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108 Chapter 3. Bent functions and algebraic curvesTheorem 3.2.17 ([275, Theorem 4.4.1]). Let H be an imaginary hyperelliptic curve of genus gdefined over F 2 m. There exists an algorithm to compute the cardinality of H inbit operations and O(g 4 m 3 ) memory.O(g 3 m 3 (g 2 + log 2 m log log m) log gm log log gm)A slightly stronger result applies for Artin–Schreier curves.Theorem 3.2.18 ([275, Theorem 4.3.1]). Let H be an Artin–Schreier curve of genus g definedover F 2 m. There exists an algorithm to compute the cardinality of H inbit operations and O(g 3 m 3 ) memory.O(g 3 m 3 (g 2 + log 2 m log log m) log gm log log gm)In particular, the complexities of these algorithms are polynomial in the genus of the curve.
Chapter 4Efficient characterizations forbentnessNo constaba el nombre del heresiarca, pero sí lanoticia de su doctrina, formulada en palabras casiidénticas a las repetidas por él, aunque — tal vez— literariamente inferiores. Él había recordado:Copulation and mirrors are abominable. El textode la Enciclopedia decía: «Para uno de esosgnósticos, el visible universo era una ilusión o(más precisamente) un sofisma. Los espejos y lapaternidad son abominables (mirrors andfatherhood are abominable) porque lo multiplicany lo divulgan».FiccionesJorge Luis Borges [22]Contents4.1 Hyper-bentness characterizations . . . . . . . . . . . . . . . . . . . . 1104.1.1 Monomial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.1.2 Functions with multiple trace terms . . . . . . . . . . . . . . . . . . . 1114.2 Reformulation in terms of cardinalities of curves . . . . . . . . . . 1144.2.1 Kloosterman sums and elliptic curves . . . . . . . . . . . . . . . . . . 1144.2.2 Exponential sums and hyperelliptic curves . . . . . . . . . . . . . . . . 1154.3 Divisibility of binary Kloosterman sums . . . . . . . . . . . . . . . . 1194.3.1 Classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.2 Using torsion of elliptic curves . . . . . . . . . . . . . . . . . . . . . . 1204.4 Finding specific values of binary Kloosterman sums . . . . . . . . 1214.4.1 Generic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.4.2 Zeros of binary Kloosterman sums . . . . . . . . . . . . . . . . . . . . 1224.4.3 Implementation for the value 4 . . . . . . . . . . . . . . . . . . . . . . 1224.5 Experimental results for m even . . . . . . . . . . . . . . . . . . . . 124
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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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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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