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 http://dx.doi.org/10.1090/conm/521 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 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. 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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 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 Titles in This Series 527 Ricardo Casta˜no-Bernard, Yan Soibelman, and Ilia Zharkov, Editors, Mirror symmetry and tropical geometry, 2010 526 Helge Holden and Kenneth H. Karlsen, Editors, Nonlinear partial differential equations and hyperbolic wave phenomena, 2010 525 Manuel D. Contreras and Santiago D´ıaz-Madrigal, Editors, Five lectures in complex analysis, 2010 524 Mark L. Lewis, Gabriel Navarro, Donald S. Passman, and Thomas R. Wolf, Editors, Character theory of finite groups, 2010 523 Aiden A. Bruen and David L. Wehlau, Editors, Error-correcting codes, finite geometries and cryptography, 2010 522 Oscar Garc´ıa-Prada, Peter E. Newstead, Luis Alverez-C´´ onsul, Indranil Biswas, Steven B. Bradlow, and Tom´asL. G´omez, Editors, Vector bundles and complex geometry, 2010 521 David Kohel and Robert Rolland, Editors, Arithmetic, geometry, cryptography and coding theory 2009, 2010 520 Manuel E. Lladser, Robert S. Maier, Marni Mishna, and Andrew Rechnitzer, Editors, Algorithmic probability and combinatorics, 2010 519 Yves F´elix, Gregory Lupton, and Samuel B. Smith, Editors, Homotopy theory of function spaces and related topics, 2010 518 Gary McGuire, Gary L. Mullen, Daniel Panario, and Igor E. Shparlinski, Editors, Finite fields: Theory and applications, 2010 517 Tewodros Amdeberhan, Luis A. Medina, and Victor H. Moll, Editors, Gems in experimental mathematics, 2010 516 Marlos A.G. Viana and Henry P. Wynn, Editors, Algebraic methods in statistics and probability II, 2010 515 Santiago Carrillo Men´endez and Jos´eLuisFern´andez P´erez, Editors, Mathematics in finance, 2010 514 Arie Leizarowitz, Boris S. Mordukhovich, Itai Shafrir, and Alexander J. Zaslavski, Editors, Nonlinear analysis and optimization II, 2010 513 Arie Leizarowitz, Boris S. Mordukhovich, Itai Shafrir, and Alexander J. Zaslavski, Editors, Nonlinear analysis and optimization I, 2010 512 Albert Fathi, Yong-Geun Oh, and Claude Viterbo, Editors, Symplectic topology and measure preserving dynamical systems, 2010 511 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, II, 2010 510 Mario Bonk, Jane Gilman, Howard Masur, Yair Minsky, and Michael Wolf, Editors, In the Tradition of Ahlfors-Bers, V, 2010 509 Primitivo B. Acosta-Hum´anez and Francisco Marcell´an, Editors, Differential algebra, complex analysis and orthogonal