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Arithmetic, Geometry, Cryptography and Coding Theory 2009 CONTEMPORARY MATHEMATICS 521 Arithmetic, Geometry, Cryptography and Coding Theory 2009 12th Conference on Arithmetic, Geometry, Cryptography and Coding Theory March 30–April 3, 2009 Marseille, France Geocrypt Conference April 27–May 1, 2009 Pointe-à-Pitre, Guadeloupe, France European Science Foundation Exploratory Workshop Curves, Coding Theory, and Cryptography March 25–29, 2009 Marseille, France David Kohel Robert Rolland Editors American Mathematical Society Arithmetic, Geometry, Cryptography and Coding Theory 2009 This page intentionally left blank CONTEMPORARY MATHEMATICS 521 Arithmetic, Geometry, Cryptography and Coding Theory 2009 12th Conference on Arithmetic, Geometry, Cryptography and Coding Theory March 30–April 3, 2009 Marseille, France Geocrypt Conference April 27–May 1, 2009 Pointe-à-Pitre, Guadeloupe, France European Science Foundation Exploratory Workshop Curves, Coding Theory, and Cryptography March 25–29, 2009 Marseille, France David Kohel Robert Rolland Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 11G10, 11G15, 11G20, 14G10, 14G15, 14G50, 14H05, 14H10, 14H45, 14Q05. Library of Congress Cataloging-in-Publication Data International Conference “Arithmetic, Geometry, Cryptography and Coding Theory” (2009 : Mar- seille, France) Arithmetic, geometry, cryptography, and coding theory 2009 : Geocrypt, April 27–May 1, 2009, Point-`a-Pitre, Guadeloupe : 12th Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, March 30–April 3, 2009, Marseille, France : European Science Foundation Exploratory Workshop on Curves, Coding Theory, and Cryptography, March 25–29, 2009, Marseille, France / David Kohel, Robert Rolland, editors. p. cm. — (Contemporary mathematics ; v. 521) Includes bibliographical references. ISBN 978-0-8218-4955-2 (alk. paper) 1. Arithmetical algebraic geometry—Congresses. 2. Coding theory—Congresses. 3. Cryp- tography—Congresses. I. Kohel, David R., 1966– II. Rolland, Robert. III. European Science Foundation. Exploratory Workshop on Curves, Coding Theory, and Cryptography (2009 : Mar- seille, France) IV. Title. QA242.5.I58 2009 516.35—dc22 2010010568 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface vii Differentially 4-uniform functions Yves Aubry and Franc¸ois Rodier 1 Computing Hironaka’s invariants: Ridge and directrix Jer´ emy´ Berthomieu, Pascal Hivert and Hussein Mourtada 9 Nondegenerate curves of low genus over small finite fields Wouter Castryck and John Voight 21 Faster side-channel resistant elliptic curve scalar multiplication Alexandre Venelli and Franc¸ois Dassance 29 Non lin´earit´edesfonctionsbool´eennes donn´ees par des polynˆomes de degr´e binaire 3 d´efinies sur F2m avec m pair Eric Ferard´ and Franc¸ois Rodier 41 A note on a maximal curve Arnaldo Garcia and Henning Stichtenoth 55 Computing Humbert surfaces and applications David Gruenewald 59 Genus 3 curves with many involutions and application to maximal curves in characteristic 2 Enric Nart and Christophe Ritzenthaler 71 Uniqueness of low genus optimal curves over F2 Alessandra Rigato 87 Group order formulas for reductions of CM elliptic curves Alice Silverberg 107 Families of explicit isogenies of hyperelliptic Jacobians Benjamin Smith 121 Computing congruences of modular forms and Galois representations modulo prime powers Xavier Taixes´ i Ventosa and Gabor Wiese 145 v This page intentionally left blank Preface The 12th conference Arithmetic, Geometry, Cryptography and Coding The- ory (AGC2T 12) took place in Marseille at the Centre International de Recontres Math´ematiques (CIRM) from 30 March to 3 April 2009. This biennial conference has been a major event in applied arithmetic geometry for nearly a quarter cen- tury, organized by the research group Arithm´etique et Th´eorie de l’Information of the Institut de Math´ematiques de Luminy. There were more than 40 research talks and 80 participants from sixteen countries. This year the AGC2T was preceded by a three-day Exploratory Workshop funded by the European Science Foundation on Curves, Coding Theory, and Cryptography, which brought some 30 researchers together for expository lectures and discussions on the arithmetic of curves and ap- plications. We especially thank the speakers Dan