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Table A.3: Coefficients for d = 3, normalized by (1/27) = (1/3 3 )4ˆ 3 4ˆ 2 4ˆ 1 4ˆ 0(2, 0, 0) 1 (2, 0) −1 (2, ) 1(1, 1, 0) 2 (1, 1) −2(1, 0, 0) −11 (1, 0) 5 (1, ) 1(0, 0, 0) 64/3 (0, 0) −4/3 (0, ) −2/3 () 35/3Table A.4: Coefficients for d = 4, normalized by (1/81) = (1/3 4 )4ˆ 4 4ˆ 3 4ˆ 2 4ˆ 1 4ˆ 0(3, 0, 0, 0) −1/3 (3, 0, 0) 1/3 (3, 0) −1/3 (3, ) 1/3(2, 1, 0, 0) −1 (2, 1, 0) 1 (2, 1) −1(1, 1, 1, 0) −2 (1, 1, 1) 2(2, 0, 0, 0) 23/3 (2, 0, 0) −14/3 (2, 0) 5/3 (2, ) 4/3(1, 1, 0, 0) 46/3 (1, 1, 0) −28/3 (1, 1) 10/3(1, 0, 0, 0) −416/9 (1, 0, 0) 119/9 (1, 0) 16/9 (1, ) 11/9(0, 0, 0, 0) 1808/27 (0, 0, 0) −80/27 (0, 0) −28/27 (0, ) −26/27 () 971/27Table A.5: Coefficients for d = 5, normalized by (1/243) = (1/3 5 )4ˆ 5 4ˆ 4 4ˆ 3 4ˆ 2 4ˆ 1 4ˆ 0(4, 0, 0, 0, 0) 1/12 (4, 0, 0, 0) −1/12 (4, 0, 0) 1/12 (4, 0) −1/12 (4, ) 1/12(3, 1, 0, 0, 0) 1/3 (3, 1, 0, 0) −1/3 (3, 1, 0) 1/3 (3, 1) −1/3(2, 2, 0, 0, 0) 1/2 (2, 2, 0, 0) −1/2 (2, 2, 0) 1/2 (2, 2) −1/2(2, 1, 1, 0, 0) 1 (2, 1, 1, 0) −1 (2, 1, 1) 1(1, 1, 1, 1, 0) 2 (1, 1, 1, 1) −2(3, 0, 0, 0, 0) −59/18 (3, 0, 0, 0) 41/18 (3, 0, 0) −23/18 (3, 0) 5/18 (3, ) 13/18(2, 1, 0, 0, 0) −59/6 (2, 1, 0, 0) 41/6 (2, 1, 0) −23/6 (2, 1) 5/6(1, 1, 1, 0, 0) −59/3 (1, 1, 1, 0) 41/3 (1, 1, 1) −23/3(2, 0, 0, 0, 0) 161/4 (2, 0, 0, 0) −69/4 (2, 0, 0) 13/4 (2, 0) 7/4 (2, ) 9/4(1, 1, 0, 0, 0) 161/2 (1, 1, 0, 0) −69/2 (1, 1, 0) 13/2 (1, 1) 7/2(1, 0, 0, 0, 0) −9421/54 (1, 0, 0, 0) 1933/54 (1, 0, 0) 209/54 (1, 0) 79/54 (1, ) 119/54(0, 0, 0, 0, 0) 16832/81 (0, 0, 0, 0) −560/81 (0, 0, 0) −160/81 (0, 0) −92/81 (0, ) −142/81 () 8881/81
Table A.6: Coefficients for d = 6, normalized by (1/729) = (1/3 6 )4ˆ 6 4ˆ 5 4ˆ 4 4ˆ 3 4ˆ 2 4ˆ 1 4ˆ 0(5, 0, 0, 0, 0, 0) −1/60 (5, 0, 0, 0, 0) 1/60 (5, 0, 0, 0) −1/60 (5, 0, 0) 1/60 (5, 0) −1/60 (5, ) 1/60(4, 1, 0, 0, 0, 0) −1/12 (4, 1, 0, 0, 0) 1/12 (4, 1, 0, 0) −1/12 (4, 1, 0) 1/12 (4, 1) −1/12(3, 2, 0, 0, 0, 0) −1/6 (3, 2, 0, 0, 0) 1/6 (3, 2, 0, 0) −1/6 (3, 2, 0) 1/6 (3, 2) −1/6(3, 1, 1, 0, 0, 0) −1/3 (3, 1, 1, 0, 0) 1/3 (3, 1, 1, 0) −1/3 (3, 1, 1) 1/3(2, 2, 1, 0, 0, 0) −1/2 (2, 2, 1, 0, 0) 1/2 (2, 2, 1, 0) −1/2 (2, 2, 1) 1/2(2, 1, 1, 1, 0, 0) −1 (2, 1, 1, 1, 0) 1 (2, 1, 1, 1) −1(1, 1, 1, 1, 1, 0) −2 (1, 1, 1, 1, 1) 2(4, 0, 0, 0, 0, 0) 1 (4, 0, 0, 0, 0) −3/4 (4, 0, 0, 0) 1/2 (4, 0, 0) −1/4 (4, 0) 0 (4, ) 1/4(3, 1, 0, 0, 0, 0) 4 (3, 1, 0, 0, 0) −3 (3, 1, 0, 0) 2 (3, 1, 0) −1 (3, 1) 0(2, 2, 0, 0, 0, 0) 6 (2, 2, 0, 0, 0) −9/2 (2, 2, 0, 0) 3 (2, 2, 0) −3/2 (2, 2) 0(2, 1, 1, 0, 0, 0) 12 (2, 1, 1, 0, 0) −9 (2, 1, 1, 0) 6 (2, 