃcacy of pruning. To give an idea of the eﬃcacy of pruning, we can use the notion, from [19], of expected enumeration cost for a given lattice basis {bi } and for vectors of norm A (a function EnumerationCost is available in Magma [5]): d d π d−j+1 k=j A/ bk 2 . (2) Γ 1 + (d − j + 1)/2 j=1

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A typical enumeration cost for our bases with N80 was around 1023 . This is the expected number of nodes of the KFP tree. For comparison, the implementation in Magma [5] has a traversal rate of about 7.5 million nodes per second. By comparing this enumeration cost estimate to the expected 7.5·1012 vectors of norm 10, we ﬁnd that more than 1010 nodes are expected to be searched for each vector found. In the case of the pruned enumeration, the jth summand in (2) should be multiplied by the volume of the truncated hypersphere {(zj , . . . , zd ) : ∀i ≥ j, k≥i zk2 ≤ Pi }. By estimating these volumes with a Monte-Carlo rejection method (uniformly sampling points in the full hypersphere and counting how many belong to the truncation), we expect our pruning to gain a factor of around 104 here, at the cost of missing about 60% of the short vectors. These speedup and miss ratios are not constant across all levels of the search: they seem to be closer to 100 and 25% respectively for the levels of our interest (due to the early abort and perturbation strategy described below). 6.3

Switching Bases

The early stages of the tree search can have a signiﬁcantly better chance of providing short vectors, due primarily to the relative paucity of “uninteresting” branches that tend to become more numerous at higher levels. In practice, we would ﬁnd 105 vectors in about 30 minutes, for a ratio of about 150 000 nodes searched for each vector found, more than an order of magnitude lower than the above estimate, even with the pruning included. Every 15-30 minutes we would switch the basis by applying a random permutation to the coordinates of the current basis, and then multiplying by a random upper triangular matrix with ones on the diagonal and oﬀ-diagonal entries in {−1, 0, +1}. We then re-apply LLL (with a δ-value nearly 1) to the perturbed basis, and then LLL with deep insertions [38]. Overall, this takes only a few seconds. This basis switching also makes parallelisation essentially trivial. A second reason for periodically changing the basis is that (a phenomenon we found experimentally) there are some bases which “hide” many of the orbits, in the sense that every vector in such an orbit would not be found until we reach one of the latter stages. We currently have no explanation of this.

7

Conclusion and Related Work

We implemented the above in a combination of Magma [5] and C. As we typically found 105 vectors of norm 10 in about 30 minutes, the estimated time was around 52 days. Using 14 processors in parallel, it took us about 4 days in April 2009. 7.1

Software to Check Our Data

A veriﬁcation of our proof can be done in much less time than the computation itself. We provide software12 that takes less than 10 hours to verify that N80 12

The code is checkit80.c (to be run with arguments “10 ﬁlename ”) and the data is LAT80.n10.sc16.bz2 in the directory http://magma.maths.usyd.edu.au/~watkins

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is indeed extremal. The input consists of 15 298 526 entries that correspond to coordinate vectors in the T16 basis of Section 4. The following checks are run: – Each entry lexicographically follows its predecessor, −1 – Each entry has norm 160 and is integral when multiplied by T16 , – Each entry is lexicographically the ﬁrst in its orbit. The ﬁrst condition ensures that all entries are distinct, while the last ensures that each corresponds to a distinct orbit, with the middle condition implying that the vectors have norm 10 and are in N80 . We can also list the 483 orbits with nontrivial stabiliser, whose provenance can be checked separately. 7.2

Three Lattices of Dimension 72

The work in progress [1] investigates three lattices of dimension 72. Two of these are 2-neighbours of a lattice constructed via the extended quadratic residue code over F3 , and the other involves a code over Z/4Z. None of these turned out to be extremal (minimal norm of 8), and indeed, we know of no extremal lattice of this dimension. In fact, a recent preprint of Griess [17] claims to be the ﬁrst to prove a minimal norm as large as 6 for an even unimodular lattice of dimension 72. 7.3

Other Candidate Lattices for Extremality in Dimension 80

In [3], the authors note three other candidates for extremality amongst even unimodular lattices in dimension 80. One candidate comes from a cyclo-quaternionic construction given in [31, Remark 5.2], and its automorphism group contains SL2 (F41 ) ⊗ S˜3 , which is of comparable size to our SL2 (F79 ). We do not see how to facilitate the calculation of canonical orbit representatives as readily as in our case, but the fact that canonicalising took only about 5% of our running time indicates that our methods could work in this case, with suﬃcient eﬀort. The other two candidates come from a cyclotomic construction explored in [4], and have an automorphism group containing the general aﬃne linear group ∗ F+ 41 F41 . Our initial opinion is that the automorphism group (even if augmented by an order 4 element) is too small for our method to work well here. Acknowledgments. We thank the authors of [1], with whom we started this research, and S. R. Donnelly who shared some of his ideas with us. We also thank the anonymous reviewers for their recommendation to add a proof that the automorphism group of N80 diﬀers from those of L80 and M80 . The present work is part of the Australian Research Council Discovery Project DP0880724 “Integral lattices and their theta series”.

References 1. Abel, Z., Elkies, N.D., Kominers, S.D.: On 72-dimensional lattices (in preparation) 2. Aschbacher, M.: On the maximal subgroups of the ﬁnite classical groups. Invent. Math. 76(3), 469–514 (1984), http://dx.doi.org/10.1007/BF01388470

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35. Quebbemann, H.-G.: Zur Klassiﬁkation unimodularer Gitter mit Isometrie von Primzahlordnung (German) [On the classiﬁcation of unimodular lattices with an isometry of prime order]. J. Reine Angew. Math. 326, 158–170 (1981), http://resolver.sub.uni-goettingen.de/purl?GDZPPN002198681; Quebbemann, H.-G.: Unimodular lattices with isometries of large prime order. II. Math. Nachr. 156, 219–224 (1992), http://dx.doi.org/10.1002/mana.19921560114 36. Quebbemann, H.-G.: Atkin-Lehner eigenforms and strongly modular lattices. Enseign. Math. 43(1-2), 55–65 (1997), http://retro.seals.ch/digbib/view?rid=ensmat-001:1997:43::263 37. Rains, E.M.: New asymptotic bounds for self-dual codes and lattices. IEEE Trans. Inform. Theory 49(5), 1261–1274 (2003), http://dx.doi.org/10.1109/TIT.2003.810623 38. Schnorr, C.P., Euchner, M.: Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. Math. Program. 66, 181–191 (1994), http://dx.doi.org/10.1007/BF01581144 39. Schnorr, C.P., H¨ orner, H.H.: Attacking the Chor-Rivest cryptosystem by improved lattice reduction. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 1–12. Springer, Heidelberg (1995), http://dx.doi.org/10.1007/3-540-49264-X_1 40. Schulze-Pillot, R.: Quadratic residue codes and cyclotomic lattices. Arch. Math. (Basel) 60(1), 40–45 (1993), http://dx.doi.org/10.1007/BF01194237 41. Weisfeiler, B.: On the size of structure of ﬁnite linear groups. Notes from 1984, Parts 1-17, A1-A10, totalling 91 typewritten and 63 handwritten pages, http://weisfeiler.com/boris/papers/papers.html 42. Zagier, D.: Elliptic modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of modular forms. Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, Universitext, June 2004, x+266 pp. Springer, Berlin (2008)

A

Appendix: Proof That M80 and N80 Are Not Isometric

We wish to show that M80 is not isometric to our lattice N80 . Bachoc and Nebe list a subgroup of Aut(M80 ) of order 212 34 52 , while we have S ∼ = SL2 (F79 ) as a subgroup of Aut(N80 ). We wish to show that there is no ﬁnite matrix group in GL80 (Z) that is a supergroup of both of these (possibly after conjugation). We let G be such a putative supergroup, and note that [G : S] ≥ 27 33 5. From a classical theorem of Minkowski [29] on the modular reduction of matrix groups, we have injective maps ιp : G → GL80 (Fp ) for all odd primes p. By taking a gcd over all odd p this gives a bound of #G 2198 358 524 714 118 136 175 194 233 292 312 372 412 ·43·47·53·59·61·67·71·73·79, though here we really a divisibility result at a speciﬁc prime.13 only need such We write H = ι7 G ∩ SL80 (Z) , and since every matrix in S ∼ = SL2 (F79 ) has determinant 1 we have ι7 (S) ⊆ H. As every matrix in G has determinant ±1, 13

We note in passing that the best upper bound on the size of a ﬁnite matrix group is due to Feit [10], relying on unpublished notes of Weisfeiler [41].

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we get [ι7 (G) : H] ≤ 2, and since [G : S] > 4 and ι7 is injective, this implies that [H : ι7 (S)] > 2. The use of a theorem of Aschbacher (see below) now implies that 7780 #H, which contradicts the above bound. Thus G cannot exist, and so M80 and N80 are not isometric. Indeed, this argument almost shows that S is maximal ﬁnite in GL80 (Z), though a low-index extension could still exist. We now use Aschbacher’s theorem [2] on maximal subgroups of ﬁnite classical groups (see also [23]). Let l be an odd prime (to be speciﬁed below) and suppose that ιl (S) ⊂ H ⊆ SL80 (Fl ). We note that S splits into a pair of conjugate √ absolutely irreducible unitary 40-dimensional representations deﬁned over Q( −79). We know that H lies in some maximal (proper) subgroup of SL80 (Fl ), and the theorem of Aschbacher lists the possibilities. For any inert prime l that does not divide #S, we can eliminate class 1 of Aschbacher since ιl (S) acts irreducibly (we could consider split primes also, but choosing an inert prime simpliﬁes the argument slightly). Classes 2 and 4-7 are not possible simply because 79 must divide #H. This leaves subgroups of class 3 (splitting as above) or class 8 (inclusions of classical groups), or class 9 (other simple groups, handled below). The inclusions of classical groups give us G80 (Fl ) for G = Sp, SO± and SU40 (Fl ), while the splitting of class 3 yields SL40 (Fl2 ).2. where the notation indicates that we have a 2-extension – in this case, we continue the analysis after replacing H by H ∩ SL40 (Fl2 ), where this subgroup has index at most 2 in H. We iteratively apply Aschbacher’s theorem to each classical group obtained; either H is isomorphic to this classical group, or is contained in a maximal subgroup of it. We again use 79|#H, and ﬁnd that the only possible maximal subgroup of Sp80 (Fl ) that could contain H is SU40 (Fl ).2, and similarly with the others. Any maximal subgroup chain of classical groups must end here, since H contains ιl (S) and S → SU40 (Fl ) is absolutely irreducible. So we end in one of the following cases: H is isomorphic to one of SU40 (Fl ). or SL40 (Fl2 ). with = 1, 2, or G80 (Fl ) with G = Sp, SO± , SL; or [H : ιl (S)] = 2, in correspondence to a 2-extension as above; or (sometimes called “class 9” for Aschbacher) we have PSL2 (F79 ) ⊂ K ⊂ P, where K is simple and P is the associated simple group of one of the above classical groups. There is sundry general knowledge for this latter situation, but for us a caseby-case analysis (with l = 7 for concreteness) using the known orders of the ﬁnite simple groups is suﬃcient to show that no such K can exist.14 We conclude that either [H : ι7 (S)] = 2, or that H contains a copy of SU40 (F7 ) and so 7780 #H.

14

One can also proceed via degrees of representations, and D. F. Holt indicated to us that the tables of Hiss and Malle [20] should suﬃce for this.

Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number John Voight Department of Mathematics and Statistics University of Vermont 16 Colchester Ave Burlington, VT 05401, USA [email protected]

Abstract. We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve deﬁned over a totally real ﬁeld with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real ﬁeld of odd degree. We conclude with two examples which illustrate the eﬀectiveness of our algorithms.

