here - Sites personnels de TELECOM ParisTech - TÃ©lÃ©com ParisTech

More documents

Recommendations

Info

182 Chapter 6. Complex multiplication in higher generay 2 = f(x) where f(x) is a separable polynomial of degree 6. If we denote its root by α 1 , . . . , α 6 ,then the homogeneous Igusa invariants can be defined using the compact notation of Streng [252,II.2.1] as∑I 2 = a 2 6 (12) 2 (34) 2 (56) 2 ,15∑I 4 = a 4 6 (12) 2 (23) 2 (31) 2 (45) 2 (56) 2 (64) 2 ,10∑I 6 = a 6 6 (12) 2 (23) 2 (31) 2 (45) 2 (56) 2 (64) 2 (14) 2 (25) 2 (36) 2 ,I 10 = a 10660∏(α i − α j ) 2 .i
6.4. Class polynomials for genus 2 183and Streng [252] for details; and to the paper of Freeman, Stevenhagen and Streng [102] for theordinary case, and that of Hitt O’Connor et al. [134] for the p-rank one case.We have seen that the CM order is preserved by the Galois action, so the correspondingIgusa class polynomials computed over all isomorphism classes of complex abelian varieties withcomplex multiplication by a given order will have rational coefficients. Moreover, invertible idealsare sent onto invertible ideals, so that we can restrict ourselves to such ideals, even though thefollowing algorithms can be naturally extended to orbits of non-invertible proper ideals, and getfactors of the Igusa class polynomials.The first algorithm we propose is thus a straightforward generalization of the constructionof the Hilbert class polynomial in dimension 1: one first enumerates all isomorphism classes ofp.p.a.v. for a given order O and given by an invertible ideal and then computes the correspondinginvariants analytically with sufficient precision. The Igusa class polynomials, which have rationalcoefficients, can then be recognized from their approximations. Precise description and analysisof such an algorithm have been done by Streng in his Ph.D. thesis [252] in the case of maximalorders and readily extend to the case of non-maximal orders. The enumeration of isomorphismclasses is described in Algorithm 6.2 which corresponds to the algorithm proposed by Streng inthe case of maximal orders [252, Algorithm II.3.1]. Its correctness is a consequence of the resultspresented in Section 6.3. A high-level description of the algorithm is given in Algorithm 6.3Algorithm 6.2: Computation of representatives of the isomorphism classes of p.p.a.v. withCM by an order O in a primitive CM quartic field and given by invertible idealsInput: A primitive quartic CM field K and an order OOutput: A triple (Φ, a, ξ) for each isomorphism class of p.p.a.v. with CM by O1 Compute a complete set T of representatives of the CM types Φ on K2 Compute a complete set U of representatives of O ∗ K 0/ N K/K0 (O ∗ )3 Compute a complete set I of representatives of the Picard group of O4 Initialize an empty list L = []5 foreach Ideal a ∈ I do6 if a ∗ a −1 is principal and generated by an element ξ ∈ K such that ξ 2 ∈ K 0 and ξ istotally negative then7 Append (a, ξ) to A8 Initialize an empty list M = []9 foreach Pair (a, ξ) ∈ L and unit u ∈ U do10 Append (Φ, a, uξ) to M where Φ is the unique CM type Φ such that I(φuξ) > 011 return The triples (Φ, a, uξ) ∈ M such that Φ ∈ Twhich is nothing but an extension of the algorithm of Streng [252, Algorithm II.11.1].A second version of this algorithm using the explicit description of the main theorem ofcomplex multiplication over the reflex field was developed by Streng [252, Chapter III]. The ideais that, if σ ∈ Aut(C/K r ), then it does not modify the type Φ and acts as multiplication by thereflex norm N Φ r(s −1 ) of the corresponding idèle s. This action can be explicitly described asmultiplication by N Φ r(a −1 ) where a is an ideal of O K r. Therefore, it is enough to find a triple(Φ, a, ξ) and then to compute the action of Pic(O K r) on it to build irreducible components of theIgusa class polynomials over K r corresponding to a single orbit under Aut(C/K r ). In fact, it canbe shown that any K r 0-conjugate of an Igusa invariant is also a K r -conjugate [252, Corollary I.9.3].Therefore, these irreducible components of the Igusa class polynomials have coefficients in K r 0.One should then recognize the coefficients in K r 0.
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 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 39 and 40:
Page 41 and 42:
Page 43:
Page 46 and 47:
20 Chapter 2. On a conjecture about
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 119:
Part IIBent functions and pointcoun
Page 122 and 123:
96 Chapter 3. Bent functions and al
Page 124 and 125:
98 Chapter 3. Bent functions and al
Page 126 and 127:
100 Chapter 3. Bent functions and a
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
110 Chapter 4. Efficient characteri
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 155:
Part IIIComplex multiplication and
Page 158 and 159: 132 Chapter 5. Complex multiplicati
Page 185 and 186: Chapter 6Complex multiplication in
Page 187 and 188: 6.1. Abelian varieties 1616.1 Abeli
Page 189 and 190: 6.1. Abelian varieties 163Theorem 6
Page 191 and 192: 6.1. Abelian varieties 165A homomor
Page 193 and 194: 6.1. Abelian varieties 1676.1.5 Hom
Page 195 and 196: 6.2. Class groups and units 169The
Page 197 and 198: 6.2. Class groups and units 171in [
Page 199 and 200: 6.3. Complex multiplication 173If t
Page 201 and 202: 6.3. Complex multiplication 175•
Page 203 and 204: 6.3. Complex multiplication 177We h
Page 205 and 206: 6.3. Complex multiplication 179pola
Page 207: 6.4. Class polynomials for genus 2
Page 211: Appendices
Page 214 and 215: Table A.3: Coefficients for d = 3,
Page 216 and 217: Table A.8: Coefficients for d = 8,
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 238 and 239: 212 IndexForm class group . . . . .
Page 240 and 241: 214 IndexKKloosterman sum . . . . .
Page 243 and 244: Résumé longFauĆ.Werd’ iĚ beru
Page 245 and 246: 1. Des fonctions booléennes et d
Page 253 and 254: 2. Fonctions courbes et comptage de
Page 259 and 260:
2. Fonctions courbes et comptage de
Page 261 and 262:
3. Multiplication complexe et polyn
Page 263 and 264:
Page 265 and 266:
Page 267:
show all

here - Sites personnels de TELECOM ParisTech - TÃ©lÃ©com ParisTech

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?