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xivContents2.3.6 A combinatorial proposition of independent interest . . . . . . . . . . . . 332.4 A block splitting pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 General situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Combining variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3 One block: d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.4 A helpful constraint: min i (α i ) ≥ B − 1 . . . . . . . . . . . . . . . . . . . . 402.4.5 Analytic study: d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.6 Extremal value: β i = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 A closed-form expression for f d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.2 Splitting the sum into atomic parts . . . . . . . . . . . . . . . . . . . . . . 502.5.3 The residual term TX d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5.4 A polynomial expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.5.5 The coefficients a d,n(i 1,...,i n). . . . . . . . . . . . . . . . . . . . . . . . . . . 602.5.6 An additional relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.6 Asymptotic behavior: β i → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.6.1 The limit f d (∞, . . . , ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.6.2 The limit f d (1, ∞, . . . , ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.7 An inductive approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.7.1 Overflow and inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.7.2 Adding 0’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.7.3 Adding 1’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.8 Other works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.8.1 Cusick et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.8.2 Carlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.8.3 Towards a complete proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.9 Efficient test of the Tu–Deng conjecture . . . . . . . . . . . . . . . . . . . . . . . 862.9.1 The Tu–Deng algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.9.2 Necklaces and Lyndon words . . . . . . . . . . . . . . . . . . . . . . . . . 882.9.3 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89II Bent functions and point counting on algebraic curves 933 Bent functions and algebraic curves 953.1 Bent functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.1.1 Boolean functions in polynomial form . . . . . . . . . . . . . . . . . . . . 963.1.2 Walsh–Hadamard transform . . . . . . . . . . . . . . . . . . . . . . . . . . 973.1.3 Binary exponential sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.1.4 Dickson polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2 Algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2.1 Elliptic curves over perfect fields . . . . . . . . . . . . . . . . . . . . . . . 993.2.2 Elliptic curves over finite fields . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.3 Hyperelliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.2.4 Point counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Contentsxv4 Efficient characterizations for bentness 1094.1 Hyper-bentness characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.1.1 Monomial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.1.2 Functions with multiple trace terms . . . . . . . . . . . . . . . . . . . . . 1114.2 Reformulation in terms of cardinalities of curves . . . . . . . . . . . . . . . . . . 1144.2.1 Kloosterman sums and elliptic curves . . . . . . . . . . . . . . . . . . . . 1144.2.2 Exponential sums and hyperelliptic curves . . . . . . . . . . . . . . . . . . 1154.3 Divisibility of binary Kloosterman sums . . . . . . . . . . . . . . . . . . . . . . . 1194.3.1 Classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.2 Using torsion of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . 1204.4 Finding specific values of binary Kloosterman sums . . . . . . . . . . . . . . . . . 1214.4.1 Generic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.4.2 Zeros of binary Kloosterman sums . . . . . . . . . . . . . . . . . . . . . . 1224.4.3 Implementation for the value 4 . . . . . . . . . . . . . . . . . . . . . . . . 1224.5 Experimental results for m even . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124III Complex multiplication and class polynomials 1295 Complex multiplication and elliptic curves 1315.1 Further background on elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 1325.1.1 Morphisms between algebraic curves . . . . . . . . . . . . . . . . . . . . . 1325.1.2 Divisors of algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.1.3 Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.1.4 Reduction of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.2 Elliptic curves over the complex numbers . . . . . . . . . . . . . . . . . . . . . . 1395.2.1 Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2.2 Orders in imaginary quadratic fields . . . . . . . . . . . . . . . . . . . . . 1435.2.3 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.3 Elliptic curves with complex multiplication . . . . . . . . . . . . . . . . . . . . . 1475.3.1 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.2 Hilbert class polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3.3 The main theorem of complex multiplication . . . . . . . . . . . . . . . . 1495.3.4 Computing the Hilbert class polynomial . . . . . . . . . . . . . . . . . . . 1525.3.5 Shimura’s reciprocity law and class invariants . . . . . . . . . . . . . . . . 1535.4 Elliptic curves in cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.4.1 Curves with a given number of points . . . . . . . . . . . . . . . . . . . . 1535.4.2 The MOV/Frey–Rück attack . . . . . . . . . . . . . . . . . . . . . . . . . 1555.4.3 Identity-based cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . 1566 Complex multiplication in higher genera 1596.1 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.1.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . 1616.1.2 Theta functions and Riemann forms . . . . . . . . . . . . . . . . . . . . . 1616.1.3 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.1.4 Picard variety and polarizations . . . . . . . . . . . . . . . . . . . . . . . 1646.1.5 Homomorphisms and the Rosati involution . . . . . . . . . . . . . . . . . 1676.2 Class groups and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
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: ContentsList of symbols and notatio
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
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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38 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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