The development and implementation of algorithms to compute with automorphic forms has emerged as a major topic in explicit arithmetic geometry. The ﬁrst such computations were carried out for elliptic modular forms, and now very large and useful databases of such forms exist [2,13,14]. Recently, eﬀective algorithms to compute with Hilbert modular forms over a totally real ﬁeld F have been advanced. The ﬁrst such method is due to Demb´el´e [4,5], who worked initially under the assumption that F has even degree n = [F : Q] and strict class number 1. Exploiting the Jacquet-Langlands correspondence, systems of Hecke eigenvalues can be identiﬁed inside spaces of automorphic forms on B × , where B is the quaternion algebra over F ramiﬁed precisely at the inﬁnite places of F —whence the assumption that n is even. Demb´el´e then provides a computationally eﬃcient theory of Brandt matrices associated to B. This method was later extended (in a nontrivial way) to ﬁelds F of arbitrary class number by Demb´el´e and Donnelly [6]. When the degree n is odd, a diﬀerent algorithm has been proposed by Greenberg and the author [8], again under the assumption that F has strict class number 1. This method instead locates systems of Hecke eigenvalues in the (degree one) cohomology of a Shimura curve, now associated to the quaternion algebra B ramiﬁed at all but one real place and no ﬁnite place. This method uses in a critical way the computation of a fundamental domain and a reduction theory for the associated quaternionic unit group [16]; see Section 1 for an overview. In this article, we extend this method to the case where F has arbitrary (strict) class number. Our main result is as follows; we refer the reader to Sections 1 and 2 for precise deﬁnitions and notation. G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 357–371, 2010. c Springer-Verlag Berlin Heidelberg 2010

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Theorem 1. There exists an (explicit) algorithm which, given a totally real field F of degree n = [F : Q], a quaternion algebra B over F ramified at all but one real place, an ideal N of F coprime to the discriminant D of B, and a weight k ∈ (2Z>0 )n , computes the system of eigenvalues for the Hecke operators Tp with p DN and the Atkin-Lehner involutions Wpe with pe DN acting on the space of quaternionic modular forms SkB (N) of weight k and level N for B. In other words, there exists an explicit ﬁnite procedure which takes as input the ﬁeld F , its ring of integers ZF , a quaternion algebra B over F , an ideal N ⊂ ZF , and the vector k encoded in bits (each in the usual way), and outputs a ﬁnite set of number ﬁelds Ef ⊂ Q and sequences (af (p))p encoding the Hecke eigenvalues for each cusp form constituent f in SkB (N), with af (p) ∈ Ef . From the Jacquet-Langlands correspondence, applying the above theorem to the special case where D = (1) (and hence n = [F : Q] is odd), we have the following corollary. Corollary 2. There exists an algorithm which, given a totally real field F of odd degree n = [F : Q], an ideal N of F , and a weight k ∈ (2Z>0 )n , computes the system of eigenvalues for the Hecke operators Tp and Atkin-Lehner involutions Wpe acting on the space of Hilbert modular cusp forms Sk (N) of weight k and level N. This corollary is not stated in its strongest form: in fact, our methods overlap with the methods of Demb´el´e and his coauthors whenever there is a prime p which exactly divides the level; see Remark 5 for more detail. Combining these methods, Donnelly and the author [7] are systematically enumerating tables of Hilbert modular forms, and the details of these computations (including the dependence on the weight, level, and class number, as well as a comparison of the runtime complexity of the steps involved) will be reported there [7], after further careful optimization. A third technique to compute with automorphic forms, including Hilbert modular forms, has been advanced by Gunnells and Yasaki [9]. They instead use the theory of Vorono˘ı reduction and sharbly complexes; their work is independent of either of the above approaches. This article is organized as follows. In Section 1, we give an overview of the basic algorithm of Greenberg and the author which works over ﬁelds F with strict class number 1. In Section 2, using an adelic language we address the complications which arise over ﬁelds of arbitrary class number, and in Section 3 we make this theory concrete and provide the explicit algorithms announced in Theorem 1. Finally, in Section 4, we consider two examples, one in detail; our computations are performed in the computer system Magma [1]. The author would like to thank Steve Donnelly and Matthew Greenberg for helpful discussions as well as the referees for their comments. The author was supported by NSF Grant No. DMS-0901971.

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1

359

An Overview of the Algorithm for Strict Class Number 1

In this section, we introduce the basic algorithm of Greenberg and the author [8] with a view to extending its scope to base ﬁelds of arbitrary class number; for further reading, see the references contained therein. Let F be a totally real ﬁeld of degree n = [F : Q] with ring of integers ZF . Let × × F+× be the group of totally positive elements of F and let Z× F,+ = ZF ∩ F+ . Let B be a quaternion algebra over F of discriminant D. Suppose that B is split at a unique real place v1 , corresponding to an embedding ι∞ : B → B ⊗ R ∼ = M2 (R), and ramiﬁed at the other real places v2 , . . . , vn . Let O(1) ⊂ B be a maximal order and let × × O(1)× + = {γ ∈ O(1) : v1 (nrd(γ)) > 0} = {γ ∈ O(1) : nrd(γ) ∈ ZF,+ }

denote the group of units of O(1) with totally positive reduced norm. Let × + Γ (1) = ι∞ (O(1)× + /ZF ) ⊂ PGL2 (R) ,

so that Γ (1) acts on the upper half-plane H = {z ∈ C : Im(z) > 0} by linear fractional transformations. Let N ⊂ ZF be an ideal coprime to D, let O = O0 (N) × be an Eichler order of level N, and let Γ = Γ0 (N) = ι∞ (O0 (N)× + /ZF ). n Let k = (k1 , . . . , kn ) ∈ (2Z>0 ) be a weight vector; for example, the case k = (2, . . . , 2) of parallel weight 2 is of signiﬁcant interest. Let SkB (N) denote the ﬁnite-dimensional C-vector space of quaternionic modular forms of weight k and level N for B. Roughly speaking, a form f ∈ SkB (N) is an analytic function f : H → Wk (C) which is invariant under the weight k action by the group γ ∈ Γ , where Wk (C) is an explicit right B × -module [8, (2.4)] and Wk (C) = C when k is parallel weight 2. The space SkB (N) comes equipped with the action of Hecke operators Tp for primes p DN and Atkin-Lehner involutions Wpe for prime powers pe DN. The Jacquet-Langlands correspondence [8, Theorem 2.9] (see Hida [10, Proposition 2.12]) gives an isomorphism of Hecke modules ∼

SkB (N) − → Sk (DN)D-new , where Sk (DN)D-new denotes the space of Hilbert modular cusp forms of weight k and level DN which are new at all primes dividing D. Therefore, as Hecke modules one can compute equivalently with Hilbert cusp forms or with quaternionic modular forms. We compute with the Hecke module SkB (N) by identifying it as a subspace in the degree one cohomology of Γ (1), as follows. Let Vk (C) be the subspace of the algebra C[x1 , y1 , . . . , xn , yn ] consisting of those polynomials q which are homogeneous in (xi , yi ) of degree wi = ki − 2. Then Vk (C) has a right action of the group B × given by n γ −wi /2 q((x1 y1 )γ 1 , . . . , (xn yn )γ n ) (1) (det γi ) q (x1 , y1 , . . . , xn , yn ) = i=1

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for γ ∈ B × , where denotes the standard involution (conjugation) on B and γi = vi (γ) ∈ M2 (C). By the theorem of Eichler and Shimura [8, Theorem 3.8], we have an isomorphism of Hecke modules + ∼ SkB (N) − → H 1 Γ, Vk (C) where the group cohomology H 1 denotes the (ﬁnite-dimensional) C-vector space of crossed homomorphisms f : Γ → Vk (C) modulo coboundaries and + denotes the +1-eigenspace for complex conjugation. By Shapiro’s lemma [8, §6], we then have a further identiﬁcation + ∼ SkB (N) − → H 1 Γ, Vk (C) ∼ (2) = H 1 (Γ (1), V (C))+ , Γ (1)

where V (C) = CoindΓ Vk (C). In the isomorphism (2), the Hecke operators act as follows. Let p be a prime of ZF with p DN and let Fp denote the residue class ﬁeld of p. Since F has strict class number 1, by strong approximation [15, Theor`eme III.4.3] there exists π ∈ O such that nrd π is a totally positive generator for p. It follows that there × are elements γa ∈ O+ , indexed by a ∈ P1 (Fp ), such that × × O+ πO+ =

a∈P1 (F

× O+ αa

(3)

p)

where αa = πγa . Let f : Γ (1) → V (C) be a crossed homomorphism, and let γ ∈ Γ (1). The decomposition (3) extends to O(1) as × O(1)× O(1)× + πO(1)+ = + αa . a∈P1 (Fp ) 1 Thus, there are elements δa ∈ O(1)× + for a ∈ P (Fp ) and a unique permutation ∗ 1 γ of P (Fp ) such that αa γ = δa αγ ∗ a (4)

for all a. We then deﬁne f | Tp : Γ (1) → V (C) by (f | Tp )(γ) = f (δa )αa .

(5)

a∈P1 (Fp )

The space SkB (N) similarly admits an action of Atkin-Lehner operators Wpe for primes pe DN. From this description, we see that the Hecke module H 1 (Γ (1), V (C))+ is amenable to explicit computation. First, we compute a ﬁnite presentation for Γ (1) with a minimal set of generators G and a solution to the word problem for the computed presentation using an algorithm of the author [16]. Given such a set of generators and relations, one can explicitly ﬁnd a basis for the C-vector space H 1 (Γ (1), V (C)) [8, §5].

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We then compute the action of the Hecke operator Tp on H 1 (Γ (1), V (C)). We ﬁrst compute a splitting ιp : O → M2 (ZF,p ). The elements αa in (4) are then generators with totally positive reduced norm of the left ideals

xy + Op (6) Ia = Oι−1 p 00 and are obtained by principalizing the ideals Ia ; here again we use strong approximation and the hypothesis that F has strict class number 1. Then for each a ∈ P1 (Fp ) and each γ ∈ G, we compute the permutation γ ∗ [8, Algorithm 5.8] and the element δa = αa γα−1 γ ∗ a ∈ Γ (1) as in (4). Using the solution to the word problem, we then write δa as a word in the generators G for Γ (1), and then for a basis of crossed homomorphisms f we compute f | Tp by computing (f | Tp )(γ) ∈ V (C) for each γ ∈ G as in (5). In a similar way, we compute the action of complex conjugation and the Atkin-Lehner involutions. We then decompose the space H 1 (Γ, V (C)) under the action of these operators into Hecke irreducible subspaces, and from this we compute the systems of Hecke eigenvalues using linear algebra.

2

The Indefinite Method with Arbitrary Class Number

In this section, we show how to extend the method introduced in the previous section to the case where F has arbitrary class number [8, Remark 3.11]. We refer the reader to Hida [11] for further background. 2.1

Setup

We carry over the notation from Section 1. Recall that O = O0 (N) is an Eichler order of level N in the maximal order O(1) ⊂ B. Let H± = {z ∈ C : Im(z) = 0} = C \ R be the union of the upper and lower half-planes. Then via ι∞ , the group B × acts on H± by linear fractional transformations. In this generality, we ﬁnd it most elucidating to employ adelic notation. Let = lim Z/nZ and let denote tensor with Z over Z. Consider the double coset Z ←−n × /O × ), X(C) = B × \(H± × B × /O × by left multiplication via the diagonal embedding. where B × acts on B Then X(C) has the structure of a complex analytic space [3] which fails to be compact if and only if B ∼ = M2 (Q), corresponding to the classical case of elliptic modular forms—higher class number issues do not arise in this case, so from now we assume that B is a division ring. We again write SkB (N) for the ﬁnite-dimensional C-vector space of quaternionic modular forms of weight k and level N: here, again roughly speaking, a quaternionic modular form of weight k ∈ (2Z>0 )n and level N for B is an analytic function × /O × → Wk (C) f : H± × B which is invariant under the weight k action of B × , with Wk (C) as in Section 1.

