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214 IndexKKloosterman sum . . . . . . . . . . 98, 114Divisibility . . . . . see Divisibility ofKloosterman sumsGeneric search algorithm . . . . . 121using elliptic curves . . . . . . . . 114Value 0 . . . . . . . . . . . . . . . 122Value 4 . . . . . . . . . . . . . 122, 124Kronecker congruence relation . . 149, 179Kronecker symbol . . . . . . . . . 105, 145LLaguerre polynomial . . . . . . . . . . . 74Lattice . . . . . . . . . . . . . . . . 103, 175Frobenius basis . see Symplectic basisin the complex numbers . . . . . 140Multiplier ring . . . . . . . . . 140, 143Period matrix . . . . . . . . . . . 163Symplectic basis . . . . . . . . . . 163Local field . . . . . . . . . . . . . . . . 138Lyndon word . . . . . . . . . . . . . . . 88MMain theorem of complex multiplicationfor elliptic curves . . . . . . . . . 152over the rationals . . . . . . . . . 180over the reflex field . . . . . . . . 179Mesnager criterion I . . . . . . . . . . 111using elliptic curves . . . . . . . . 115Mesnager criterion II . . . . . . . . . . 113using hyperelliptic curves . . . 117, 118Modular function . . . . . . . 142, 149, 153Modular group . . . . . . . . . . . . . 141Modular integerBinary not . . . . . . . . . . . . . . 22Block splitting pattern . . . . . 35, 37Carries . . . . . . . . . . . . . . . . 23Hamming weight . . . . . . . . . . 21Length . . . . . . . . . . . . . . . . 80Modular polynomial . . . . . . . . . . 154MorphismDegree . . . . . . . . . . . . . . . 133Inseparable . . . . . . . . . . . . . 133Ramification index . . . . . . . . 133Separable . . . . . . . . . . . . . 133Multinomial coefficient . . . . . . . . . . 48NN-systems . . . . . . . . . . . . . . . . 153Necklace . . . . . . . . . . . . . . . . . . 88Aperiodic necklace . see Lyndon wordIterative generation . . . . . . . . . 89Normalization kernel . . . . . . . . . . 169Number field . . . . . . . . . . . . . . 103Imaginary quadratic . . . . . . . 103Ring of integers . . . . . . . . . . 143OOrder . . . . . . . . . . . . . . . . . . 103Codifferent . . . . . . . see Trace dualConductor . . . . . 105, 143, 169, 172Discriminant . . . . . . . . . . . . 105in an imaginary quadratic field105, 143Integral closure . . . . . . . . . . 168Maximal . . . . . . . . . . . . . . 160Normalization . . see Integral closurePPairing . . . . . . . . . . . . . . . 136, 155Ate pairing . . . . . . . . . . . . . 155Eta pairing . . . . . . . . . . . . . 155Tate pairing . . . . . . . . . . . . 136Weil pairing . . . . . . . . . . 137, 156Perfect field . . . . . . . . . . . . . . . . 99Pochhammer symbol . . . . . . . . . . . 68Poincaré upper halfplane . . . . . . . 140Fundamental domain . . . . . . . 141Point countingl-adic algorithms . . . . . . . . . 107SEA algorithm . . . . . . . . . 107p-adic algorithms . . . . . . . . . 107Canonical lift methods . . . 107, 121Cohomological methods . . . . 107Deformation theory methods . 107Proved cases of the Tu–Deng conjectureAsymptotic case . . . . . . . . . . . 68Cyclotomic case . . . . . . . . . . . 33Extremal case . . . . . . . . . . 45, 46One block case . . . . . . . . . . . . 39Tang–Carlet–Tang conjecture . . . 24Two blocks case . . . . . . . . . . . 45Zero case . . . . . . . . . . . . . . . 30QQuaternion algebra . . . . . . . . . . . 103RReflex field . . . . . . . . . . . . . . . 173Riemann form . . . . . 162, 163, 165, 168Pfaffian . . . . . . . . . . . . . . . 163Rising factorial . . . . . . . . . . . . . . 68
Index 215SSemi-bent function . . . . . . . . . . . . 98Semicharacter . . . . . . . . . . . . . . 162Shimura reciprocity law . . . . . . 153, 184Shimura–Taniyama formula . . . . . . 179Siegel upper half-space . . . . . . . . . 163Stirling number . . . . . . . . . . . . . . 54Stream cipher . . . . . . . . . . . . . . . . 7Combiner model . . . . . . . . . . . . 9Filter model . . . . . . . . . . . . . . 9Symmetric cryptosystem . . . . . . . . . . 7One-time pad . . . . . . . . . . . . . 8S-box . . . . . . . . . . . . . . . . . 96Stream cipher . . . see Stream cipherSymplectic group . . . . . . . . . . . . 166TTensor product . . . . . . . . . . . . . 103Theta function . . . . . . . . . . . 153, 161Associated Riemann form . . . . 162Equivalence . . . . . . . . . . . . 162Factor of automorphy see MultiplicatorMultiplicator . . . . . . . . . . . . 162Riemann theta function . . . . . . 181Theta constant . . . . . . . . 181, 184Theta null value . . see Theta constantTrace dual . . . . . . . . . . . . . . . . 176Transfer-matrix method . . . . . . . . . 86Tu–Deng conjecture . . . . . . . 12, 21, 29Block splitting pattern . . . . . 35, 37Closed-form expression . . . . . . . 47Constrained case . . . . . . . . . . 40Extremal conjecture . . . . . . . . . 47Jin et al. generalization . . . 16, 21, 23Reformulation using carries . . . . 31Tang–Carlet–Tang conjecture . 15, 21Tang–Carlet–Tang generalization15, 21,23Tu–Deng algorithm . . . . . . . . . 88UUnit group . . . . . . . . . . . . . . . 171WWalsh–Hadamard transform . . . . . . . 97Fast Walsh–Hadamard transform . 98,122, 125
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2012-ENST-003EDITE de ParisDoctorat
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Jean-Pierre Flori: Boolean function
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AbstractThe core of this thesis is
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AcknowledgmentsVladimir. — Quand
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ContentsList of symbols and notatio
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Contentsxv4 Efficient characterizat
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List of figures1.1 The filter model
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List of symbols and notationGeneral
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List of symbols and notationxxil(t)
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List of symbols and notationxxiiidi
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List of symbols and notationxxvu
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2 Introduction1 Mathematics and cry
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4 Introductiongiving more general b
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Chapter 1Boolean functions incrypto
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1.1. Cryptographic criteria for Boo
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1.2. Families of Boolean functions
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20 Chapter 2. On a conjecture about
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Part IIBent functions and pointcoun
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96 Chapter 3. Bent functions and al
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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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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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Page 218 and 219: 192 Bibliography[13] Elwyn Ralph Be
Page 220 and 221: 194 Bibliography[42] Claude Carlet,
Page 222 and 223: 196 Bibliography[72] Cunsheng Ding,
Page 224 and 225: 198 Bibliography[102] David Freeman
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Page 228 and 229: 202 Bibliography[161] Philippe Lang
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Page 232 and 233: 206 Bibliography[222] Oscar Seymour
Page 234 and 235: 208 Bibliography[254] Marco Streng.
Page 236 and 237: 210 Bibliography[286] Chia-Fu Yu. T
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