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70 Chapter 2. On a conjecture about addition modulo 2 k − 1Finally, if d = 1, then ( )d−1+jd−1 = 1, so that the sum becomesand, if d = 2, then ( )d−1+jd−1 = j + 1, so thatP 2 = 1 4 2P 1 = 1 14 1 − 1/4 = 1 3 ;∞ ∑j=0(j + 1) 24 j = 1 4(= 1 214( 2)4 1 −1 3+4= 2 27 + 1 9 = 527 .14∞∑j=0(1 −14When the number of blocks d goes as well to infinity, f d (∞, . . . , ∞) converges towards 1/2.Indeed 1 ≤ P3 d d ≤ 1 + 3 14 d 4converges towards 0 as d goes to infinity. As we show below, it does2 dso monotonically so that f d (∞, . . . , ∞) goes to 1/2 monotonically as well.A first step towards proving the monotonicity of P d in d is to study the special case d = 1. Inthis case the value P [X 1 = k] has indeed a short closed-form expression.j 24 j) 2)Lemma 2.6.6. For d = 1,Proof. Indeed, for k ≥ 0,P [X 1 = k] = 13 · 2 |k| .P [X 1 = k] = P [G 1 = k + H 1 ]∞∑= P [G 1 = i] P [H 1 = k + i]=i=0∞∑i=01 12 i+1 2 k+i+1 = 1 ∑∞ 12 k+2 4 i= 1 42 k+2 3 = 1 312 k .In the general case d ≥ 1, it can also be proven quite directly that the maximal value ofP [X d = k] is attained for k = 0.Lemma 2.6.7. For d ≥ 1 and k ≠ 0,P [X d = k] der the real Hilbert space H = l 2 (Z, R) of square-summable sequences. It is equippedwith norm preserving translation operators τ k defined by (τ k u) j = u j+k for a sequence u =(u j ) j∈Z ∈ H. Consider now the sequence u (d) ∈ H defined byu (d)ji=0[ d∑] [ d∑]= P G i = j = P H i = ji=1i=1whose exact values are given in Lemma 2.6.4 for j ≥ 0, and u (d)j = 0 for j < 0.
2.6. Asymptotic behavior: β i → ∞ 71Then, as shown at the beginning of the proof of Proposition 2.6.5, we haveP [X d = k] =[∞∑ d∑] [ d∑]P G i = j P H i = j + k = 〈u (d) , τ k u (d) 〉i=1i=1j=0where 〈·, ·〉 is the scalar product of H. We now use the Cauchy–Schwarz inequality and the factthat τ k is norm preserving to conclude that√P [X d = k] = 〈u (d) , τ k u (d) 〉 < 〈u (d) , u (d) 〉〈τ k u (d) , τ k u (d) 〉 = 〈u (d) , u (d) 〉 = P [X d = 0] .(Remark that the Cauchy–Schwarz inequality is strict here because u (d) and τ k u (d) are notproportional when k ≠ 0.)Combining Lemmas 2.6.6 and 2.6.7, we then get the monotonicity of P d in d.Proposition 2.6.8. For d ≥ 1,P d > P d+1 .Proof.P d+1 = P [X d+1 = 0] = P [X 1 + X d = 0]==
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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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120 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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