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xxivList of symbols and notationFrac(O) Semigroup of fractional ideals of O . . . . . . . . . . . . . . . . . . . . . . . 143H g Siegel upper half-space of genus g . . . . . . . . . . . . . . . . . . . . . . . . 163Λ A lattice in V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161N Φ Norm type associated with Φ . . . . . . . . . . . . . . . . . . . . . . . . . . 173O An order in a number field K . . . . . . . . . . . . . . . . . . . . . . . . . . 168O K Maximal order of a number field K . . . . . . . . . . . . . . . . . . . . . . . 168Â Dual variety of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164ω(x, y) A Riemann form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Õ Normalization, or integral closure, of O . . . . . . . . . . . . . . . . . . . . 168PF(O) Group of principal ideals of O . . . . . . . . . . . . . . . . . . . . . . . . . . 144pf(ω) Pfaffian of ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Φ A CM type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Pic(O) Picard group of O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Pic(A) Picard group of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Prop(O) Semigroup of proper ideals of O . . . . . . . . . . . . . . . . . . . . . . . . 144ψ L Polarization associated with a line bundle L . . . . . . . . . . . . . . . . . . 164SP 2g (Q) Symplectic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166θ(z) A theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Tr Φ Type trace associated with Φ . . . . . . . . . . . . . . . . . . . . . . . . . . 173A An abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161E A CM algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172E r Reflex field of a CM algebra E . . . . . . . . . . . . . . . . . . . . . . . . . 173g Dimension of an abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . 161H(O) Kronecker class number of O . . . . . . . . . . . . . . . . . . . . . . . . . . 144H(x, y) A Hermitian form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162H prop (O) Proper class number of O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144L(H, α) Line bundle associated with H and α . . . . . . . . . . . . . . . . . . . . . 165R An order in a CM algebra E . . . . . . . . . . . . . . . . . . . . . . . . . . 174R(Λ) The multiplier ring of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . 140S O Normalization kernel of O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
List of symbols and notationxxvu † Rosati involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167V A g-dimensional complex vectorspace . . . . . . . . . . . . . . . . . . . . . 161X A complex torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
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 and 20: List of symbols and notationGeneral
Page 21 and 22: List of symbols and notationxxil(t)
Page 23: List of symbols and notationxxiiidi
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
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48 Chapter 2. On a conjecture about
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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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