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48 Chapter 2. On a conjecture about addition modulo 2 k − 1where ( )li 1,...,i nis a multinomial coefficient and bd,nl,mis defined asn−mb d,nl,m = ∑( n − mii=0) d−n∑( ) d − njj=0⎛⎝ ∑ k≥1∑k j≥0,j∈I∪J,1≤j≤m(l + S − m)!l!2 k [ ] ⎞ h − k ⎠ ∏ A kj∏ A kj − 3 kj=0(h − k)! l + S − m k j ! k j !j∈J j∈Im∏j=1C kj−1|k j − 1|!.Within the above expression for b d,nl,m, the following notation is used:• I = {m + 1, . . . , m + i};• J = {n + 1, . . . , n + j};• S = ∑ j∈I∪J,1≤j≤m k j;• h = d − m − j − i;and⎧⎨ A j + Bj+1j+1if j > 0 ,C j = −⎩13 6if j = 0 ,1 if j = −1 .Here, A i is a sum of Eulerian numbers and B i a Bernoulli number; both are described inSubsection 2.5.3.Finally, we prove in Subsection 2.5.6 an additional property predicted experimentally.Proposition 2.5.3. For 0 < j ≤ i,a d,ni.e. the value of b d,nl,mdoes not depend on m.2.5.1 Experimental resultsFor d = 1, by Theorem 2.4.8, we have(i,j,...) = i + 1 a d,nj(i+1,j−1,...) ;f 1 (β 1 ) = 2 3 4−β1 + 1 3 .The case d = 2 has been treated in Subsection 2.4.5 and leads to a similar expression:f 2 (β 1 , β 2 ) = 11 ( 227 + 4−β1 9 β 1 − 2 )27( 2+ 4 −β2 9 β 2 − 2 ) ( 20+ 4 −β1−β22727 − 2 )9 (β 1 + β 2 ) .In both these cases, f d has the correct form and has been shown to verify Conjecture 1.2.2.The tables in Appendix A give the normalized coefficients a d,n(i 1,...,i n)of the multivariatepolynomials Pd n for the first few d’s. All of these data were computed symbolically using Sage [250],Pynac [268] and Maxima [267]. As a byproduct of this work, the interface between Sage [250]
2.5. A closed-form expression for f d 49and Maxima [267] was completely rewritten 5 . The functions used for internal comparison inPynac [268] were also completely rewritten 6 . Graphs of some functions derived from f d are givenin Figures 2.3, 2.4 and 2.5.Figure 2.3: Graph of f d (β i ) for β i = 1, i ≠ 1Figure 2.4: Graph of f d (β i ) for β i = 2, i ≠ 1Moreover, looking at the tables in Appendix A, some additional properties seem to be verified.Here are some examples. The value of a d,d(1,...,1,0)is easy to predict:a d,d(1,...,1,0) = (−1)d+1 2 ;5 http://trac.sagemath.org/sage_trac/ticket/73776 http://trac.sagemath.org/sage_trac/ticket/9880
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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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Page 28 and 29: 2 Introduction1 Mathematics and cry
Page 30 and 31: 4 Introductiongiving more general b
Page 33 and 34: Chapter 1Boolean functions incrypto
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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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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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