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Decomposing the Double Coset Space

× By Eichler’s theorem of norms, we have nrd(B × ) = F(+) where × = {a ∈ F × : vi (a) > 0 for i = 2, . . . , n} F(+)

is the subgroup of elements of F which are positive at all real places which are × ∼ ramiﬁed in B. In particular, B × /B+ = Z/2Z, where × = {γ ∈ B × : v1 (nrd(γ)) > 0} = {γ ∈ B : nrd(γ) ∈ F+× }. B+ × acts on the upper half-plane H, therefore we may identify The group B+ × × /O × ). \(H × B X(C) = B+

Now we have a natural (continuous) projection map × × × \B /O , X(C) → B+

and by strong approximation [15, Theor`eme III.4.3] the reduced norm gives a bijection + × × × ∼ × ∼ nrd : B+ \B /O − → F+× \F × /Z (7) F = Cl ZF , where Cl+ ZF denotes the strict class group of ZF , i.e. the ray class group of ZF with modulus equal to the product of all real (inﬁnite) places of F . The space X(C) is therefore the disjoint union of Riemann surfaces indexed by Cl+ ZF , which we identify explicitly as follows. Let the ideals b ⊂ ZF form F be such that b Z F ∩ ZF = b. a set of representatives for Cl+ ZF , and let b ∈ Z For expositional simplicity, choose b = ZF and β = 1 for the representatives of × such that the trivial class. By strong approximation (7), there exists β ∈ B nrd(β) = b. Therefore × × ). X(C) = B+ (H × βO (8) [b]

We have a map × × ) → O× \H (H × βO B+ β,+

× ) → z (z, βO β−1 ∩ B and O× = O× ∩ B × , so that O = O. where Oβ = βO + 1 × β,+ × β + For each β, let Γ = ι∞ O /Z ⊂ PGL2 (R) . Then the Eichler-Shimura β

β,+

F

isomorphism on each component in (8) gives an identiﬁcation of Hecke modules ∼ SkB (N) − → H 1 (Γβ , Vk (C))+ , (9) β

let where + denotes the +1-eigenspace for complex conjugation. For each β, −1 O(1)β = βO(1)β ∩ B be the maximal order containing the Eichler order Oβ ,

Automorphic Forms on Shimura Curves Γ (1)β

× and let Γ (1)β = ι∞ (O(1)× (C) = CoindΓ /ZF ). Further, let Vβ β,+ β Shapiro’s lemma applied to each summand in (9) gives ∼ → H 1 (Γ (1)β , Vβ (C))+ . SkB (N) −

363

Vk (C). Then

(10)

β

2.3

Hecke Operators

H 1 (Γ (1)β , Vβ (C)) in F be the following way. Let p be a prime ideal of ZF with p DN, and let p ∈ Z F ∩ ZF = p. We consider the β -summand in (10), corresponding such that p Z to the ideal class [b ]. Let f : Γ (1)β → Vβ (C) be a crossed homomorphism: we will then obtain a new crossed homomorphism f | Tp : Γ (1) → V (C), where β In the description (10), the Hecke operators Tp act on

β

β

β

corresponds to the ideal class of [pb ] among the explicit choices made above. , be such that nrd( ) = p. Then there are elements γ a ∈ O Let

∈ O β β 1 indexed by a ∈ P (Fp ), such that × × α ×

(11) O O O = a β

β

a∈P1 (Fp )

β

where α a = γa . × , we conclude that there exist unique Let γ ∈ Γβ . Extending (11) to O(1) β × and a unique permutation γ ∗ of P1 (Fp ) such that elements δa ∈ O(1) β

γ ∗ a α a γ = δa α for a ∈ P1 (Fp ). Thus we have a )γ = (β β−1 )δa α γ ∗ a = δa (β β−1 α γ ∗ a ). (β β−1 α where δa = (β β−1 )δa (β β−1 )−1 . β−1 has right order O . has left order O and similarly O Recall that β O β β -ideal Therefore, we may consider the left O β

β O β−1 O α O β β a

(12)

noting that the left and right orders in each case match up, so the product is

compatible. Next, recall that the elements β , β, have reduced norms corresponding to the ideal classes [b ], [pb ], and [p], respectively. Thus the reduced norm of the left ideal (12) has a trivial ideal class. Therefore, by strong approximation (applied now to left ideals of the order Oβ ), for each a ∈ P1 (Fp ), there × exist elements πa ∈ Oβ ∩ B+ such that β β−1 α O a ∩ B = Oβ πa . β

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Hence there exists a unique permutation γ ∗ of P1 (Fp ) such that πa γ = δa πγ ∗ a with δa ∈ O× → Vβ (C) is then ,+ . The new crossed homomorphism f | Tp : Γβ β deﬁned by the formula (f | Tp )(γ) = f (δa )πa a∈P1 (Fp )

for γ ∈ Γβ . 2.4

Complex Conjugation and Atkin-Lehner Involutions

We now deﬁne an operator W∞ which acts by complex conjugation. Let Cl(+) ZF denote the ray class group of ZF with modulus equal to the real (inﬁnite) places of F which are ramiﬁed in B. Then we have a natural map Cl+ ZF → Cl(+) ZF ; this map is an isomorphism if and only if there exists a unit u ∈ Z× F which satisﬁes v1 (u) < 0 and vi (u) > 0 for the other real places vi (i = 2, . . . , n) of F , otherwise the kernel of this map is isomorphic to Z/2Z. Let [m] ∈ Cl+ ZF generate the kernel of this map. Let f : Γ (1)β → Vβ be a crossed homomorphism, and let β correspond to the ideal class [b m−1 ]; we will deﬁne the complex conjugate crossed homomorphism β O β−1 ∩ B has reduced norm (f | W∞ ) : Γ (1)β → Vβ (C). The left Oβ -ideal O β + corresponding to the ideal class [m] ∈ Cl ZF , so there exists a generator μ ∈ Oβ of this ideal such that v1 (nrd(μ )) < 0 but vi (nrd(μ )) > 0 for i = 2, . . . , n. Then given γ ∈ Γ (1)β , we deﬁne

(f | W∞ )(γ) = f (μ γμ−1 )μ . Finally, we deﬁne the Atkin-Lehner involutions Wpe for pe DN. Let p corre F . Then there exists an element π spond to p ∈ Z ∈ Oβ which generates the unique two-sided ideal of Oβ of reduced norm generated by pe . The element 2 ∈ O× F × . Let β correspond to the ideal class [pb ]. π normalizes O and π β

β

× Then as above, by strong approximation there exists an element μ ∈ Oβ ∩ B+ π ∩ B = O μ . Given f : Γ (1) → V , we then deﬁne such that Oβ β β β β β (f | Wpe ) : Γ (1)β → Vβ (C) by

(f | Wp )(γ) = f (μ γμ−1 )μ for γ ∈ Γ (1)β .

3

Algorithmic Methods

In this section, we take the adelic description of Section 2 and show how to compute with it explicitly, proving Theorem 1.

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Our algorithm takes as input a totally real ﬁeld F of degree [F : Q] = n, a quaternion algebra B over F split at a unique real place, an ideal N ⊂ ZF coprime to the discriminant D of B, a vector k ∈ (2Z>0 )n , and a prime p DN, and outputs the matrix of the Hecke operator Tp acting on the space H = +

1 (in the notation of Section 2) with respect to some ﬁxed H Γ (1)β , Vβ (C) β basis which does not depend on p. From these matrices, one decomposes the space H into Hecke-irreducible subspaces by the techniques of basic linear algebra. Our algorithm follows the form given in the overview in Section 1, so we describe our algorithm in steps, with a description of each step along the way. Step 1 (Compute a splitting field): Let K → C be a Galois number ﬁeld containing F which splits B: for example, we can take the normal closure of any quadratic ﬁeld contained in B. Since all computations then occur inside K ⊂ C, we may work then with coeﬃcient modules over K using exact arithmetic. (This step is only necessary if k is not parallel weight 2, for otherwise the action of B × factors through K = Q.) Step 2 (Compute ideal class representatives): Compute a set of representatives [b] for the strict class group Cl+ ZF with each b coprime to pDN. (See Remark 4 below.) Compute a maximal order O(1) ⊂ B. For each representative ideal b, compute a right O(1)-ideal Jb such that nrd(Jb ) = b and let O(1)b be the left order of Jb . (In the notation of Section 2, the right O(1)-ideals Jb represent the elements and O(1)b = O(1) .) β, β Step 3 (Compute presentations for the unit groups): Compute an embedding ι∞ : B → M2 (R) corresponding to the unique split real place. × For each b, compute a ﬁnite presentation for Γ (1)b = ι∞ (O(1)× b,+ /ZF ) consisting of a (minimal) set of generators Gb and relations Rb together with a solution to the word problem for the computed presentation [16]. (Note that the algorithm stated therein [16, Theorem 3.2] is easily extended from units of reduced norm 1 to totally positive units.) For eﬃciency, we start by computing such a presentation with generators G associated to the order O(1) and then for each order O(1)b we begin with the elements in hand formed by short products of elements in G which happen to lie in O(1)b (to aid in the search for units [16, Algorithm 3.2]; note that O(1)∩O(1)b is an Eichler order of level b in O(1)b ). Step 4 (Compute splitting data): Compute a splitting ιN : O(1) → O(1) ⊗ZF ZF,N ∼ = M2 (ZF,N ). Note that since b is coprime to N, we have O(1) ⊗ ZF,N = O(1)b ⊗ ZF,N for all b, so ιN also gives rise to a splitting for each O(1)b . For each b, compute the Eichler order Ob ⊂ O(1)b of level N with respect to ιN . Next, for each b, compute representatives for the left cosets of the group × Γb = ι∞ (Ob,+ /Z× F ) inside Γ (1)b [8, Algorithm 6.1]. Finally, identify Γ (1)b

V (K)b = CoindΓb

Vk (K)

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as a K-vector space given by copies of Vk (K) indexed by these cosets, and compute the permutation action of the representatives of these cosets on this space. In practice, it is more eﬃcient to identify the above coset representatives with elements of P1 (ZF / N) and thereby work directly with the coeﬃcient module V (K)b ∼ = K[P1 (ZF /N)] ⊗ Vk (K). Step 5 (Compute a basis for cohomology): Identify the space of crossed homo morphisms b Z 1 (Γ (1)b , V (K)b ) with its image under the inclusion Z 1 (Γb , V (K)b ) →

V (K)b

g∈Gb

f → (f (g))g∈Gb

consisting of those f ∈ g∈Gb V (K)b which satisfy the relations f (r) = 0 for r ∈ Rb . Compute the space of principal crossed homomorphisms B 1 (Γ (1)b , V (K)b ) in a similar way, and thereby compute using linear algebra a K-basis for the quotient H 1 (Γ (1)b , V (K)b ) = Z 1 (Γ (1)b , V (K)b )/B 1 (Γ (1)b , V (K)b ) for each b. Let H = b H 1 (Γ (1)b , V (K)b ). Step 6 (Compute representatives for left ideal classes): Compute a splitting ιp : O(1) → M2 (ZF,p ). For each ideal b , perform the following steps. First, compute the ideal b with ideal class [b] = [pb ]. Compute the left ideals Ia =

Oι−1 p

xy + Op 00

indexed by the elements a = (x : y) ∈ P1 (Fp ) and then compute the left Ob ideals Ia = Jb J b Ia . × Compute totally positive generators πa ∈ Ob ∩ B+ for Ob πa = Ia [12]. ∗ Now, for each γ ∈ Gb , compute the permutation γ of P1 (Fp ) [8, Algorithm 1 5.8] and then the elements δa = πa γπγ−1 ∗ a for a ∈ P (Fp ); write each such element δa as a word in Gb and from the formula (f | Tp )(γ) =

f (δa )πa

a∈P1 (Fp )

with f in a basis for the b -component of cohomology as in Step 5 compute the induced crossed homomorphism f | Tp in the b-component. Step 7 (Compute the blocks of the intermediate matrix): Assemble the matrix T with rows and columns indexed as in Step 5 with blocks in the (b, b ) position given by the output of Step 6: this matrix describes the action of Tp on H. Step 8 (Decompose H into ±-eigenspaces for complex conjugation): Determine the representative ideal m (among the ideals b) which generates the kernel of the map Cl+ ZF → Cl(+) ZF .

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For each ideal b , perform the following steps. Compute the ideal b such that [b] = [b m−1 ], and compute a generator μ with Ob μ = Jb J b such that v(nrd(μ )) < 0. For each γ ∈ Gb , from the formula

(f | W∞ )(γ) = f (μ γμ−1 )μ , for f in a basis for the b -component of cohomology as in Step 5 compute the induced crossed homomorphism f | Tp in the b-component. Assemble the matrix with blocks in the (b, b ) position given by this output: this matrix describes the action of complex conjugation W∞ on H. Compute a K-basis for the +1-eigenspace H + of H for W∞ . Finally, compute the matrix T + giving the action of Tp restricted to H + and return T + . This completes the description of the algorithm. In a similar way, one computes the Atkin-Lehner involutions, replacing Step 6 with the description given in Section 2.4, similar to the computation of complex conjugation in Step 8. Remark 3. Note that Steps 1 through 3 do not depend on the prime p nor the level N and Steps 4, 5, and 8 do not depend on the prime p, so these may be precomputed for use in tabulation. Remark 4. To arrange uniformly that the ideals b representing the classes in Cl+ ZF are coprime to the prime p in advance for many primes p, one has several options. One possibility is to choose suitable ideals b of large norm in advance. Another option is to make suitable modiﬁcations “on the ﬂy”: if p is not coprime to b, we simply choose a diﬀerent ideal c coprime to p with [b] = [c], a new ideal Jc with nrd(Jc ) = c, and compute an element ν ∈ Ob such that νOb ν −1 = Oc . Conjugating by ν where necessary, one can then transport the computations from one order to the other so no additional computations need to take place.