1, 1) −3(1, 1, 1, 1, 0, 0) 24 (1, 1, 1, 1, 0) −18 (1, 1, 1, 1) 12(3, 0, 0, 0, 0, 0) −745/36 (3, 0, 0, 0, 0) 391/36 (3, 0, 0, 0) −145/36 (3, 0, 0) 7/36 (3, 0) 23/36 (3, ) 55/36(2, 1, 0, 0, 0, 0) −745/12 (2, 1, 0, 0, 0) 391/12 (2, 1, 0, 0) −145/12 (2, 1, 0) 7/12 (2, 1) 23/12(1, 1, 1, 0, 0, 0) −745/6 (1, 1, 1, 0, 0) 391/6 (1, 1, 1, 0) −145/6 (1, 1, 1) 7/6(2, 0, 0, 0, 0, 0) 4840/27 (2, 0, 0, 0, 0) −6319/108 (2, 0, 0, 0) 365/54 (2, 0, 0) 323/108 (2, 0) 61/27 (2, ) 485/108(1, 1, 0, 0, 0, 0) 9680/27 (1, 1, 0, 0, 0) −6319/54 (1, 1, 0, 0) 365/27 (1, 1, 0) 323/54 (1, 1) 122/27(1, 0, 0, 0, 0, 0) −83909/135 (1, 0, 0, 0, 0) 26503/270 (1, 0, 0, 0) 1246/135 (1, 0, 0) 643/270 (1, 0) 271/135 (1, ) 1243/270(0, 0, 0, 0, 0, 0) 640 (0, 0, 0, 0, 0) −448/27 (0, 0, 0, 0) −112/27 (0, 0, 0) −16/9 (0, 0) −44/27 (0, ) −98/27 () 2993/9Table A.7: Coefficients for d = 7, normalized by (1/2187) = (1/3 7 )4ˆ 7 4ˆ 6 4ˆ 5 4ˆ 4 4ˆ 3 4ˆ 2 4ˆ 1 4ˆ 0(6, 0, 0, 0, 0, 0, 0) 1/360 (6, 0, 0, 0, 0, 0) −1/360 (6, 0, 0, 0, 0) 1/360 (6, 0, 0, 0) −1/360 (6, 0, 0) 1/360 (6, 0) −1/360 (6, ) 1/360(5, 1, 0, 0, 0, 0, 0) 1/60 (5, 1, 0, 0, 0, 0) −1/60 (5, 1, 0, 0, 0) 1/60 (5, 1, 0, 0) −1/60 (5, 1, 0) 1/60 (5, 1) −1/60(4, 2, 0, 0, 0, 0, 0) 1/24 (4, 2, 0, 0, 0, 0) −1/24 (4, 2, 0, 0, 0) 1/24 (4, 2, 0, 0) −1/24 (4, 2, 0) 1/24 (4, 2) −1/24(3, 3, 0, 0, 0, 0, 0) 1/18 (3, 3, 0, 0, 0, 0) −1/18 (3, 3, 0, 0, 0) 1/18 (3, 3, 0, 0) −1/18 (3, 3, 0) 1/18 (3, 3) −1/18(3, 2, 1, 0, 0, 0, 0) 1/6 (3, 2, 1, 0, 0, 0) −1/6 (3, 2, 1, 0, 0) 1/6 (3, 2, 1, 0) −1/6 (3, 2, 1) 1/6(2, 2, 2, 0, 0, 0, 0) 1/4 (2, 2, 2, 0, 0, 0) −1/4 (2, 2, 2, 0, 0) 1/4 (2, 2, 2, 0) −1/4 (2, 2, 2) 1/4(2, 2, 1, 1, 0, 0, 0) 1/2 (2, 2, 1, 1, 0, 0) −1/2 (2, 2, 1, 1, 0) 1/2 (2, 2, 1, 1) −1/2(2, 1, 1, 1, 1, 0, 0) 1 (2, 1, 1, 1, 1, 0) −1 (2, 1, 1, 1, 1) 1(1, 1, 1, 1, 1, 1, 0) 2 (1, 1, 1, 1, 1, 1) −2(5, 0, 0, 0, 0, 0, 0) −17/72 (5, 0, 0, 0, 0, 0) 67/360 (5, 0, 0, 0, 0) −49/360 (5, 0, 0, 0) 31/360 (5, 0, 0) −13/360 (5, 0) −1/72 (5, ) 23/360(4, 1, 0, 0, 0, 0, 0) −85/72 (4, 1, 0, 0, 0, 0) 67/72 (4, 1, 0, 0, 0) −49/72 (4, 1, 0, 0) 31/72 (4, 1, 0) −13/72 (4, 1) −5/72(3, 2, 0, 0, 0, 0, 0) −85/36 (3, 2, 0, 0, 0, 0) 67/36 (3, 2, 0, 0, 0) −49/36 (3, 2, 0, 0) 31/36 (3, 2, 0) −13/36 (3, 2) −5/36(3, 1, 1, 0, 0, 0, 0) −85/18 (3, 1, 1, 0, 0, 0) 67/18 (3, 1, 1, 0, 0) −49/18 (3, 1, 1, 0) 31/18 (3, 1, 1) −13/18(2, 2, 1, 0, 0, 0, 0) −85/12 (2, 2, 1, 0, 0, 0) 67/12 (2, 2, 1, 0, 0) −49/12 (2, 2, 1, 0) 31/12 (2, 2, 1) −13/12(2, 1, 1, 1, 0, 0, 0) −85/6 (2, 1, 1, 1, 0, 0) 67/6 (2, 1, 1, 1, 0) −49/6 (2, 1, 1, 1) 31/6(1, 1, 1, 1, 1, 0, 0) −85/3 (1, 1, 1, 1, 1, 0) 67/3 (1, 1, 1, 1, 1) −49/3(4, 0, 0, 0, 0, 0, 0) 1595/216 (4, 0, 0, 0, 0, 0) −947/216 (4, 0, 0, 0, 0) 461/216 (4, 0, 0, 0) −137/216 (4, 0, 0) −25/216 (4, 0) 25/216 (4, ) 137/216(3, 1, 0, 0, 0, 0, 0) 1595/54 (3, 1, 0, 0, 0, 0) −947/54 (3, 1, 0, 0, 0) 461/54 (3, 1, 0, 0) −137/54 (3, 1, 0) −25/54 (3, 1) 25/54(2, 2, 0, 0, 0, 0, 0) 1595/36 (2, 2, 0, 0, 0, 0) −947/36 (2, 2, 0, 0, 0) 461/36 (2, 2, 0, 0) −137/36 (2, 2, 0) −25/36 (2, 2) 25/36(2, 1, 1, 0, 0, 0, 0) 1595/18 (2, 1, 1, 0, 0, 0) −947/18 (2, 1, 1, 0, 0) 461/18 (2, 1, 1, 0) −137/18 (2, 1, 1) −25/18(1, 1, 1, 1, 0, 0, 0) 1595/9 (1, 1, 1, 1, 0, 0) −947/9 (1, 1, 1, 1, 0) 461/9 (1, 1, 1, 1) −137/9(3, 0, 0, 0, 0, 0, 0) −23021/216 (3, 0, 0, 0, 0, 0) 9611/216 (3, 0, 0, 0, 0) −2573/216 (3, 0, 0, 0) −37/216 (3, 0, 0) 163/216 (3, 0) 251/216 (3, ) 739/216(2, 1, 0, 0, 0, 0, 0) −23021/72 (2, 1, 0, 0, 0, 0) 9611/72 (2, 1, 0, 0, 0) −2573/72 (2, 1, 0, 0) −37/72 (2, 1, 0) 163/72 (2, 1) 251/72(1, 1, 1, 0, 0, 0, 0) −23021/36 (1, 1, 1, 0, 0, 0) 9611/36 (1, 1, 1, 0, 0) −2573/36 (1, 1, 1, 0) −37/36 (1, 1, 1) 163/36(2, 0, 0, 0, 0, 0, 0) 130997/180 (2, 0, 0, 0, 0, 0) −11399/60 (2, 0, 0, 0, 0) 1301/90 (2, 0, 0, 0) 262/45 (2, 0, 0) 63/20 (2, 0) 653/180 (2, ) 443/45(1, 1, 0, 0, 0, 0, 0) 130997/90 (1, 1, 0, 0, 0, 0) −11399/30 (1, 1, 0, 0, 0) 1301/45 (1, 1, 0, 0) 524/45 (1, 1, 0) 63/10 (1, 1) 653/90(1, 0, 0, 0, 0, 0, 0) −173017/81 (1, 0, 0, 0, 0, 0) 109904/405 (1, 0, 0, 0, 0) 18719/810 (1, 0, 0, 0) 3709/810 (1, 0, 0) 1039/405 (1, 0) 280/81 (1, ) 8387/810(0, 0, 0, 0, 0, 0, 0) 476416/243 (0, 0, 0, 0, 0, 0) −9856/243 (0, 0, 0, 0, 0) −2240/243 (0, 0, 0, 0) −784/243 (0, 0, 0) −512/243 (0, 0) −676/243 (0, ) −1970/243 () 244403/243
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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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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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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
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Page 197 and 198: 6.2. Class groups and units 171in [
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Page 205 and 206: 6.3. Complex multiplication 179pola
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Page 209 and 210: 6.4. Class polynomials for genus 2
Page 211: Appendices
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
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3. Multiplication complexe et polyn
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