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xviiiLists of figures, tables and algorithms2.7 Timings for checking the Tu–Deng conjecture . . . . . . . . . . . . . . . . . . . . 914.1 Families of hyper-bent and semi-bent functions for K m (a) = 0 . . . . . . . . . . . 1114.2 Families of (hyper-)bent functions for K m (a) = 4 . . . . . . . . . . . . . . . . . . 1124.3 Families of semi-bent functions for K m (a) = 4 . . . . . . . . . . . . . . . . . . . . 1124.4 Test of bentness for m even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.5 The fourteen cyclotomic classes such that K 16 (a) = 4 . . . . . . . . . . . . . . . . 127A.1 Coefficients a 1,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.2 Coefficients a 2,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.3 Coefficients a 3,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.4 Coefficients a 4,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.5 Coefficients a 5,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.6 Coefficients a 6,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189A.7 Coefficients a 7,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189A.8 Coefficients a 8,n(i 1,...,i n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190List of algorithms2.1 Tu–Deng algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.2 Iterative generation of necklaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1 Finding the value 4 of binary Kloosterman sums for m odd . . . . . . . . . . . . . 1234.2 Testing bentness for m even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 Miller’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.2 Computation of the Hilbert class polynomial . . . . . . . . . . . . . . . . . . . . . 1525.3 The CM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.4 Cornacchia’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.5 The MOV/Frey–Rück Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.1 Computation of the Picard group of a non-maximal order . . . . . . . . . . . . . . 1726.2 Computation of the isomorphism classes of p.p.a.v. with CM by a given order . . 1836.3 Computation of the Igusa class polynomials of a non-maximal order . . . . . . . . 184
List of symbols and notationGeneralN Semiring of non-negative integers . . . . . . . . . . . . . . . . . . . . . . . xxiiiZ Ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiQ Field of rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiR Field of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiC Field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiK, L A perfet field or a number field . . . . . . . . . . . . . . . . . . . . . . . . xxiiiK An algebraic closure of K . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiia, b Fractional ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiip, q Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiF q Finite field with q elements . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii#S Cardinality of the set S . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiip A prime number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiq A prime number’s power . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiil A prime number different from p . . . . . . . . . . . . . . . . . . . . . . . xxiiiCombinatorics( )li 1,...,i n( nk)〈 ki〉[ ij]Multinomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Eulerian number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Unsigned Stirling number of the first kind . . . . . . . . . . . . . . . . . . . 54A i Sum of Eulerian numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B i Bernoulli number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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: List of figures1.1 The filter model
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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42 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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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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