4

Examples

In this section, we compute with two examples to demonstrate the algorithm outlined in Section 3. Throughout, we use the computer system Magma [1]. Our ﬁrst and most detailed example is concerned with the smallest totally real cubic ﬁeld F with the property that the dimension of the space of Hilbert cusp forms of parallel weight 2 and level (1) is greater than zero and the strict class number of F is equal to 2. This ﬁeld is given by F = Q(w) where w satisﬁes the equation f (w) = w3 − 11w − 11 = 0. The discriminant of F is equal to 2057 = 112 17, and ZF = Z[w]. The roots of f in R are −2.602 . . . , −1.131 . . . , and 3.73 . . . , and we label the real places v1 , v2 , v3 of F into R according to this ordering. We deﬁne the sign of a ∈ F to be the triple sgn(a) = (sgn(vi (a)))3i=1 ∈ {±1}3 . The unit group of F is generated by the elements −1, w + 1 with sgn(w + 1) = (1, −1, −1), and the totally positive unit −w2 + 2w + 12.

368

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We begin by ﬁnding a quaternion algebra B with D = ZF which is ramiﬁed at

w + 1, −1 all but one real place [8, Algorithm 4.1]. We ﬁnd the algebra B = F ramiﬁed only at v1 and v2 , generated by i, j subject to i2 = w + 1, j 2 = −1, and ji = −ij. For forms of parallel weight 2, Step 1 is trivial: we can take K = Q. Next, in Step 2 we compute ideal class representatives. The nontrivial class in Cl+ (ZF ) is represented by the ideal b = (w2 − 2w − 6)ZF , which is principal but does not possess a totally positive generator, since sgn(−w2 + 2w + 6) = (−1, 1, −1) and there is no unit of ZF with this sign. We note that N(b) = 7. Next, we compute a maximal order O = O(1); it is generated over ZF by i and the element k = (1 + (w2 + 1)i + ij)/2. Next, we ﬁnd that the right O-ideal Jb generated by w2 − 2w − 6 and the element (5 + (w2 + 5)i + ij)/2 = 2 + 2i + k has nrd(Jb ) = b. Next, in Step 3 we compute presentations for the unit groups. We take the splitting B → M2 (R)

0 1 s 0 , i, j → −1 0 0 −s

where s = v3 (w + 1). We then compute a fundamental domain for Γ = Γ (1) [16], given below.

We ﬁnd that Γ = Γ (1) is the free group on the generators α, β, γ1 , . . . , γ7 subject to the relations γ12 = γ22 = γ33 = γ42 = γ53 = γ62 = γ72 = αβα−1 β −1 γ1 · · · γ7 = 1.

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For example, we have 2α = (w2 − 14) + (2w2 − 4w − 13)i + (−2w2 + 5w + 9)j + (−4w2 + 8w + 26)ij. The groups Γ and Γb have isomorphic presentations. In particular, we note that both Γ and Γb have genus 1, so we conclude that dim S2 (1) = 1 + 1 = 2. We illustrate the computation of Hecke operators with the primes p3 = (w + 2)ZF of norm 3 and p5 = (w + 3)ZF of norm 5. Note that p3 is nontrivial in Cl+ (ZF ) whereas p5 is trivial. Step Step 4 requires no work, since we work with forms of level (1). In Step 5 we compute with a basis for cohomology, and here we see directly that H 1 (Γ, Q) ∼ = Hom(Γ, Q) ∼ = Zfα ⊕ Zfβ where fα , fβ are the characteristic functions for α and β. We have a similar description for H 1 (Γb , Q). Next, in Step 6 we compute representatives of the left ideal classes. For p3 , for example, for I[1:0] ⊂ O we ﬁnd that Jb I[1:0] = Ob ((w + 1) + i + ij) and for I[1:1] ⊂ Ob we have Jb I[1:1] = O(w + 1 − i + ij); we thereby ﬁnd elements πa , πa for a ∈ P1 (Fp3 ). For the generators γ = α, β of O and Ob , we compute the permutations γ ∗ of P1 (Fp3 ); we ﬁnd for example that α∗ is the identity and π[1:0] α = δ[1:0] π[1:0] with δ[1:0] ∈ Ob , namely, 14δ[1:0] = (7w2 − 98) + (−23w2 + 40w + 167)i+

(−25w2 + 59w + 103)j + (−2w2 + 5w + 20)ij. We then write δ[1:0] as a word in the generators for Γb of length 23. Repeating these steps (reducing a total of 64 units), we assemble the block matrix in Step 7 as the matrix ⎞ ⎛ 0020 ⎜0 0 0 2 ⎟ ⎟ T p3 | H = ⎜ ⎝2 0 0 0 ⎠ . 0200

In a similar way, we ﬁnd that Tp5 is the identity matrix. Finally, in Step 8 we compute the action of complex conjugation. Here we have simply μ = i (whereas μb is more complicated), and thereby compute that ⎞ ⎛ 1 1 0 0 ⎜0 −1 0 0 ⎟ ⎟ W∞ | H = ⎜ ⎝0 0 1 1 ⎠ . 0 0 0 −1 + We verify that W∞ commutes

with Tp3 (and Tp5 ). We conclude that Tp3 | H = 10 02 . and Tp5 | H + = 01 20

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We then diagonalize the space H + , which breaks up into two one-dimensional eigenforms f and g, and compute several more Hecke operators: we list in Table 1 below a generator for the prime p, its norm N p, and the Hecke eigenvalues ap (f ) and ap (g) for the cusp forms f, g. Table 1. Hecke eigenvalues for the Hilbert cusp forms for F = Q(w) with w3 − 11w − 11 = 0 of level (1) and parallel weight 2 p w+2 w+3 2 2w + 7 w w2 − w − 8 w−3 2w2 − 5w − 10 w2 − 3w − 2 w2 − 6 w+4 2w2 − 3w − 16 w2 − 2w − 9 w2 + w − 3

Np 3 5 8 9 11 17 17 23 25 29 31 37 41 49

ap (f ) 2 1 −5 −2 0 −5 −5 2 −9 9 −2 −3 −5 −10

ap (g) −2 1 −5 2 0 5 −5 −2 −9 −9 −2 3 5 10

We note that the primes generated by w and w − 3 are ramiﬁed in F . By work of Deligne [3], the curves X = X(1) and Xb are deﬁned over the + + strict class √ ﬁeld F of F , and Gal(F /F ) permutes them. We compute that + 2 F = F ( −3w + 8w + 12). Therefore the Jacobian Jf , corresponding to the cusp form f , is a modular elliptic curve over F + with #J(Fp ) = N p + 1 − af (p) with everywhere good reduction. The form g is visibly a quadratic twist of f by the character corresponding to the extension F + /F . Unfortunately, this curve does not have any apparent natural torsion structure which would easily allow for its identiﬁcation as an explicit curve given by a sequence of coeﬃcients [6, §4]. As a second and ﬁnal example, we compute with a quaternion algebra deﬁned over therefore ramiﬁed at a ﬁnite prime. We take F = √ a quadratic ﬁeld and √ Q( 65), with ZF = Z[(1 + 65)/2]. The ﬁeld F has # Cl(F ) = # Cl+ (F ) = 2. We compute the space S = S2 (p5 )p5 -new of Hilbert cuspidal new forms of parallel weight 2 and level p5 , where p5 is the unique prime in ZF of norm 5. We compute that dim S = 10, and that the space S decomposes into Heckeirreducible subspaces of dimensions 2, 2, 3, 3. For example, the characteristic polynomial of Tp2 for p2 either prime above 2 factors as (T 2 − 2T − 1)(T 2 + 2T − 1)(T 6 + 11T 4 + 31T 2 + 9). Remark 5. By the Jacquet-Langlands correspondence, the space S2 (p5 )p5 -new also occurs in the space of quaternionic modular forms for an Eichler order of

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level p5 in the deﬁnite quaternion algebra ramiﬁed at the the two real places of F and no ﬁnite place, and therefore is amenable to calculation by the work of Demb´el´e and Donnelly. We use this overlap to duplicate their computations (as well as ours) and thereby give some compelling evidence that the results are correct since they are computed in entirely diﬀerent ways.

References 1. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997) 2. Cremona, J.: The elliptic curve database for conductors to 130000. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 11–29. Springer, Heidelberg (2006) 3. Deligne, P.: Travaux de Shimura. S´eminaire Bourbaki, Lecture notes in Math. 244(389), 123–165 √ 4. Demb´el´e, L.: Explicit computations of Hilbert modular forms on Q( 5). Experiment. Math. 14(4), 457–466 (2005) 5. Demb´el´e, L.: Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76(258), 1039–1057 (2007) 6. Demb´el´e, L., Donnelly, S.: Computing Hilbert modular forms over ﬁelds with nontrivial class group. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 371–386. Springer, Heidelberg (2008) 7. Donnelly, S., Voight, J.: Tables of Hilbert modular forms and elliptic curves over totally real ﬁelds (in preparation) 8. Greenberg, M., Voight, J.: Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Math. Comp. (accepted) 9. Gunnells, P., Yasaki, D.: Hecke operators and Hilbert modular forms. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 387–401. Springer, Heidelberg (2008) 10. Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. American Journal of Mathematics 103(4), 727–776 (1981) 11. Hida, H.: Hilbert modular forms and Iwasawa theory. Clarendon Press, Oxford (2006) 12. Kirschmer, M., Voight, J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. (SICOMP) 39(5), 1714–1747 (2010) 13. Stein, W.A.: Modular forms database (2004), http://modular.math.washington.edu/Tables 14. Stein, W.A., Watkins, M.: A database of elliptic curves—ﬁrst report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002) 15. Vign´eras, M.-F.: Arithm´etique des alg`ebres de quaternions. LNM, vol. 800. Springer, Berlin (1980) 16. Voight, J.: Computing fundamental domains for coﬁnite Fuchsian groups. J. Th´eorie Nombres Bordeaux 21(2), 467–489 (2009)

Improved Primality Proving with Eisenstein Pseudocubes Kjell Wooding and H.C. Williams Institute for Security, Privacy and Information Assurance, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4, Canada [email protected], [email protected]

Abstract. In August 2002, Agrawal, Kayal, and Saxena described an unconditional, deterministic algorithm for proving the primality of an integer N . Though of immense theoretical interest, their technique, even incorporating the many improvements that have been proposed since its publication, remains somewhat slow for practical application. This paper describes a new, highly eﬃcient method for certifying the primality of an integer N ≡ 1 (mod 3), making use of quantities known as Eisenstein pseudocubes. This improves on previous attempts, including the peudosquare-based approach of Lukes et al., and the pseudosquare improvement proposed by Berrizbeitia, et al.

1

Motivation

In [1], Lukes et al., building on the ideas of Hall [2], Shanks [3, p. 414], and Selfridge and Weinberger [4], described a highly eﬃcient method for proving the primality of an integer N using quantities known as pseudosquares. Their test requires a table of least pseudosquares, denoted M2,x , of suﬃcient size to ensure that N < M2,x . If such a table is available, their method certiﬁes the primality of an integer N using only (log N )3+o(1) operations. In [5], Berrizbeitia et al. introduced a conjecturally more eﬃcient test, relying on quantities they termed pseudocubes, denoted M3,x . Though expected to outperform the pseudosquare-based method asymptotically, this test required a 2/3 table of pseudocubes of suﬃent size to ensure that N < M3,x . In [6], we provided numerical data to support the conjectured asymptotic improvement. In the same paper, however, we pointed out that it is unlikely we will obtain pseudocubes large enough to realize the theoretical gains. Recent results of Sorenson [7] further support both the asymptotic beneﬁt and the practical limitations of this method. In this paper, we propose an alternate deﬁnition of pseudocube — the Eisenstein pseudocube — with a conjectured growth rate better than that of the pseudosquares. Furthermore, we propose an algorithm for proving primality of integers N ≡ 1 (mod 3) that eliminates the troublesome 2/3 exponent of Berrizbeitia’s method. In the process, we supply numerical evidence to support the argument that, both asymptotically and practically, proving primality using G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 372–384, 2010. c Springer-Verlag Berlin Heidelberg 2010

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373

Eisenstein pseudocubes will soon be more eﬃcient than the pseudosquare test for primes N ≡ 1 (mod 3).

