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86 Chapter 2. On a conjecture about addition modulo 2 k − 12.8.3 Towards a complete proofThe numbers for which P t,k is the nearest to the bound of the conjecture seem to be the oneswhich verify the constraint min(α i ) ≥ k − w H (t) − 1, and especially the ones which also satisfyβ i = 1 for all i. Moreover, puncturing a 1 out of a binary string seems to make P t,k smaller mostof the time.We consequently hope to be able to completely solve the conjecture using one of the followingstrategies:• Show that any number gives a smaller set than an extremal one by induction (i.e. bypuncturing 1’s, even so that different blocks merge).• Show that the conjecture is true for every number which verifies the constraint min(α i ) ≥k − w H (t) − 1, and then that the other numbers give smaller sets by induction (i.e. bypuncturing 1’s, but without merging different blocks).2.9 Efficient test of the Tu–Deng conjecture2.9.1 The Tu–Deng algorithmIn this subsection we present the reformulation of Conjecture 1.2.2 by Tu and Deng [264, Appendix]using the transfer-matrix method [249, Subsection 4.7]. Their idea is to build the automatadescribing the addition bit after bit of two modular integers modulo 2 k − 1 for any integer k.Its states are given by triplets corresponding to the two current bits and the previous carry. Inparticular, this automata does not depend on k.For a fixed t ∈ ( Z/(2 k − 1)Z ) ∗, let D be the directed graph corresponding to this automata.Its vertices are the triplets 8 (a, t, r) ∈ {0, 1} 3 . There exists an edge from (a, t, r) to (a ′ , t ′ , r ′ ) ifand only if t + 2r ′ − a − r ∈ {0, 1} and its weight is x r′ where x is a polynomial indeterminate. Inparticular, all edges to a given node have the same weight, but this is also true for all the verticesfrom a given node.If (a, b) ∈ ( Z/(2 k − 1)Z ) 2verifies a + b = t, we can define a closed path in D as follows. For0 ≤ j ≤ k − 1, denote by v j the vertex v j = (a j , t j , r j−1 ) where r j−1 is 1 if a carry comes fromthe j − 1-st bit when adding a and b and 0 otherwise. Then, (a, b) corresponds to the closed pathΓ : v 0 → v 1 → · · · → v k−1 → v 0 of weight w H (Γ) = x r(a,b) .The vertices of D are mapped to the range of integers [0, 7] via the following map: (a, t, r) ↦→4a + 2t + r. The directed graph D is depicted in Figure 2.6 where edges from the nodes labeled0, 2, 6 and 3 have weight 1 and edges from the nodes label-led 1, 5, 7 and 4 have weight x; itsadjacency matrix A is given in Figure 2.7.Let V i be the set of vertices (a, t, r) such that t = i ∈ {0, 1}, i.e. V 0 = {0, 1, 4, 5} andV 1 = {2, 3, 6, 7}. Denote by B i,j the submatrices of A made of the entries at the intersection ofthe rows and columns respectively specified by V i and V j for i, j ∈ {0, 1}. Then, B 0,0 = B 0,1 andB 1,0 = B 1,1 are respectively given by the following (4, 4)-matrices B and C:⎛⎞ ⎛⎞1 0 1 0 1 0 1 0B = ⎜0 x 0 x⎟⎝0 x 0 x⎠ , C = ⎜1 0 1 0⎟⎝1 0 1 0⎠ .0 x 0 x 0 x 0 x8 There is a slight inconsistency in our notation here: t and a denote both modular integers and one of theirbits. For simplicity, we allow ourselves to make that abuse of notation.
2.9. Efficient test of the Tu–Deng conjecture 87250 43761Figure 2.6: Directed graph D⎛⎞1 0 1 0 1 0 1 00 x 0 x 0 x 0 x1 0 1 0 1 0 1 01 0 1 0 1 0 1 00 x 0 x 0 x 0 x⎜0 x 0 x 0 x 0 x⎟⎝1 0 1 0 1 0 1 0⎠0 x 0 x 0 x 0 xFigure 2.7: Adjacency matrix A of directed graph D
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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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Page 62 and 63: 36 Chapter 2. On a conjecture about
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136 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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