Bernstein, Claus Diem, Ralf Gerk- mann, Hendrik Hubrechts, Ian Kimming, Tanja Lange, Gabriele Nebe, Christophe Ritzenthaler, Patrick Sol´e, and Gabor Wiese for their lectures, and all participants of both events for creating a stimulating research environment. Less than one month later, on a different continent, the ATI group, together with the eRISCS laboratory of the Universit´e de la Mediterran´ee, Marseille and the AOC laboratory (Analyse, Optimisation, Contrˆole) of the Universit´e des Antilles et de la Guyane, assembled 34 participants for the first Geocrypt conference from 27 April to 1 May 2009, in Pointe-`a-Pitre, Guadeloupe. We thank Yves Aubry, Stephane Ballet, Vicent Cossart, Noam Elkies, Everett Howe, Marc Girault, Marc Joye, Gilles Lachaud, Kristin Lauter, Heeralal Janwa, Gary McGuire, Christophe Ritzenthaler, Fran¸cois Rodier, Karl Rubin, Ren´e Schoof, Alice Silverberg, Peter Stevenhagen, and John Voight for their mathematical contributions, making this both an enjoyable and informative extension of the AGC2T conference. We also thank Microsoft Research for financial support as well as R´egisBlache for the occasion of his habilitation defense to make this possible. The 12 articles of this volume represent a selection of research presented at this trilogy of events in the spring of 2009. vii This page intentionally left blank Contemporary Mathematics Volume 521, 2010 Differentially 4-uniform functions Yves Aubry and Fran¸cois Rodier Abstract. We give a geometric characterization of vectorial Boolean func- tions with differential uniformity ≤ 4. This enables us to give a necessary condition on the degree of the base field for a function of degree 2r − 1tobe differentially 4-uniform. 1. Introduction F Fm We are interested in vectorial Boolean functions from the 2-vectorial space 2 to itself in m variables, viewed as polynomial functions f : F2m −→ F2m over the m field F2m in one variable of degree at most 2 − 1. For a function f : F2m −→ F2m , we consider, after K. Nyberg (see [16]), its differential uniformity δ(f)= max{x ∈ F2m | f(x + α)+f(x)=β}. α=0,β This is clearly a strictly positive even integer. Functions f with small δ(f) have applications in cryptography (see [16]). Such functions with δ(f) = 2 are called almost perfect nonlinear (APN) and have been extensively studied: see [16]and[9] for the genesis of the topic and more recently [3]and[6] for a synthesis of open problems; see also [7] for new constructions and [20] for a geometric point of view of differential uniformity. Functions with δ(f) = 4 are also useful; for example the function x −→ x−1, which is used in the AES algorithm over the field F28 , has differential uniformity 4 on F2m for any even m. Some results on these functions have been collected by C. Bracken and G. Leander [4, 5]. We consider here the class of functions f such that δ(f) ≤ 4, called differentially 4-uniform functions. We will show that for polynomial functions f of degree d = r 2 − 1 such that δ(f) ≤ 4 on the field F2m ,thenumberm is bounded by an expression depending on d. The second author demonstrated the same bound in thecaseofAPNfunctions[17, 18]. The principle of the method we apply here was already used by H. Janwa et al. [13] to study cyclic codes and by A. Canteaut [8] to show that certain power functions could not be APN when the exponent is too large. 2000 Mathematics Subject Classification. 11R29,11R58,11R11,14H05. Key words and phrases. Boolean functions, almost perfect nonlinear functions, varieties over finite fields. c 2010 Americanc 0000 Mathematical (copyright Societyholder) 1 2 YVES AUBRY AND FRANC¸OIS RODIER Henceforth we fix q =2m. In order to simplify our study of such functions, let us recall the following elementary results on differential uniformity; the proofs are straightforward: Proposition 1. (i) Adding a polynomial whose monomials are of degree 0 or a power of 2 to a function f does not change δ(f). (ii) For all a, b and c in Fq, such that a =0 and c =0 we have δ(cf(ax + b)) = δ(f). (iii) One has δ(f 2)=δ(f). Hence, without loss of generality, from now on we can assume that f is a polynomial mapping from Fq to itself which has neither terms of degree a power of 2 nor a constant term, and which has at least one term of odd degree.
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