2

Eisenstein Pseudocubes √

Let ω be a primitive cube root of unity; i.e. ω = −1+2 3i , and consider the ring of Eisenstein integers, Z[ω]. Recall [8, Chap. 9] that Z[ω] is a unique factorization domain with a norm given by N (α) = αα, and six units: ±1, ±ω, ±ω 2. There are three types of primes in Z[ω]: (1 − ω), which lies over 3; the inert rational primes q ≡ −1 (mod 3) with norm q 2 ; and the primes π of norm ππ = p ≡ 1 (mod 3) where p is prime in Z. We say that an element α ∈ Z[ω] is primary if α ≡ −1 (mod 3).1 It is straightforward to show that every prime in Z[ω] except (1 − ω) has exactly one primary associate. For any α, π ∈ Z[ω] with π prime, N (π) = 3, we can deﬁne the cubic residue α character of α modulo π, denoted π , as follows: 3 1. α = 0 if π | α π 3 α 2. π ≡ α(N (π)−1)/3 (mod π) otherwise, where α ∈ {1, ω, ω 2 }. π 3

3

The properties of this symbol are well-known. See, for example [8]. We can extend the notion of cubic residue character to include non-primes as follows. If α, τ ∈ Z[ω] with 3 | N (τ ), we deﬁne 1 if τ is a unit of Z[ω], α = k α τ 3 otherwise i=1 πi 3

k

where τ = i=1 πi and all πi ∈ Z[ω] are prime. Finally, recall the Cubic Reciprocity Law (CRL), as it applies to to the cubic Jacobi symbol [5, §2.3]: Theorem 1. (Cubic Let α, β be primary in Z[ω] and of coprime Reciprocity) β α norm = 3. Then β = α . 3

3

We are now in a position to deﬁne an Eisenstein pseudocube. Definition 1. Let p be a fixed rational prime. Define μp = a + bω ∈ Z[ω], a, b ∈ Z to be an element of Z[ω] of minimal norm such that: 1. 2. 3. 4.

μp is primary gcd (a, b) = 1 q = 1 for all rational primes q ∈ Z, q ≤ p μp 3 μp not a cube in Z[ω].

We will call μp a minimal Eisenstein pseudocube (or simply an Eisenstein pseudocube) for the prime p. 1

That is to say, if we write α = a + bω, a ≡ −1 (mod 3) and 3 | b.

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Congruence Criteria for Eisenstein Pseudocubes

One technique for eﬃciently computing a table of Eisenstein pseudocubes μp = xp + yp ω, is that of congruential sieving. In order to use this technique, we must ﬁrst establish a set of acceptable residue conditions Sq on μp for each of the primes q ≤ p corresponding to the requirements of Deﬁnition 1. There are 3 cases to consider, one for each type of prime in the Eisenstein integers. 3.1

Case 1: q ≡ −1 (mod 3)

In this case, q is inert and primary. μp is by deﬁnition primary, we can Since μ q = qp , and obtain the desired residue invoke cubic reciprocity: 1 = μp 3

3

conditions by simply computing μp ≡ (m + nω)3 (mod q) for all 0 ≤ m, n < q; i.e. the residue classes given by xp ≡ m3 − 3mn2 + n3 (mod q) yp ≡ 3mn(m − n) (mod q). There are q2 − 1 3

(1)

such solutions modulo q. Example 1. The set of acceptable residues for Eisenstein pseudocubes modulo 5 is given by S5 = {(1 + 0ω), (2 + 0ω), (3 + 0ω), (4 + 0ω), (3 + 1ω), (1 + 2ω), (4 + 3ω), (2 + 4ω)}. 3.2

Case 2: q = 3

Observe that −3ω = (1 − ω)2 . By the bimultiplicity of the cubic Jacobi symbol, 2 3 ω(1 − ω) = . μp 3 μp 3 k Write μp = xp + yp ω = (−1)k−1 i=1 αi where αi = ri + si ω are primary primes; i.e. 3 | si and ri ≡ −1 (mod 3). From the properties of the cubic Jacobi symbol, we know that 2(ri +1) ri +1+si ω 3 , and αωi = ω 3 giving 3

1−ω μp

= 3

k i=1

ω

2(ri +1) 3

= ω2

k

i=1 (ri +1)/3

,

1−ω αi

= 3

Improved Primality Proving with Eisenstein Pseudocubes

and hence Thus

ω μp

ω(1−ω) μp

=

3

ω

ri +1+si 3

=ω

k

i=1 (ri +1)/3+

k

i=1

si /3

,

i=1

3

k

375

=ω

k

i=1

si /3

3 μp

.

2

= ω3

k

i=1

si

.

(2)

3

n n−1 Lemma 1. Let μp = xp + yp ω = (−1) i=1 αi where αi = rni + si ω are prin n−1 mary primes. Then xp ≡ (−1) r (mod 9) and y ≡ p i=1 i i=1 si (mod 9). Proof. If n = 1, the statement is trivially true. Let αj = rj + sj ω, αk = rk + sk ω be primary; i.e. rj ≡ rk ≡ −1 (mod 3) and sj ≡ sk ≡ 0 (mod 3). Writing si = 3Si , ri = −1 + 3Ri for some Si , Ri ∈ Z, observe that −(rk + sk ω)(rj + sj ω) = −(rk + 3Sk ω)(rj + 3Sj ω) ≡ −rk rj − 3(Sj rk + Sk rj )ω ≡ −rk rj − 3(−Sk + 3Rj Sk − Sj + 3Rk Sj )ω ≡ −rk rj + (sk + sj )ω (mod 9) n n−1 which ω) ≡ (−1)n−1 i=1 (ri + si n primary. Thus, by induction, (−1) n is again n n−1 i=1 ri + i=1 si ω (mod 9), so writing μp = xp + yp ω = (−1) i=1 αi where αi = ri + si ω are primary primes xp ≡ (−1)n−1 yp ≡

n

n

ri

(mod 9),

i=1

si

(mod 9)

i=1

as desired.

k (mod 9), soyp /3 si /3 (mod 3). Combin ≡ i=1 k 2 3 i=1 si = ω 2yp /3 . Clearly, 3 ing these facts with Equation 2, we obtain μp = ω 3 y 3 = 1 ⇐⇒ 3 | 3p which, when combined with the requirement that μp be μp 3 primary, gives the requisite congruence conditions: 3 = 1 ⇐⇒ 9 | yp and xp ≡ −1 (mod 3). μp 3

From Lemma 1, yp ≡

k

i=1 si

Example 2. The set of acceptable residues for Eisenstein pseudocubes modulo 9 is given by S9 = {(2 + 0ω), (5 + 0ω), (8 + 0ω)}.

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Case 3: q ≡ 1 (mod 3)

3.3

We can write q = πq πq where πq = a + bω and πq is primary. Of course, πq is also primary. Lemma 2. Let q be a rational prime, μqp = 1 and q = πq πq with πq ∈ Z[ω] 3 μ μ prime and primary, then μqp = 1 if and only if πpq = πpq . 3

3

3

Proof. Recall q = πq πq , and that πq , πq , and μp are all primary. From cubic reciprocity and the properties of the cubic Jacobi symbol [8, §9.3] we have that

−1 μp μp μp μp = πq 3 πq 3 πq 3 πq 3 3 3 3 3 3 μ μ And thus it is clear that μqp = 1 if and only if πpq = πpq . q μp

If

=

q μp

πq μp

πq μp

=

μp πq

μp πq

=

3

3

3

= 1, then from Lemma 2 and the properties of the cubic reciprocity 3

q−1

q−1

symbol, μp 3 ≡ μp 3 (mod πq ). By complex conjugation, we have also that q−1 q−1 μp μp = , and hence μp 3 ≡ μp 3 (mod πq ). Combining these facts, we πq πq 3

obtain

3

q μp

q−1

= 1 ⇐⇒ μp 3 ≡ μp

q−1 3

(mod q).

(3)

3

Writing μp = xp +yp ω, we will now endeavour to reduce (3) to a set of congruence conditions on xp and yp . Note that when q is small, these congruence conditions can be computed by exhaustion. A more elegant algorithm, however, can be obtained from the theory of Lucas sequences. First, observe that if q | yp then (3) reduces to the trivial xp ≡ xp (mod q); i.e. x + 0ω ⊂ Sq for x = 1, . . . , q − 1. For the remaining case, consider the recurring sequences Sn (x, y), Tn (x, y) ∈ Z[x, y] given by:2 S1 (x, y) = x T1 (x, y) = y Sn + Tn ω = (S1 + T1 ω)n with Sn , Tn ∈ Z. Clearly, we have also that Sn + Tn ω 2 = (S1 + T1 ω 2 )n . By subtraction, (ω − ω 2 )Tn = (S1 + T1 ω)n − (S1 + T1 ω 2 )n , and thus writing α = μp = xp + yp ω, β = μp = xp + yp ω 2 , we have Tn =

αn − β n , ω − ω2

(4)

a recurrent sequence whose properties are described in [9]. We may parameterize this recurrence by writing G = α + β, H = αβ, and observing that Tn (G, H) is 2

For simplicity, we will usually write Sn and Tn for Sn (x, y) and Tn (x, y), respectively.

Improved Primality Proving with Eisenstein Pseudocubes

given by the second-order recurrence: Tn+2 = GTn+1 − HTn . From (3), 1 if and only if q | (α

q−1 3

q−1 3

−β ) and hence from (4), q = 1 ⇐⇒ q | T q−1 (G, H). 3 μp 3

377 q μp

= 3

(5)

Since only the case q | yp remains, we can rewrite (5) in terms of a single variable by deﬁning zp ≡ xp yp−1 (mod q). Now (xp + yp ω)(q−1)/3 ≡ (xp + yp ω 2 )(q−1)/3 (mod q) if and only if (zp + ω)(q−1)/3 ≡ (zp + ω 2 )(q−1)/3 (mod q). Setting α = zp + ω, β = zp + ω 2 in (4), we obtain q = 1 ⇐⇒ q | T q−1 (G , H ) (6) 3 μp 3 where G = 2zp − 1, H = zp2 − zp + 1. Since this relationship involves only one variable, we are in eﬀect considering polynomials Tn (x) where T0 (x) = 0,

T1 (x) = 1

Tn+1 (x) = (2x − 1)Tn x − (x2 − x + 1)Tn−1 (x) for a ﬁxed x ∈ Z. By induction, we see that Tn (x) is a polynomial over Z with coeﬃcients of degree n − 1 and leading coeﬃcient n. n −β n In fact, Tn (x) = Un (G , H ) where Un is the Lucas function, Un = αα−β , 2 G = α + β = 2x − 1, H = αβ = x − x + 1, and hence α = (x + ω), β = (x + ω 2 ). By drawing on the rich theory of Lucas functions, we can obtain both an eﬃcient algorithm for computing the acceptable congruence conditions on xp , yp (mod q), and the number of acceptable residues for the prime q. To obtain the candidate solutions zp satisfying (6), compute T q−1 (x) for all 3 0 ≤ x < q by the method described in [3, §4.4], retaining solutions for which T q−1 (x) ≡ 0 (mod q). Each zp obtained in this fashion can then be used to 3 produce (q − 1) acceptable values of μp by evaluating xp = 1, 2, . . . , q − 1 and computing the corresponding yp = xp zp (mod q)—a procedure illustrated in Example 3. To obtain a count of these solutions, observe that in (6), we can write Δ = (α −β)2 = (2x − 1)2 − 4(x2 − x + 1) = −3. If q is a prime ≡ 1 (mod 3) then = 1. Thus if x ∈ Z and q | x2 − x + 1 then q | Tq− (x) [3, Equation = Δ q

4.3.3]. It follows that the polynomial Tq−1 (x) of degree q − 2 has precisely q − 2 distinct zeros modulo q. Now T q−1 (x) ∈ Z[x], and so it divides Tq−1 (x) as, from 3 the theory of Lucas functions [3, Equation 4.2.45], we have T3n (2x − 1, x2 − x + 1) = 3Tn ((x2 − x + 1)n − Tn2 ). It follows that T q−1 (x) has exactly q−1 3 − 1 distinct zeros modulo q. 3 By combining the cases when q | yp and q | yp , we see that there are (q − 1)2 q−1 − 1 (q − 1) + (q − 1) = 3 3 acceptable residues for a prime q ≡ 1 (mod 3).

