Arithmetic, Geometry, Cryptography and Coding Theory 2009 PDF
Preview Arithmetic, Geometry, Cryptography and Coding Theory 2009
C ONTEMPORARY M ATHEMATICS 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 C ONTEMPORARY M ATHEMATICS 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. Primary11G10, 11G15, 11G20,14G10, 14G15, 14G50,14H05, 14H10, 14H45, 14Q05. Library of Congress Cataloging-in-Publication Data InternationalConference“Arithmetic,Geometry,CryptographyandCodingTheory”(2009: Mar- seille,France) Arithmetic,geometry,cryptography,andcodingtheory2009: Geocrypt,April27–May1,2009, Point-`a-Pitre,Guadeloupe: 12thConferenceonArithmetic,Geometry,Cryptography,andCoding Theory, March 30–April 3, 2009, Marseille, France : European Science Foundation Exploratory WorkshoponCurves,CodingTheory,andCryptography,March25–29,2009,Marseille,France/ DavidKohel,RobertRolland,editors. p.cm. —(Contemporarymathematics;v.521) Includesbibliographicalreferences. ISBN978-0-8218-4955-2(alk.paper) 1. Arithmetical algebraic geometry—Congresses. 2. Coding theory—Congresses. 3. Cryp- tography—Congresses. I.Kohel,DavidR.,1966– II.Rolland,Robert. III.EuropeanScience Foundation. Exploratory WorkshoponCurves, Coding Theory, andCryptography (2009: Mar- seille,France) IV.Title. QA242.5.I58 2009 516.3(cid:2)5—dc22 2010010568 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- 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 commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2010bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://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 J´er´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´e des fonctions bool´eennes donn´ees par des polynˆomes de degr´e binaire 3 d´efinies sur F avec m pair 2m Eric F´erard 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 F 2 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 Taix´es 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 byathree-day ExploratoryWorkshopfundedbytheEuropeanScienceFoundation on Curves, Coding Theory, and Cryptography, which brought some 30 researchers togetherforexpositorylecturesanddiscussions onthearithmeticofcurvesandap- plications. WeespeciallythankthespeakersDanBernstein,ClausDiem,RalfGerk- 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 withtheeRISCS laboratoryoftheUniversit´edelaMediterran´ee,Marseilleandthe 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´egis Blache for the occasion of his habilitation defense to make this possible. The12articlesofthisvolumerepresentaselectionofresearchpresentedatthis trilogy of events in the spring of 2009. vii This page intentionally left blank CCoonntteemmppoorraarryyMMaatthheemmaattiiccss Volume521,2010 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 conditiononthedegreeofthebasefieldforafunctionofdegree2r−1tobe differentially4-uniform. 1. Introduction WeareinterestedinvectorialBooleanfunctionsfromtheF -vectorialspaceFm 2 2 to itself in m variables, viewed as polynomial functions f : F −→ F over the 2m 2m fieldF inonevariableofdegreeatmost2m−1. Forafunctionf :F −→F , 2m 2m 2m we consider, after K. Nyberg (see [16]), its differential uniformity δ(f)= max (cid:3){x∈F |f(x+α)+f(x)=β}. 2m α(cid:2)=0,β This is clearly a strictly positive even integer. Functionsf withsmallδ(f)haveapplicationsincryptography(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 (cid:4)−→ x−1, F which is used in the AES algorithm over the field , has differential uniformity 4 28 F on for any even m. Some results on these functions have been collected by C. 2m Bracken and G. Leander [4, 5]. Weconsiderheretheclassoffunctionsf suchthatδ(f)≤4,calleddifferentially 4-uniform functions. We will show that for polynomial functions f of degree d = 2r − 1 such that δ(f) ≤ 4 on the field F , the number m is bounded by an 2m expression depending on d. The second author demonstrated the same bound in thecaseofAPNfunctions[17, 18]. Theprincipleofthemethodweapplyherewas 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. 2000MathematicsSubjectClassification. 11R29,11R58,11R11,14H05. Keywordsandphrases. Booleanfunctions,almostperfectnonlinearfunctions,varietiesover finitefields. (cid:2)c2010 Americ(cid:2)acn00M00at(hceompyartiigcahltShoolcdieetry) 11
The list of books you might like
