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212 IndexForm class group . . . . . . . . . 146Form class semigroup . . . . . . . 146Picard group . . . . . see Class groupProper class semigroup . . . . 144, 169Class numberClass number . . . 104, 126, 144, 146Kronecker class number . 104, 126, 144,146Proper class number . . . . . . . 144Class polynomial . . . . . . . . . . 153, 154Hilbert class polynomial . . . . . 148Igusa class polynomial . . . . 182, 184CM abelian variety . . . . . . . . . 174, 175a-multiplication . . . . . . . . . . 178a-transform . . . . . . . . . . . . 178Classification up to isomorphism . 178Dual . . . . . . . . . . . . . . . . 176Polarization . . . . . . . . . . . . 177Principal . . . . . . . . . . . . 174, 179Simple . . . . . . . . . . . . . . . 174Type . . . . . . . . . . . . . . 174, 175CM algebra . . . . . . . . . . . . . . . 172CM field . . . . . . . . . . . . . . . . . 172CM method . . . . . . . . . . . . . 154, 156CM type . . . . . . . . . . . . . . . . 173Equivalence . . . . . . . . . . . . 173Primitive . . . . . . . . . . . . 173, 174Simple . . . . . . . . . . see PrimitiveType norm . . . . . . . . . . . . . 173Type trace . . . . . . . . . . . . . 173Type transfer . . . . . . . . . . . 180Complementary lattice . . . see Trace dualComplex Lie group . . . . . . . . . 142, 161Exponential map . . . . . . . . . 161Complex torus . . . . . . . . 140, 161, 175Dual torus . . . . . . . . . . . . . 165Eisenstein series . . . . . . . . . . 142Riemann conditions . . . . . . . . 163Weierstraß ℘-function . . . . . . . 142Computation of the Hilbert class polynomialComplex analytic method . . . . 152CRT method . . . . . . . . . . . . 153p-adic method . . . . . . . . . . . 152Computation of the Igusa class polynomialsComplex analytic method . . . . 184CRT method . . . . . . . . . . . . 184p-adic method . . . . . . . . . . . 184Cryptographic property of Boolean functions8Algebraic degree . . . . . . . . . 10, 97Algebraic immunity . . . . . . . . . 10Balancedness . . . . . . . . . . . . . . 9Bentness . . . . . . see Bent functionHyper-bentnesssee Hyper-bent functionNonlinearity . . . . . . . . . . . . . 11Resiliency . . . . . . . . . . . . 9, 10Semi-bentness see Semi-bent functionCyclotomic character . . . . . . . . . . 180Cyclotomic class . . . . . . . . . . . 22, 88Coset leader . . see Cyclotomic leaderCyclotomic leader . . . . . 88, 97, 117Equivalence . . . . . . . . . . . . . 22Cyclotomic coset . . . see Cyclotomic classCyclotomic polynomial . . . . . . . . . 156DDedekind η function . . . . . . . . . . 153Dedekind ring . . . . . . . . . . . . 143, 168Dickson polynomial . . . . . . . . 99, 113Dillon criterion . . . . . . . . . . . . . 110using elliptic curves . . . . . . . . 114Discriminant . . . . . . . . . . . . . . 156of a binary quadratic form . . . . 146of a number field . . . . . . . . . 145of a Weierstraß equation . . . 100, 140of an order . . . . . . . . . . . 104, 105Divisibility of Kloosterman sumsClassical approach . . . . . . . . . 119using elliptic curves . . . . . . . . 120Divisor . . . . . . . . . . . . . . . . . 134Algebraic equivalence . . . . . . . 164Ample . . . . . . . . . . . . . . . 164Degree . . . . . . . . . . . . . . . 134Group of . . . . . . . . . . . . . . 134Linear equivalence . . . . . . . 134, 164of a function . . . . . . . . . . . . 134Principal . . . . . . . . . . . . . . 134Support . . . . . . . . . . . . . . 134Double η quotient . . . . . . . . . . . 153EElliptic curve . . . . . . . . . . 99, 114, 161Addition law . . . . . . . . . . . . 100Bad reduction . . . . . . . . . 139, 149Canonical lift . . . . . . . . . . . 107Chord-and-tangent lawsee Addition lawCM curve . . . . . . . . . . . . . 154Discriminant . . . . . . . . . . 100, 140