(7)

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Example 3. Consider the case q = 7. We can derive the acceptable residue conditions on μp as follows. If q | yp , then (x + 0ω) is acceptable for x = 1, . . . , (q − 1). If q | yp then from (6), we have that

7 μp

3

= 1 ⇐⇒ 7 | T 7−1 (G , H ) = 3

T2 (G , H ). Further, T2 (G , H ) = G T1 (G , H )−HT0(G , H ) = G −0 = 2zp −1 and hence, 7 = 1 ⇐⇒ 7 | 2zp − 1. μp 3 Thus, zp ≡ 4 (mod 7). Since we deﬁned zp = xp yp−1 (mod q), xp ≡ 4yp (mod 7), and we can obtain all solutions by running xp through all nonzero residue classes (modulo 7) and computing yp ≡ 4−1 xp ≡ 2xp (mod 7); i.e. xp 1 2 3 4 5 6 . yp ≡ 2xp (mod 7) 2 4 6 1 3 5 Combining these solutions with the trivial case (q | yp ), we obtain a complete set of solutions (modulo 7): S7 = {(1 + 0ω), (2 + 0ω), (3 + 0ω), (4 + 0ω), (5 + 0ω), (6 + 0ω), (4 + 1ω), (1 + 2ω), (5 + 3ω), (2 + 4ω), (6 + 5ω), (3 + 6ω)}.

4

Eisenstein Pseudocubes and Primality Testing

Eisenstein pseudocubes may be employed to prove primality for integers N ≡ 1 (mod 3) via the following theorem [10]. Theorem 2. (Berrizbeitia, 2003, personal correspondence) Let ν = a + bω be a primary element of Z[ω], where gcd (a, b) = 1, ν is not a unit, prime, or perfect power in Z[ω], and N (ν) < N (μp ). Then there must exist a rational prime q ≤ p such that

q ν

3

≡ q (N (ν)−1)/3 (mod ν).

Recall that if N ≡ 1 (mod 3) and N is a prime in Z, then N = νν, where ν is a primary prime in Z[ω]. Furthermore, if q is any rational prime, then N −1 q ≡q 3 (mod ν). ν 3 If we have a table of Eisenstein pseudocubes available to us, Berrizbeitia’s result gives us a means to certify the primality of N ≡ 1 (mod 3); i.e. 1. Test that N is not a perfect power; e.g. via [11]. 2. Find a primary ν ∈ Z[ω] such that N (ν) = N . This can be done eﬃciently using Cornacchia’s algorithm [12, §1.5.2] via the method of Williams [13, §5]. If this step fails, then N is composite.3 3

Cornacchia’s algorithm requires the evaluation of a square root modulo N , and hence, usually requires a factorization of N . For our purposes, however, we simply assume that N is prime in this step. If Cornacchia fails, it is because N was composite, which is exactly what we set out to determine.

Improved Primality Proving with Eisenstein Pseudocubes

379

3. From a precomputed table of Eisenstein pseudocubes, choose μp ∈ Z[ω] of minimal norm such that N < (μp ). N −1 N q ≡ q 3 (mod ν). If the test succeeds for 4. For each prime q ≤ p, test ν 3 all q, then N is prime. Step 1 of this algorithm requires (log N )1+o(1) operations. Cornacchia’s algorithm (Step 2) essentially consists of a GCD computation ((log N )2+o(1) operations), and the computation of a square root modulo a prime ((log N )3+o(1) ). Step 3 is a merely a table lookup. Step 4 appears to be the most computationally intensive component of the algorithm, requiring a series of modular exponentiations (each requiring (log N )2+o(1) operations). The precise number of exponentiations is dependent on the expected growth rate of the Eisenstein pseudocubes, something which we will now attempt to estimate.

5

Eisenstein Pseudocube Growth Rate

Let pi denote the ith prime (p1 = 2), and let Sp denote the set of acceptable residues modulo p for the Eisenstein pseudocubes as developed in Section 3. Writing p = pn , and denoting by (a, b) the Eisenstein integer a + bω, we know that S2 = {(1, 0)} , S9 = {(2, 0), (5, 0), (8, 0)} , and p Sp = (a, b) ∈ Z × Z = 1, a + bω 3

p−1 p−1 ≤ a, b ≤ − 2 2

for p > 3

Recall from Equations (1) and (7) that we expect ⎧ (p − 1)2 ⎪ ⎪ ⎨ if p ≡ 1 (mod 3) 3 |Sp | = 2 ⎪ ⎪ ⎩ (p − 1) if p ≡ 2 (mod 3) 3 acceptable residues modulo p. Writing

S1 = p≡1

(mod 3)

S2 = p≡2

(mod 3)

(p − 1)2 3

H1 =

(p2 − 1) 3

H2 =

p≡1

p≡2

p

(mod 3)

p

(mod 3)

for primes p ≤ pn , and invoking the Chinese Remainder Theorem we see that there are S = 3S1 S2 solutions satisfying the congruence criteria of the Eisenstein pseudocubes in the region −H/2 ≤ a, b < H/2, where H = 9H1 H2 .

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K. Wooding and H.C. Williams

Assume the S solutions μ = a + bω are equidistributed in the region −H/2 ≤ a, b < H/2. By a similar argument to that of Lukes et al. [1], we expect the solution of minimal norm, denoted by μp , to be given by a ≈ b ≈ √HS ; i.e. H2 . S

N (μp ) ≈

(8)

Consider the primes p = pn as n → ∞. Making an assumption that the primes are distributed equally between p ≡ 1 (mod 3) and p ≡ 2 (mod 3), we can approximate H 2 /S as follows. Write H12 = S1 H22 = S2

p≡1

3p2 , and (p − 1)2

(mod 3) p≤x

p≡2

3p2 . − 1)

H12 ≈ 3π(x)/2 S1 ≈ 3π(x)/2

1 p≤x 1−1/p

p≡1

p≤x

(10)

(p2

(mod 3) p≤x

From Mertens’s Theorem [14, p. 351], becomes

(9)

(mod 3) p≤x

e−γ log x

∼

p p−1

as x → ∞, so (9)

2

p p−1

γ π(x)/2

log x. ∼e 3 For (10), recall that p≤x 1 − p12 = ζ(2) = H22 ∼ 3π(x)/2 S2

π2 6

as x → ∞.4 Hence

6 π2

Putting these together, and writing n = π(x), c = N (μpn ) ≈

27eγ π

√

6

, we obtain

(9H1 H2 )2 ∼ c3n log pn 3S1 S2

as n → ∞. Thus, we expect (log N )1+o(1) exponentiations in Step 4 of our primality proving algorithm, for a combined (randomized) complexity of (log N )3+o(1) operations.5 4 5

See, for example, [15, Theorem 1.4.1]. The randomized nature of the algorithm stems solely from the requirement for a quadratic nonresidue in Cornacchia’s algorithm. Finding this quadratic nonresidue requires, on average, two evaluations of a Jacobi symbol.

Improved Primality Proving with Eisenstein Pseudocubes

381

90

80

70

60

50

40

30

20 log(M2,p/ log p) f(x)= 0.676694 x + 4.55216 2/3 ln(M3,q / (ln q)4/3) g(x)=0.707752x + 0.633985 ln(N(mup) / ln p)

10

h(x)=1.05557x + 3.79531 0 0

10

20

30

40 n

50

Fig. 1. Growth Rates

60

70

80

382

6

K. Wooding and H.C. Williams

Experimental Results

Our experiment followed the same basic approach as [6]. To test our hypotheses, a table of Eisenstein pseudocubes was developed using the Calgary Scalable Sieve (CASSIE), a software toolkit for congruential sieving on the University of Calgary’s Advanced Cryptography Laboratory (ACL) Beowulf cluster [6]. First, a series of small, non-normalized runs were performs in order to obtain Eisenstein pseudocubes for values of p ≤ 109. Once these runs were completed, a large parallel job was executed. This larger job evaluated all candidate solutions with N (μp ) ≤ 264 . To parallelize this job, the 11520 acceptable residues formed by Table 1. Eisenstein Pseudocube Results p 18 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61, 67 71 73 79, 83 89 97 101 103, 107 109 113 127 131 137 139 149 151 157

10 15 21 596 2127 5736 9708 14102

1 4 11 82 115 2507 3393 9175 21408 81221 70670 84695 44850 03669 62708 34194 82344 28178

N (μp ) 247 643 5113 13507 39199 1 07803 3 60007 39 04969 61 07191 103 18249 273 33067 991 79467 5329 97833 22785 22747 27417 02809 85007 66499 15475 53813 94233 48797 46210 13649 18103 60731 90827 69801 26375 28481 67688 29893 90619 32079 66151 53761 04348 13749 56547 47279 97583 41459 06441 31739 04110 19739 93471 77659 17235 68077 31706 25921

μp 11 + 18ω 29 + 18ω 71 + 72ω 23 + 126ω 227 + 90ω −181 + 198ω 653 + 126ω 443 + 2160ω −1669 + 1170ω 3617 + 2520ω 6023 + 3366ω 4973 + 11466ω −15451 + 11088ω 54017 + 17514ω 47477 + 56160ω 66887 + 156510ω 235061 + 107172ω −139813 + 253764ω −267733 + 744120ω 1227419 + 761670ω 5052689 + 4961880ω −2127709 + 4462200ω 10322861 + 8601732ω 3056387 + 15918570ω −27791551 + 1366560ω 109364777 + 13014540ω −114717193 + 19952010ω 160585853 + 126202050ω 845355437 + 667764090ω −724036477 + 954969030ω 696254903 + 2666049750ω 2979509543 + 3236384556ω 3671532959 + 3833807040ω

Improved Primality Proving with Eisenstein Pseudocubes

383

combining the solution candidates for moduli 18, 5, 7, and 11 were each used as a normalization modulus.6 Each of these jobs required approximately 8000 CPUseconds. Using 250 processing nodes, the complete job required approximately 4.25 days to complete, obtaining Eisenstein pseudocubes μp for p ≤ 157. These results are summarized in Table 1.

7

Analysis and Conclusions

In Figure 1, Eisenstein pseudocube growth is shown as a function of n, where pn is the nth prime. The straight line represents the least squares line ﬁtted to this data, and is given by: y = 1.05557x + 3.79531 a result that is remarkably consistent with the slope predicted by the argument of Section 5; i.e. log 3 = 1.09861. As a basis for comparison, classical pseudocube and pseudosquare results (including the recent work of Sorenson [7]) are also shown. Two conclusions may be drawn from these results. First, even with the relatively modest amount of computing power used to compute our table of Eisenstein pseudocubes, we have already produced a test that is more eﬃcient than the pseudocube method originally proposed by Berrizbeitia, et al. Second, we would expect that with a reasonable amount of computational investment, the Eisenstein pseudocube primality proving method will eventually be more eﬃcient than existing methods involving the pseudosquares.

8

Summary

In this paper, we have adapted a theorem of Berrizbeitia to produce a highly eﬃcient primality proving algorithm for integers N ≡ 1 (mod 3), making use of quantities known as Eisenstein pseudocubes. In addition to theoretical contributions, we have compiled a table of these quantities using an extensive twodimensional sieve calculation, and oﬀered numerical evidence for a conjectured growth rate: N (μpn ) ∼ c3n log pn as n → ∞.

References 1. Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Mathematics of Computation 65(213), S25–S27, 361–372 (1996) 2. Hall, M.: Quadratic residues in factorization. Bulletin of the American Mathematical Society 39, 758–763 (1933) 6

The normalization optimization, ﬁrst proposed by Lehmer in [16], is described in some detail in [6, §3.2].

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´ 3. Williams, H.C.: Edouard Lucas and Primality Testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. Wiley Interscience, Hoboken (1998) 4. Williams, H.C.: Primality testing on a computer. Ars Combinatoria 5, 127–185 (1978) 5. Berrizbeitia, P., M¨ uller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004) 6. Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006) 7. Sorenson, J.P.: Sieving for pseudosquares and pseudocubes in parallel using doublyfocused enumeration and wheel datastructures. In: Hanrot, G., Morain, F., Thom´e, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 331–339. Springer, Heidelberg (2010) 8. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, Heidelberg (1990) 9. Williams, H.C.: Some properties of a special set of recurring sequences. Paciﬁc Journal of Mathematics 77(1), 273–285 (1978) 10. Wooding, K.: The Sieve Problem in One- and Two-Dimensions. PhD thesis, The University of Calgary, Calgary, AB (April 2010), http://math.ucalgary.ca/~ hwilliam/files/wooding10thesis.pdf 11. Bernstein, D.J.: Detecting perfect powers in essentially linear time. Mathematics of Computation 67, 1253–1283 (1998) 12. Cohen, H.: A Course in Computational Algebraic Number Theory, 4th edn. Springer, Heidelberg (1993) 13. Williams, H.C.: An m3 public-key encryption scheme. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 358–368. Springer, Heidelberg (1986) 14. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979) 15. Crandall, R., Pomerance, C.: Prime numbers: A computational Perspective, 2nd edn. Springer, New York (2005) 16. Lehmer, D.H.: The sieve problem for all-purpose computers. Mathematical Tables and Other Aids to Computation 7(41), 6–14 (1953)

Hyperbolic Tessellations Associated to Bianchi Groups Dan Yasaki Department of Mathematics and Statistics University of North Carolina at Greensboro, Greensboro, NC 27412, USA d [email protected]

Abstract. Let F/Q be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.

