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100 Chapter 3. Bent functions and algebraic curvesa quantity called its discriminant 5 In fact, a Weierstraß equation describes the affine part of asmooth, or non-singular, projective curve — which is then an elliptic curve — if and only if itsdiscriminant is non-zero [244, Proposition III.1.4]. The affine parts of such curves defined overthe real numbers, with discriminants zero and non-zero, are depicted in Figures 3.1, 3.2, 3.3, 3.4,and 3.5.Figure 3.1: The elliptic curve E : y 2 = x 3 + 1, ∆ = −432Over an algebraically closed field, elliptic curves are classified up to isomorphism by thej-invariant, which can be defined as a rational function of the coefficients of the curve 6 [244,Proposition III.1.4]. Furthermore, for any j 0 in K, the algebraic closure of K, there is a curvedefined over K(j 0 ) with j-invariant j 0 [244, Proposition III.1.4].For a given extension L of K, a point is said to be rational if its coordinates lie in L (andnot in a larger extension). There is a composition law, usually additively denoted, on the set ofrational points giving it a group structure [244, Section III.2]. It can be explicitly described bythe so-called “chord-and-tangent” law and is depicted in Figure 3.6. The point at infinity O E isthe neutral element for the addition law. Multiplication by an integer n on E, that we denote by[n], can then be naturally defined.The following result shows that any rational map between curves is the composition of atranslation and a homomorphism.Definition 3.2.2 (Isogeny). Let E 1 and E 2 be two elliptic curves defined over K and φ amorphism between them. We say that φ is an isogeny if φ(O E1 ) = O E2 .5 The expression of the discriminant using the coefficients a 1 , a 3 , a 2 , a 4 and a 6 is quite complicated and we willnot give it here. A simpler expression exists in characteristic different from 2 and 3 and is given in Definition 5.2.1.6 The expression of the j-invariant using the coefficients a 1 , a 3 , a 2 , a 4 and a 6 is also quite complicated andwill not be given here. A simpler expression exists as well in characteristic different from 2 and 3 and is given inDefinition 5.2.2.
3.2. Algebraic curves 101Figure 3.2: The singular curve E : y 2 = x 3 ,∆ = 0Figure 3.3: The singular curve E : y 2 =x 3 − 3x + 2, ∆ = 0Figure 3.4: The elliptic curve E : y 2 =x 3 − 2x, ∆ = 512Figure 3.5: The elliptic curve E : y 2 =x 3 − 2x + 2, ∆ = −1216
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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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150 Chapter 5. Complex multiplicati
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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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