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xxList of symbols and notationBoolean functionsAI(f) Algebraic immunity of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10deg(f) Algebraic degree of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10d H (f, g) Hamming distance between f and g . . . . . . . . . . . . . . . . . . . . . . 11nl(f) Nonlinearity of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11supp(f) Support of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9w H (f) Hamming weight of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9f, g Boolean functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Tu–Deng conjectureα i Size of the i-th subblock of 1’s of t . . . . . . . . . . . . . . . . . . . . . . . 35β i Size of the i-th subblock of 0’s of t . . . . . . . . . . . . . . . . . . . . . . . 35C t,k,i Set of modular integers producing w H (t) − i carries . . . . . . . . . . . . . . 29δ i Number of 1’s in front of the end of the 0’s subblock of t. . . . . . . . . . . 35ɛ i Number of carries not occurring in the i-th block of t while adding a . . . . 37γ i Number of 0’s in front of the end of the 1’s subblock of t . . . . . . . . . . 35I t,k,i Set of inert modular integers producing w H (t) − i carries . . . . . . . . . . . 80D(·) Diagonal of a double power series . . . . . . . . . . . . . . . . . . . . . . . . 73O t,k,i Set of overflowing modular integers producing w H (t) − i carries . . . . . . . 81P t,k Fraction of modular integers in S t,k . . . . . . . . . . . . . . . . . . . . . . 29Res(f, z) Residue of a complex-valued function f at z . . . . . . . . . . . . . . . . . . 73S t,k Set of the Tu–Deng conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 29S t,v,u,k Set of the generalized Tu–Deng conjecture . . . . . . . . . . . . . . . . . . . 21w H (t) Hamming weight of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21a ≃ b Cyclotomic equivalence for modular integers . . . . . . . . . . . . . . . . . . 22a d,n(i 1,...,i n)Coefficients for the closed-form expression of f d . . . . . . . . . . . . . . . . 48B Number of 0’s of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35d Number of blocks of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35f d (β 1 , . . . , β d ) Function giving P t,k for a set of β i ’s of size d . . . . . . . . . . . . . . . . . 42G i , H i Geometrically distributed variables with parameter 1/2 . . . . . . . . . . . 68k Exponent for modulus 2 k − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 21
List of symbols and notationxxil(t) Number of bits of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80P d Probability that X d = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68r(a, t) Number of carries occurring while adding a to t . . . . . . . . . . . . . . . . 23t, a, b Modular integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21X d Difference of two sums of d geometrically distributed variables . . . . . . . 682F 1 (a, b; c; z) Gaussian hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . 67Bent functionsχ f Sign function of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Tr k r Field trace from F 2 k to F 2 r . . . . . . . . . . . . . . . . . . . . . . . . . . . 97̂χ f Walsh–Hadamard transform of f . . . . . . . . . . . . . . . . . . . . . . . . 97C n (a, b) Cubic sum associated with a and b . . . . . . . . . . . . . . . . . . . . . . . 98D r (x) Dickson polynomial of degree r . . . . . . . . . . . . . . . . . . . . . . . . . 99E a Elliptic curve associated with a . . . . . . . . . . . . . . . . . . . . . . . . . 114f a,b A Boolean function of the Mesnager family . . . . . . . . . . . . . . . . . . 113f a A Boolean function of the Charpin–Gong family . . . . . . . . . . . . . . . 113G f Artin–Schreier curve associated with f . . . . . . . . . . . . . . . . . . . . . 115g a A Boolean function associated with f a . . . . . . . . . . . . . . . . . . . . . 113H f Artin–Schreier curve associated with f . . . . . . . . . . . . . . . . . . . . . 116K n (a) Kloosterman sum associated with a . . . . . . . . . . . . . . . . . . . . . . 98o(j) Size of the cyclotomic coset of j modulo 2 n − 1 . . . . . . . . . . . . . . . . 97Elliptic curves[n] Multiplication by n on an elliptic curve . . . . . . . . . . . . . . . . . . . . 100#E Number of rational points of E . . . . . . . . . . . . . . . . . . . . . . . . . 104Ell(O) Set of elliptic curves with complex multipliation by O . . . . . . . . . . . . 147End(E) Ring of geometric endomorphisms of E . . . . . . . . . . . . . . . . . . . . 103End K (E) Ring of K-rational endomorphisms of E . . . . . . . . . . . . . . . . . . . . 103F Fundamental domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Γ(1) Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141H Poincaré upper halfplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Λ A lattice in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
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 and 12: AcknowledgmentsVladimir. — Quand
Page 13 and 14: ContentsList of symbols and notatio
Page 15 and 16: Contentsxv4 Efficient characterizat
Page 17 and 18: List of figures1.1 The filter model
Page 19: List of symbols and notationGeneral
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
Page 37 and 38: 1.2. Families of Boolean functions
Page 43: 1.2. Families of Boolean functions
Page 46 and 47: 20 Chapter 2. On a conjecture about
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44 Chapter 2. On a conjecture about
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Part IIBent functions and pointcoun
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98 Chapter 3. Bent functions and al
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100 Chapter 3. Bent functions and a
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110 Chapter 4. Efficient characteri
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Part IIIComplex multiplication and
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132 Chapter 5. Complex multiplicati
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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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