1

Introduction

Let F/Q be a number ﬁeld. The space of positive deﬁnite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into polyhedral cones corresponding to the facets of the Vorono¨ı polyhedron [1, 11, 13]. This has been computationally explored for real quadratic ﬁelds in [16, 12] and the cyclotomic ﬁeld Q(ζ5 ) in [23]. For F an imaginary quadratic ﬁeld, the polyhedral cones give rise to ideal polytopes in H3 , 3-dimensional hyperbolic space. In work of Cremona and his students [6, 7, 5, 14, 22], analogous polytopes have already been computed for class number one imaginary quadratic ﬁelds as well as a few ﬁelds with class number two and three using diﬀerent methods. The structure of the polytopes was used to compute Hecke operators on modular forms for the Bianchi groups over those ﬁelds. These polytopes were used by Goncharov [10] in his study of Euler complexes on modular curves. The data of the polytope and stabilizer could also be used to give explicit presentations of GL2 (O) using results of Macbeath and Weil [15,21]. Swan [20] has computed presentations of these groups, though √ not with the polytopes constructed here, for imaginary quadratic ﬁelds Q( d) for −d ∈ {1, 2, 3, 5, 6, 7, 11, 15, 19}. Such explicit presentations have been used to compute cohomology of Bianchi groups of small discriminant with non-trivial coeﬃcients in work of Berkove, Sengun, and Finis-Grunewald-Tirao [2, 3, 9, 19]. We remark that there are other ways to obtain the fundamental polytope data. Riley [18] wrote the ﬁrst computer implementation of Poincar´e’s Polyhedron Theorem, which works in the more general setting of geometrically ﬁnite Kleinian G. Hanrot, F. Morain, and E. Thom´ e (Eds.): ANTS-IX 2010, LNCS 6197, pp. 385–396, 2010. c Springer-Verlag Berlin Heidelberg 2010

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groups. He computed the fundamental polytopes for many Bianchi groups. From this data, he computed presentations for the Bianchi groups and calculated the rank of their abelianizations. Another method is to use reduction theory. An algorithm of Swan [20] has been very recently implemented by Rahm and Fuchs [17], who used it to compute the integral homology groups of all Bianchi groups which are over imaginary quadratic ﬁelds of class number less than three. In this paper, we investigate the structure of these ideal polytopes for a large range of imaginary quadratic ﬁelds. Our approach and implementation works for general imaginary quadratic ﬁelds, but we restrict the range to ease the computation. We compute the ideal polytope classes for all imaginary quadratic ﬁelds of class number one and two, as well as some ﬁelds of higher class number with small √ discriminant. Speciﬁcally, we compute the ideal polytopes for the ﬁelds Q( d) for square-free d, where −d ∈ {1, · · · , 100, 115, 123, 163, 187, 235, 267, 403, 427}. There is no theoretical obstruction to computing these tessellations for higher class number and higher discriminant. The structure of the paper is as follows. We set the notation for the quadratic ﬁelds and Hermitian forms in Section 2. The implementation is described in Section 3. Finally, in Section 4, we summarize some of the data collected so far. Finally, we describe a general result of Macbeath on computing group presentations for groups of homeomorphisms, illustrating one possible use√of this data. We use this technique to give an explicit presentation for GL2 (Q( −14)) in Section 5.

2

Notation and Background

√ Let F = Q( d) ⊂ C be an imaginary quadratic number ﬁeld. We always take d < 0 to be a square-free integer. Let O ⊂ F denote the ring of integers in F . Then O has a Z-basis consisting of 1 and ω, where √ 1+ d if d ≡ 1 mod 4, ω = √2 d if d ≡ 2, 3 mod 4. Let ¯· denote complex conjugation, the nontrivial Galois automorphism of F . Definition 1. A binary Hermitian form over F is a map φ : F 2 → Q of the form φ(x, y) = ax¯ x + bx¯ y + ¯b¯ xy + cy y¯, where a, c ∈ Q and b ∈ F such that φ is positive definite. By choosing a Q-basis for F , φ can be viewed as a quadratic form over Q. In particular, it follows that φ(O2 ) is discrete in Q.

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Definition 2. The minimum of φ is m(φ) =

inf

v∈O 2 \{0}

φ(v).

A vector v ∈ O2 is minimal vector for φ if φ(v) = m(φ). The set of minimal vectors for φ is denoted M (φ). Definition 3. A Hermitian form over F is perfect if it is uniquely determined by M (φ) and m(φ).

3 3.1

Implementation Cone of Hermitian Forms and Hyperbolic Space

The space of positive deﬁnite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into polyhedral cones corresponding to the facets of the Vorono¨ı polyhedron Π [11, 13, 1]. The top-dimensional cones of this decomposition correspond to perfect forms and descend to ideal polytopes in H3 , 3-dimensional hyperbolic space. Details are given below. Let G be the restriction of scalars G = ResF/Q (GL2 ). Then the group of rational points G(Q) = GL2 (F ), and the group of real points is G = G(R) GL2 (C). Let H3 be hyperbolic 3-space: H3 = {(z, t) : z ∈ C,

t ∈ R>0 }.

Then G acts on H3 by αβ · (z, t) = (z ∗ , t∗ ), γ δ z∗ =

(αz + β)(γz + δ) + (αt)(γt) |γz + δ|2 + |γ|2 t2

where

and t∗ =

|αδ − βγ|t |γz + δ|2 + |γ|2 t2

Note that diagonal matrices act trivially on H3 , and the stabilizer of the point (i, 1) is U (2). Thus one gets an identiﬁcation between H3 and the coset space GL2 (C)/(U (2) · R>0 ). A binary Hermitian form can be identiﬁed with the 4-dimensional real vector space V of Hermitian 2 × 2 matrices. The group GL2 (C) acts on this space via g · A = gAg ∗ and preserves the open cone C ⊂ V of positive deﬁnite Hermitian matrices, and the stabilizer of I is U (2). Thus one has identiﬁcation C GL2 (C)/U (2). Modding out by homotheties, one gets C/R>0 H3 .

(1)

388

3.2

D. Yasaki

Vorono¨ı Decomposition

There is a map q from O2 to the closure C¯ of C ⊂ V given by q(v) = vv ∗ . The Vorono¨ı polyhedron Π is the unbounded polytope gotten by taking the convex hull of {q(v) : v ∈ O2 \ 0}. Taking cones over the facets of Π, one gets a decomposition of C into polyhedral cones known as the Vorono¨ı decomposition of C. By (1), this decomposition descends to a tessellation of H3 by ideal polytopes. Note that the group Γ = G(Z) = GL2 (O) acts on C and preserves this decomposition. 3.3

Perfect Forms

A perfect form φ is uniquely determined by its minimum m(φ) and set of minimal vectors M (φ). By scaling, we can assume m(φ) = 1. Since each minimal vector deﬁnes a linear equation in V , and V is 4-dimensional, generically 4 minimal vectors will uniquely determine φ. Note that this does not imply that #M (φ) = 4. Indeed in many examples, one has M (φ) > 4. There is a bijection between perfect forms over F and the facets of Π. Let P be a facet of Π with vertices {w1 , . . . , wk }. Then there is a unique form φP ∈ C such that m(φP ) = 1 and {q(v) : v ∈ M (φP )} = {w1 , . . . , wk }. There is an algorithm [11] that uses this bijection to compute the GL2 (O)equivalency classes of perfect forms. The algorithm uses linear algebra and convex geometry, but requires an initial input of a perfect form. To this end, we describe the method that √we used to compute an initial perfect form. For each ﬁeld F = Q( d), we need only to ﬁnd a single perfect form to begin the algorithm. Thus we limit our search to a particular family of quadratic forms. Speciﬁcally, let S0 ⊂ C be the subset of quadratic forms φ such that 1 0 1 ⊆ M (φ). , , 1 1 0 For φ ∈ S0 , the Hermitian matrix Aφ associated to φ must have the form 1 1β , where β ∈ F with Re(β) = − and |β| < 1. Aφ = ¯ β1 2 a ∈ O2 , then If φ ∈ S0 and φ has an additional minimal vector b 1 − a1 2 + a2 2 d + a1 b1 − a2 db2 − b1 2 + b2 2 d √ 1 d, β=− + 2 2 da1 b2 − 2 da2 b1 √ √ where a = a1 + a2 d and b = b1 + b2 d. Combined with (2), this implies

2 1 − a1 2 + a2 2 d + a1 b1 − a2 db2 − b1 2 + b2 2 d d 3 − < . 2 4 (2 da1 b2 − 2 da2 b1 )

(2)

(3)

Hyperbolic Tessellations Associated to Bianchi Groups

389

Reduction theory, speciﬁcally the existence of Siegel sets, ensures that the values NF/Q (a), NF/Q (b), and NF/Q (b − a) for a solution are bounded above by a constant depending upon d. Thus we implement a brute force search over a, b ∈ O a is found satisfying (3), we beginning at 0 and moving out. When a vector b check that the corresponding form φ satisﬁes a 1 0 1 ⊆ M (φ). , , , b 1 1 0 This corresponds to a ideal polytope whose vertices contain {∞, 0, 1, ab }. Once the initial form is found, we implement the algorithm of [11] to ﬁnd all the perfect forms over F up to the action of GL2 (O) (and the corresponding structure of the Vorono¨ı polyhedron) in Magma [4]. This descends, via (1), to give a tessellation of H3 by ideal polytopes.

4

Polytope Data

In this section we collect the results of the computations of the GL2 (O)-conjugacy classes of the ideal Vorono¨ı polytopes. Example: d = −14 √ Let F = Q( √ −14). Then F has class number four and ring of integers O = Z[ω], where ω = −14. There are 9 GL2 (O)-classes of polytopes which are of 3 combinatorial types. There are 3 triangular prisms with cuspidal vertices 5 + 2ω 2 + ω 4 + 2ω P1 = ∞, 1, , , ,0 9 4 9 5 + 2ω 4 + 2ω 12 + 4ω 11 + 4ω , 1, , , , 0 , and P2 = 23 9 9 23 8 + 5ω 2 + ω 1 + ω 2 + ω 3 + 2ω 7 + 4ω P3 = , , , , , , 23 5 5 6 10 21 4.1

and 5 tetrahedra with cuspidal vertices 11 + 4ω 2 + ω 4 + 2ω T1 = , , ,0 , 23 5 9 5 + 2ω 3 + ω 12 + 4ω T2 = 1, , , , 9 5 23 11 + 4ω 2 + ω 2 + ω , , ,0 , T3 = 23 5 6 8 + 5ω 2 + ω 4 + 2ω T4 = , , , 0 , and 23 5 9 3 + ω 12 + 4ω 4+ω , 1, , , T5 = 6 5 23

390

D. Yasaki

and a square pyramid with cuspidal vertices 8 + 5ω 2 + ω 1 + ω 2 + ω , , , ,0 . S= 23 5 5 6 Given the cuspidal vertices, one can easily compute the stabilizers of each polytope. The stabilizers are all cyclic in this case. For each stabilizer, we compute a generator. The results are given in Table 1. √ Table 1. Stabilizer groups of Vorono¨ı ideal polytopes for Q( −14) Polytope Stabilizer

4.2

P1

C6

P2

C2

P3

C4

T1

C2

T2

C2

T3

C2

T4

C2

T5

C2

S

C2

Generator 1 −1 1 0 −1 0 0 −1 ω + 1 −ω + 6 2 −ω − 1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1 −1 0 0 −1

Polytope Summary

We compute √ the Vorono¨ı polytopes for all imaginary quadratic number ﬁelds F = Q( d) with class number one and two as well as higher class number for d > −100. Although there is no reason an arbitrary convex 3-dimensional polytope could not arise, in all of these cases only 8 combinatorial types show up. We give the names and F -vector ([#vertices, #edges, #faces]) for each in Table 2. We also note that the triangular dipyramid shows up in this range much less frequently than the other polytopes. In Table 3, we give the number of GL2 (O)-classes of each polytope type for F with class number one or two. In Table 4, we give the number of GL2 (O)classes of each polytope type for the remaining imaginary quadratic ﬁelds with d > −100.

