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xiiAcknowledgmentsMadore, Bertrand Meyer and Jacques Sakarovitch who answered my silly questions; and finallyIrène Charon, Laurent Decreusefond, Olivier Hudry, Antoine Lobstein and Süleyman Üstünel.From the INFRES department, I also have a thought for Jean Leneutre and Ahmed Serhrouchni— you once saved my computer during one of my lonely Sunday afternoons at Télécom ParisTech.To go on with the scientific aspect of these past three years, I must also thank the fellow membersof the “Groupe de Travail de la Butte aux Cailles” — Alain Couvreur, Luca De Feo and JérômePlût — for keeping my modest initiative alive for more than two years. I have the feeling thatwe even managed to build something more than a purely mathematical relation. Finally, I cannot omit my new colleagues from the ANSSI: Loïc Duflot, Olivier Levillain, and the members ofthe “Laboratoire de Cryptographie” — Aurélie Bauer, Thomas Fuhr, Henri Gilbert, Jean-RenéReinhard, Yannick Seurin, Joana Treger–Marim. You have warmly welcomed me and offered meexceptional conditions to finish the writing of this manuscript. I hope that the near future willsee the beginning of a long lasting collaboration between us.On the technical side, I guess most of the work I conducted during the past three years wouldhave not been possible without the use of a wide spectrum of free and open source software amongwhich I will only mention a few: the Debian operating system, the L A TEX typesetting system, theGNU Compiler Collection and the Sage project. I will do my best to contribute back to theseprojects. I am also grateful to the maintainers of the arxiv.org and iacr.org preprint serversand the numerous authors making their results freely available there. A particular thought goesto the wonderful math webpage of J.S. Milne: this is a goldmine for course notes.To conclude, I must say that I could not have survived throughout these three years in apurely mathematical world. The bo-buns I ate with Chico and Bertrand; the concerts I attendedwith Clément; the exotic countries and high routes I traveled with Moche, Rod, Titi and Val; thebeers I shared with my fellow comrades from the École Polytechnique and the BôBar — Baptiste,Milpatte and Tipiak — and my old fellow comrades from the Lycée Pasteur and elsewhere —Bousty, Flavien, Franz, Guillaume, Jérémy, Olivier and Victor. On a more personal note, I wouldlike to thank the extended Debarre family — Anne, Becky, Benoît, Clélia, Fanfette, François,Jacques, Magali, Olivier and Paul — for welcoming me into their world. The same must definitelybe said for my family — my grandmother, my mother, my uncle Albert and my sister, whobravely brought back stinking cheese from Corsica, despite the interrogative looks of the busdriver. And last but not least comes Marianne who has put up with me and my stupid ideas forall these years.
ContentsList of symbols and notationxixIntroduction 11 Mathematics and cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Computational experiments and implementation . . . . . . . . . . . . . . . . . . 23 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I Boolean functions and a combinatorial conjecture 51 Boolean functions in cryptography 71.1 Cryptographic criteria for Boolean functions . . . . . . . . . . . . . . . . . . . . . 81.1.1 The filter and combiner models . . . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Balancedness and resiliency . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Algebraic degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Algebraic immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5 Nonlinearity and bentness . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Families of Boolean functions with good cryptographic properties . . . . . . . . . 111.2.1 Trade-offs between the different criteria . . . . . . . . . . . . . . . . . . . 111.2.2 The Carlet–Feng family . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 The Tu–Deng family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.4 The Tang–Carlet–Tang family . . . . . . . . . . . . . . . . . . . . . . . . 151.2.5 The Jin et al. family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 On a conjecture about addition modulo 2 k − 1 192.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 Negation and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.3 Carries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 The case ɛ = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Proof of the conjecture of Tang, Carlet and Tang . . . . . . . . . . . . . . 242.2.2 Computing the exact gap . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 The case ɛ = +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.3 Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.5 Reformulation in terms of carries . . . . . . . . . . . . . . . . . . . . . . . 31
Page 1: 2012-ENST-003EDITE de ParisDoctorat
Page 4 and 5: Jean-Pierre Flori: Boolean function
Page 7: AbstractThe core of this thesis is
Page 11: AcknowledgmentsVladimir. — Quand
Page 15 and 16: Contentsxv4 Efficient characterizat
Page 17 and 18: List of figures1.1 The filter model
Page 19 and 20: List of symbols and notationGeneral
Page 21 and 22: List of symbols and notationxxil(t)
Page 23 and 24: List of symbols and notationxxiiidi
Page 25: List of symbols and notationxxvu
Page 28 and 29: 2 Introduction1 Mathematics and cry
Page 30 and 31: 4 Introductiongiving more general b
Page 33 and 34: Chapter 1Boolean functions incrypto
Page 35 and 36: 1.1. Cryptographic criteria for Boo
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Part IIBent functions and pointcoun
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Part IIIComplex multiplication and
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Chapter 6Complex multiplication in
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6.1. Abelian varieties 1616.1 Abeli
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6.1. Abelian varieties 163Theorem 6
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6.1. Abelian varieties 165A homomor
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6.1. Abelian varieties 1676.1.5 Hom
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6.2. Class groups and units 169The
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6.2. Class groups and units 171in [
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6.3. Complex multiplication 173If t
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6.3. Complex multiplication 175•
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6.3. Complex multiplication 177We h
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6.3. Complex multiplication 179pola
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6.4. Class polynomials for genus 2
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6.4. Class polynomials for genus 2
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Appendices
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Table A.3: Coefficients for d = 3,
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Table A.8: Coefficients for d = 8,
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192 Bibliography[13] Elwyn Ralph Be
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194 Bibliography[42] Claude Carlet,
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196 Bibliography[72] Cunsheng Ding,
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198 Bibliography[102] David Freeman
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200 Bibliography[132] Marc Hindry a
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202 Bibliography[161] Philippe Lang
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204 Bibliography[191] Willi Meier a
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206 Bibliography[222] Oscar Seymour
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208 Bibliography[254] Marco Streng.
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210 Bibliography[286] Chia-Fu Yu. T
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212 IndexForm class group . . . . .
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214 IndexKKloosterman sum . . . . .
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Résumé longFauĆ.Werd’ iĚ beru
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1. Des fonctions booléennes et d
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2. Fonctions courbes et comptage de
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3. Multiplication complexe et polyn
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