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xviContents6.2.2 Computation of the Picard and unit groups . . . . . . . . . . . . . . . . . 1706.2.3 Multiplication of fractional ideals by finite idèles . . . . . . . . . . . . . . 1716.3 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.1 CM field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.2 Reflex field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.3 CM abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.3.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.5 Dual abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.6 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.3.7 The main theorems of complex multiplication . . . . . . . . . . . . . . . . 1796.4 Class polynomials for genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.4.1 Jacobian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.4.2 Igusa invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.4.3 Igusa class polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.4.4 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Appendices 185A Coefficients of f d 187Bibliography 191Index 211Résumé long 2171 Des fonctions booléennes et d’une conjecture combinatoire . . . . . . . . . . . . . 2191.1 Fonctions booléennes en cryptographie . . . . . . . . . . . . . . . . . . . . 2191.2 D’une conjecture sur l’addition modulo 2 k − 1 . . . . . . . . . . . . . . . . 2222 Fonctions courbes et comptage de points sur les courbes algébriques . . . . . . . 2272.1 Fonctions courbes et courbes algébriques . . . . . . . . . . . . . . . . . . . 2272.2 Caractérisations efficaces de fonctions courbes . . . . . . . . . . . . . . . . 2303 Multiplication complexe et polynômes de classes . . . . . . . . . . . . . . . . . . 2353.1 Multiplication complexe et courbes elliptiques . . . . . . . . . . . . . . . . 2353.2 Multiplication complexe en genre supérieur . . . . . . . . . . . . . . . . . 238
List of figures1.1 The filter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The combiner model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Graph of f 2 (x, y) for 0 ≤ x, y ≤ 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2 Zoom on the graph of f 2 (x, y) for 1 ≤ x, y ≤ 5 . . . . . . . . . . . . . . . . . . . . 442.3 Graph of f d (β i ) for β i = 1, i ≠ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 Graph of f d (β i ) for β i = 2, i ≠ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Graph of f d (β i ) for β i = 10, i ≠ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6 Directed graph D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.7 Adjacency matrix A of directed graph D . . . . . . . . . . . . . . . . . . . . . . . 873.1 The elliptic curve E : y 2 = x 3 + 1, ∆ = −432 . . . . . . . . . . . . . . . . . . . . 1003.2 The singular curve E : y 2 = x 3 , ∆ = 0 . . . . . . . . . . . . . . . . . . . . . . . . 1013.3 The singular curve E : y 2 = x 3 − 3x + 2, ∆ = 0 . . . . . . . . . . . . . . . . . . 1013.4 The elliptic curve E : y 2 = x 3 − 2x, ∆ = 512 . . . . . . . . . . . . . . . . . . . . 1013.5 The elliptic curve E : y 2 = x 3 − 2x + 2, ∆ = −1216 . . . . . . . . . . . . . . . . 1013.6 Addition law on the elliptic curve E : y 2 = x 3 + 1 . . . . . . . . . . . . . . . . . 1023.7 The hyperelliptic curve H : y 2 = (x + 2)(x + 1)x(x − 2)(x − 5/2) of genus 2 . . . 1065.1 A complex torus of dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2 The lattice corresponding to τ = 1+i√ 23. . . . . . . . . . . . . . . . . . . . . . . 148List of tables2.1 Distributions of γ i and δ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Distributions for γ ′ i and δ′ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Values of A i for 0 ≤ i ≤ 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4 Values of f d (1, . . . , 1, ∞, . . . , ∞) for d ≥ 1 . . . . . . . . . . . . . . . . . . . . . . 792.5 Computers’ description — Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.6 Computers’ description — Part II . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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
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Page 15: Contentsxv4 Efficient characterizat
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Page 28 and 29: 2 Introduction1 Mathematics and cry
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Page 33 and 34: Chapter 1Boolean functions incrypto
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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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6.2. 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 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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3. Multiplication complexe et polyn
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