Hyperbolic Tessellations Associated to Bianchi Groups

391

Table 2. Combinatorial types of ideal polytopes that occur in this range polytope

F -vector picture

tetrahedron

[4, 6, 4]

octahedron

[6, 12, 8]

cuboctahedron

[12, 24, 14]

triangular prism

[6, 9, 5]

hexagonal cap

[9, 15, 8]

square pyramid

[5, 8, 5]

truncated tetrahedron [12, 18, 8] triangular dipyramid

5

[5, 9, 6]

Group Presentation

A general result of Macbeath [15] and analogous result of Weil [21] give a general method of computing group presentations for groups of homeomorphisms. For the convenience of the reader, we recall these results here and describe how the polytope data computed above can be used to compute explicit presentations of GL2 (OF ). Consider a connected space X acted upon by a group of homeomorphisms Γ . Let U ⊂ X be an open set such that Γ · U = X, and let Σ ⊂ Γ denote the set Σ = {g ∈ Γ : g · U ∩ U = ∅}. Let F (Σ) be the free group generated by Σ. For g ∈ Σ, let xg denote the corresponding element of F (Σ). Let W ⊂ Σ × Σ denote the set W = {(g, h) : U ∩ g · U ∩ gh · U = ∅}. Let R ⊂ F (Σ) denote the subgroup generated by xg xh x(gh)−1 for (g, h) ∈ W . Suppose π0 (X) = π1 (X) = π0 (U ) = 1. Then the subgroup R is a normal subgroup of F (Σ) and Γ F (Σ)/R. To apply this result to the polytope data computed above, choose X = H3 . Fix representatives P1 , . . . , Pk of the GL2 (O) classes of polytopes such that D = P1 ∪· · ·∪Pk is a connected set of polytopes meeting along facets. Let U ⊂ H3 be an open neighborhood of D ∩ H3 . We note that since the vertices D are at

392

D. Yasaki

Table 3. GL2 (O)-classes of Vorono¨ı ideal polytopes for class number one and two

hF

d

1 1 1 1 1 1 1 1 1

−1 0 −2 0 −3 1 −7 0 −11 0 −19 0 −43 0 −67 0 −163 11

1 0 0 0 0 0 0 1 0

0 1 0 0 0 1 0 0 1

0 0 0 1 0 1 2 2 8

0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 2 3

0 0 0 0 1 0 1 1 0

0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

−5 −6 −10 −13 −15 −22 −35 −37 −51 −58 −91 −115 −123 −187 −235 −267 −403 −427

0 0 1 0 1 0 4 0 0 0 1 1 1 1 1 1 1 2

0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0

2 0 1 3 0 4 1 8 2 7 5 5 6 4 12 13 16 19

0 1 0 1 0 0 0 1 1 2 0 2 3 1 4 5 2 4

0 0 2 1 0 2 2 8 0 6 3 4 3 9 11 10 20 24

0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0

0 0 0 1 1 5 3 10 1 47 5 3 1 18 13 24 66 65

Hyperbolic Tessellations Associated to Bianchi Groups Table 4. GL2 (O)-classes of Vorono¨ı ideal polytopes with d > −100

hF

d

3 3 3 3

−23 −31 −59 −83

0 0 0 6

1 0 1 0

0 0 1 0

1 3 3 2

0 0 0 2

1 1 2 1

0 0 0 1

0 0 0 0

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

−14 −17 −21 −30 −33 −34 −39 −46 −55 −57 −73 −78 −82 −85 −93 −97

5 5 8 6 9 20 1 32 5 33 57 69 92 56 79 95

0 0 2 0 0 0 0 1 1 1 1 1 0 0 1 0

0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1

3 2 2 6 8 3 3 5 2 10 13 11 8 17 20 19

0 1 1 4 1 1 1 0 0 3 1 4 3 0 7 3

1 3 4 4 6 6 1 9 2 14 14 18 11 28 21 19

0 1 0 0 1 1 0 0 0 2 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0

5 −47 5 5 −79 9

0 0

0 0

1 5

1 0

2 4

0 0

0 0

−26 −29 −38 −53 −61 −87

18 15 33 45 41 6

1 0 1 0 1 0

0 0 0 0 0 0

2 6 2 7 11 6

1 0 1 2 1 2

4 6 6 13 16 3

0 0 1 0 0 0

0 0 0 0 0 0

7 −71 7

1

0

4

0

4

0

0

−41 −62 −65 −66 −69 −77 −94 −95

31 81 69 67 51 81 125 12

0 0 2 1 2 1 1 0

1 0 0 1 0 0 0 0

9 7 9 9 15 9 10 4

0 2 0 4 2 2 2 0

8 7 19 12 21 26 17 9

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0

10 −74 105 10 −86 130

1 0

0 0

9 9

1 1

12 18

0 1

0 0

12 −89 136

0

0

14

1

21

1

0

6 6 6 6 6 6

8 8 8 8 8 8 8 8

393

394

D. Yasaki

inﬁnity, the set U can be chosen so that if g ∈ Σ, then g takes an edge of D to another edge of D. We remark that many redundant generators and relations are created when implementing this result, especially when the stabilizer groups of the polytopes are large. We can compensate for this using Magma’s commands for simplifying ﬁnitely-presented groups. We illustrate the technique in the example below. Example: d = −14 √ Theorem 1. Let F = Q( −14) with ring of integers O = Z[ω], where ω = √ −14. Then the following is a presentation of GL2 (O):

5.1

GL2 (O) = g1 , · · · , g8 : R1 = · · · = R22 = 1,

where

R1 = g72 ,

R2 = g82 ,

R3 = g62 ,

R4 = g32 ,

R5 = g42 ,

R6 = g22 ,

R7 = g54 ,

R8 = (g2 g1−1 )2 ,

R9 = (g4 g1 )2 , R13 = (g6 g5−2 )2 ,

R10 = g5−1 g1−3 g5−1 , R11 = (g7 g5−2 )2 , R12 = (g8 g5−2 )2 , R14 = (g4 g5−2 )2 ,

R17 = (g3 g5−1 g3 g1 g2 )2 ,

R15 = (g3 g5−2 )2 , R16 = (g6 g1−1 g5−1 )2 ,

R18 = (g3 g7 g1 g8 g1−1 )2 ,

R19 = g4 g5 g4 g1−1 g5 g1 ,

R20 = g8 g5−1 g7 g5−1 g3 g1−1 g3 g7 g3 g7 g1 g8 g3 g5 g7 g5−1 , R21 = g1 g5 g7 g5−1 g3 g1−1 g3 g7 g1 g5−1 g7 g5−1 g3 g1−1 g3 g7 , R22 = g6 g5 g7 g5−1 g3 g1−1 g3 g7 g1 g6 g1−1 g7 g3 g1 g3 g5 g7 g5 . Proof. We choose X, U , and D as described above. In fact, one can choose D to be the polytopes given in Section 4.1. Then F (Σ)/R is deﬁned by 235 generators and 3416 relations. √ We can simplify this presentation in Magma to get the presentation of GL2 (Z[ −14]) above, with 1 −1 01 , g2 = , g1 = 1 0 10 ω + 3 −ω + 1 4ω −2ω + 13 , g4 = , g3 = 6 −ω − 3 2ω + 13 −4ω −2ω − 5 2ω − 3 −5ω 3ω − 15 g5 = , g6 = , −10 2ω + 5 −3ω − 15 5ω ω + 9 −2ω − 1 −2ω − 13 4ω + 4 , g8 = . g7 = −2ω + 10 −ω − 9 ω − 14 2ω + 13 The presentation given in the theorem has torsion elements as generators. In particular, GL2 (O) is generated by elements of order 2, 4, and 6. Since any torsion-free quotient must map these generators to the identity, one immediately gets the following corollary. √ Corollary 1. GL2 (Z[ 14]) has no torsion-free quotients. √ One ﬁnds similar results for F = Q( d) for d = −1 and d = −3 in [8].

Hyperbolic Tessellations Associated to Bianchi Groups

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Acknowledgments. I thank the reviewers for their comments. I would like to thank John Cremona for helpful conversations at the beginning of this project, and Paul Gunnells for introducing me to these techniques. I thank Sebastian Pauli for his advice on the computation, Carlos Nicholas for his help with the polytopes, and Greg Bell for his help with the group presentations. Finally, I thank Steve Donnelly for helpful discussions and the Magma Group at the University of Sydney for their hospitality during a visit, in which part of this research was completed. This work was partially supported by the UNCG New Faculty grant.

References 1. Ash, A.: Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones. Math. Ann. 225(1), 69–76 (1977) 2. Berkove, E.: The mod-2 cohomology of the Bianchi groups. Trans. Amer. Math. Soc. 352(10), 4585–4602 (2000) 3. Berkove, E.: The integral cohomology of the Bianchi groups. Trans. Amer. Math. Soc. 358(3), 1033–1049 (2006) (electronic) 4. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997); Computational algebra and number theory, London (1993) 5. Bygott, J.: Modular forms and modular symbols over imaginary quadratic fields, Ph.D. thesis, Exeter University (1998) 6. Cremona, J.E., Whitley, E.: Periods of cusp forms and elliptic curves over imaginary quadratic fields. Math. Comp. 62(205), 407–429 (1994) 7. Cremona, J.E.: Periods of cusp forms and elliptic curves over imaginary quadratic fields. In: Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, pp. 29–44. Amer. Math. Soc., Providence (1994) 8. Fine, B.: The HN N and generalized free product structure of certain linear groups. Bull. Amer. Math. Soc. 81, 413–416 (1975) 9. Finis, T., Grunewald, F., Tirao, P.: The cohomology of lattices in SL(2, C). Experiment. Math. 19(1), 29–63 (2010) 10. Goncharov, A.B.: Euler complexes and geometry of modular varieties. Geom. Funct. Anal. 17(6), 1872–1914 (2008) 11. Gunnells, P.E.: Modular symbols for Q-rank one groups and Vorono˘ı reduction. J. Number Theory 75(2), 198–219 (1999) 12. Gunnells, P.E., Yasaki, D.: Hecke operators and Hilbert modular forms. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 387–401. Springer, Heidelberg (2008) 13. Koecher, M.: Beitr¨ age zu einer Reduktionstheorie in Positivit¨ atsbereichen. I. Math. Ann. 141, 384–432 (1960) 14. Lingham, M.: Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. thesis, University of Nottingham (2005) 15. Macbeath, A.M.: Groups of homeomorphisms of a simply connected space. Ann. of Math. 79(2), 473–488 (1964) 16. Ong, H.E.: Perfect quadratic forms over real-quadratic number fields. Geom. Dedicata 20(1), 51–77 (1986) 17. Rahm, A., Fuchs, M.: The integral homology of PSL2 of imaginary quadratic integers with non-trivial class group, arXiv:0903.4517 (2009)

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Author Index

Balakrishnan, Jennifer S. 16 Bernard, Aurore 32 Biasse, Jean-Fran¸cois 50 Bos, Joppe W. 66 Bradshaw, Robert W. 16 Brent, Richard P. 83 ´ Brier, Eric 96 Bruin, Nils 110

Lenstra, Arjen K. 66 Levin, Mariana 6 Lubicz, David 251

Clavier, Christophe

96

Pauli, Sebastian Pomerance, Carl

Dahmen, Sander R. Darmon, Henri 1

110

Fieker, Claus 157 Ford, David 174

Hart, William B. Ionica, Sorina

2

Nagao, Koh-ichi 285 Nebe, Gabriele 4 301 6

Regev, Oded 3 Robert, Damien 251

Elsenhans, Andreas-Stephan Enge, Andreas 142

Gama, Nicolas

McKee, James 270 Mestre, Jean-Fran¸cois

32

Siksek, Samir 316 Sorenson, Jonathan P. 331 Soukharev, Vladimir 219 Soundararajan, K. 6 Stehl´e, Damien 157, 340 Stoll, Michael 316 Sutherland, Andrew V. 142 Tibouchi, Mehdi 234 Tornar´ıa, Gonzalo 186

186 201

Jacobson Jr., Michael J. Jahnel, J¨ org 126 Jao, David 219 Joux, Antoine 201 Joye, Marc 234

126

50

Veres, Olga 174 Vergnaud, Damien Voight, John 357

Watkins, Mark 186, 340 Williams, Hugh C. 372 Wooding, Kjell 372 Yasaki, Dan

Kedlaya, Kiran S. 16 Kleinjung, Thorsten 66

234

385

Zimmermann, Paul

83