Index 213Division polynomial . . . . . . 103, 105Endomorphism ring . . . . . . . . 103Frobenius endomorphism see FrobeniusendomorphismGood reduction . . . . . . . . 139, 149Isogeny . . . . . . . . . . . . . . . 100j-invariant . . . . . . . see j-invariantJacobian variety . . . . . . . . . . 135Ordinary . . . . . . . . . . . . 104, 105Pairing . . . . . . . . . . . . . . . 135Point at infinity . . . . . . . . . . . 99Quadratic twist . . . . . . . . 105, 120Rational point . . . . . . . . . . . 100SEA algorithm . . . . . . . . . . . 107Subfield curve . . . . . . . . . . . 154Supersingular . . . . . . 104, 105, 156Torsion subgroup . . . . . . . 103, 122Weierstraß equation . . . 99, 105, 120with complex multiplication . 103, 147Endomorphism ringAnalytic representation . see Complexrepresentationas an ideal quotient . . . . . . . . 176Classification . . . . . . . . . 103, 168Complex representation . . . . . . 167of a CM abelian variety . . . . 174, 176of an elliptic curve . . . . . . . . 103Rational representation . . . . . . 167Rosati involution . . . . . . . . . 167Eulerian number . . . . . . . . . . . 48, 55Exponential sum . . . . . . . . . . . . 113Cubic sum . . . . . . . . . . . . . . 98Kloosterman sum . . . . . . . . . . 98using hyperelliptic curves . . . 115, 116FFamily of Boolean functionsCarlet and Feng . . . . . . . . . . . 12Dillon . . . . . . . . . . . . . . . . . 13Jin et al. . . . . . . . . . . . . . . . 17Tang, Carlet and Tang . . . . . . . 15Tu and Deng I . . . . . . . . . . . . 13Tu and Deng II . . . . . . . . . . . 13Tu and Deng III . . . . . . . . . . . 14Field trace . . . . . . . . . . . . . . . . 96Finite field . . . . . . . . . . . . . . . 104of even characteristic . . 99, 105, 106Fractional ideal . . . . . . . . . . . 143, 168Colon . . . . . . . . . . . see QuotientInverse . . . . . . . . . . . . . 144, 168Invertible . . . . . . . . . . . . 144, 168Principal . . . . . . . . . . . . . . 144Projective . . . . . . . . . . . . . 144Proper . . . . . . . . . . . . . 144, 168Quotient . . . . . . . . . . . . . . 144Singular . . . . . . . . . . . . . . 169Frobenius endomorphism . . . . . 104, 154Lift to characteristic zero . . . 150, 179Trace . . . . . . . . 104, 105, 107, 126GGAGA principle . . . . . . . . . . . . 166Gaussian hypergeometric series . . . 67, 69Euler’s transformation . . . . . . . 77Linear transformations . . . . . . . 77Pfaff’s transformation . . . . . . . . 77Quadratic transformation . . . . . . 71Geometrically distributed variable . 68, 78HHamming weightof a Boolean function . . . . . . 9, 96of an integer . . . . . . . . . . . . . 21Hermitian form . . . . . . . . . . . . . 162Hyper-bent function . . . . . . . . . 96, 98Charpin–Gong criterion . . . . . . 113Dillon criterion . . . . . . . . . . 110Mesnager criterion I . . . . . . . . 111Mesnager criterion II . . . . . . . 113Hyperelliptic curve . . . . . . . . . 105, 115Artin–Schreier curve . . . . . . . 106Imaginary . . . . . . . . . . . . . 106Moduli space . . . . . . . . . . . 181Point at infinity . . . . . . . . . . 106IIdèle . . . . . . . . . . . . . . . . . . . 172Igusa invariants . . . . . . . . . . . . . 181IsogenyDegree . . . . . . . . . . . . . . . 103Dual . . . . . . . . . . . . . . . . 164of abelian varieties . . . . . . . . 163of elliptic curves . . . . . . . . . . 100Trace . . . . . . . . . . . . . . 103, 104Jj-invariant . . . . . . . . 100, 105, 140, 154Integrality . . . . . . . . . . . . . 149q-expansion . . . . . . . . . . . . 152
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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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Part IIIComplex multiplication and
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Chapter 6Complex multiplication in
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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
Page 226 and 227: 200 Bibliography[132] Marc Hindry a
Page 228 and 229: 202 Bibliography[161] Philippe Lang
Page 230 and 231: 204 Bibliography[191] Willi Meier a
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
Page 240 and 241: 214 IndexKKloosterman sum . . . . .
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