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60 Chapter 2. On a conjecture about addition modulo 2 k − 1Suppose now that d > 1. ThenS d ∅ =∑J⊂{1,...,d}S d ∅,J =∑J⊂{1,...,d−1}S d ∅,J +∑{d}⊂J⊂{1,...,d}= 2 −β dS d−1∅+ 2 −β 2β d+ 2 · 2 −β d− 3dS d−13∅= 2 · 4−β d+ 1 1 ∑ ∑3 3 d−1 2 #J 4 − j∈J βj= 1 3 d ∑J⊂{1,...,d}J⊂{1,...,d−1}2 #J 4 − ∑ j∈J βjusing the induction hypothesis on d which proves the proposition for I = ∅.Suppose now that I = {j 0 + j 1 + 1, . . . , d} is not empty. It implies that d > 1 and thatS d I ==∑J⊂{1,...,j 0+j 1}∑J⊂{1,...,j 0+j 1}= 2j23 j2 d∏i=j 0+j 1+1= 2j23 j2 d∏i=j 0+j 1+1S d I,J2d∏j23 j2i=j 0+j 1+1⎛(1 − 4 −βi ) ⎝((1 − 4 −βi ) S d−j2((1 − 4 −βi ) S d−j2∑J⊂{1,...,j 0+j 1}∅− Ξ d I)I,JS d−j2I,J.− Ξ d I,J−)∑S d ∅,JJ⊂{1,...,j 0+j 1}Proposition 2.5.1 is a simple corollary to the last proposition and hence is finally proven.2.5.5 The coefficients a d,n(i 1 ,...,i n)We can now properly define the coefficients appearing in the multivariate polynomials.Definition 2.5.16. We denote by a d,n(i 1,...,i n) the coefficient of P n d (x 1, . . . , x n ) of multi-degree(i 1 , . . . , i n ) normalized by 3 d .It should be remembered that d is the index of the function f d , n represents the number of β j ’sappearing in the exponential in front of the polynomial Pdn and the i j’s the degrees (potentially0) in each of these β j ’s of a monomial appearing in Pd n . This does not depend on the ordering ofthe i j ’s because Pdn is symmetric, so we can suppose that i 1 ≥ · · · ≥ i n . Moreover, restrictions onthe degrees imply that a d,n(i = 0 as soon as ∑ n1,...,i n) j=1 i j ≥ d − 1.Lemma 2.5.17. For d ≥ 1,f d+1 (β 1 , . . . , β d , 0) = f d (β 1 , . . . , β d ) .Proof. This is obvious from the expression of f d (β 1 , . . . , β d ) as a sum.Hence, we obtain a simple recursion relation on the coefficients of P n d .Ξ d I,J⎞⎠
2.5. A closed-form expression for f d 61Corollary 2.5.18. For d ≥ 2 and 0 ≤ n < d,a d,n+1(i 1,...,i n,0) + ad,n (i 1,...,i n) = 3ad−1,n (i 1,...,i n) .We now give closed-form expressions for the coefficients a d,n(i 1,...,i n) .Here is a first simple proposition proving an experimental observation.Proposition 2.5.19. If d ≥ 2, then 7 a d,d(1,...,1,0) = (−1)d+1 2 and a d,d−1(1,...,1) = (−1)d 2.Proof. From Propositions 2.5.15 and 2.5.14, the monomials of multi-degree (1, . . . , 1, 0) in P d−1dand P d d come from Sd {d} , and within it from Sd (1,...,1,2) . MoreoverS(1,...,1,2) d = 2 ()3 (1 − 4−β d) S d−1(1,...,1) − Ξd (1,...,1,2)so it is clear that a d,d(1,...,1,0) = −ad,d−1 (1,...,1). The coefficient ad,d−1(1,...,1,0) must come from Ξd (1,...,1,2) whichis equal toΞ d (1,...,1,2) = 13 d−1 Πd (1,...,1,2) = 1d−1∏ 1 − (1 + 3β i )4 −βi3 d−1 ,3and finallya d,d−1i=0(1,...,1,0) = 2 1−3d 3 3 d−1 (−1)d−1 = (−1) d 2 .More generally, we have the following expression for a monomial of total degree d − 1.Proposition 2.5.20. Suppose that n ≥ 1 and i 1 + . . . + i n = d − 1. Then,a d,n(i 1,...,i n) = 2 (−1)n+1i 1 ! . . . i n !.Proof. We can suppose that i 1 ≥ · · · ≥ i j1 ≠ 0 > i j1+1 = 0 = · · · = i n . This notation is consistentbecause the different constraints on the degrees show that such a monomial can only appear inSX d when j 1 = # {i j | i j ≠ 0} and j 2 = d − j 1 , so that this coefficient only comes fromS d (1,...,1,2,...,2) = 2j23 j2 d∏i=j 1+1((1 − 4 −βi ) S d−j2(1,...,1) − Ξd (1,...,1,2,...,2)).Moreover, within Ξ d (1,...,1,2,...,2)it can only appear in Σd−i(1,...,1,2,...,2)when i = 0. Looking at theexpression of Π d X , we have the following expression[ ] (a d,n(i = 1,...,i n) (−1)n−j1 (−2) (−1)j1 j2 − 1 d − 1 − j 1(j 2 − 1)! d − 1 − j 1 i 1 − 1, . . . , i j1 − 1= 2 (−1)n+1(j 2 − 1)![ ] (j2 − 1j 2 − 1= 2 (−1)n+1i 1 ! . . . i j1 ! = 2 (−1)n+1i 1 ! . . . i n !7 The first equality is also true for d = 1.j 2 − 1i 1 − 1, . . . , i j1 − 1.) j1∏j=11(i j − 1) + 1) j1∏j=11(i j − 1) + 1
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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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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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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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