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156 Chapter 5. Complex multiplication and elliptic curvescontains the l-th roots of unity, i.e. the smallest integer such that l|p ord l(p) − 1, or equivalentlythe multiplicative order of p in (Z/lZ) ∗ . Then k = ord l (p)/ gcd(ord l (p), n) if q = p n . Therefore,it is ord l (p) rather than k which should be considered for the difficulty of the discrete logarithm.For cryptographic applications however, the base field is often chosen to be prime so that boththese values are equal.Finally, it should be noted that supersingular curves can not be used for classical public keycryptography.Proposition 5.4.2 ([192]). Let E be a supersingular curve. Then its embedding degree k verifiesk ≤ 6.This is not a concern for ordinary curves. Indeed, for a random curve the embedding degreeis relatively large, typically of the size of l.Proposition 5.4.3 ([10]). Let (p, E) be a random couple made of a prime number p ∈ [M/2, M]and of an elliptic curve defined over F p with a prime number of points l. Then the probabilitythat l|p k − 1 for k ≤ (log p) 2 is smaller thanc(log M) 9 (log log M) 2 /Mwhere c is an effectively computable positive constant.5.4.3 Identity-based cryptographyThe idea of identity-based cryptography, proposed by Shamir [235], is to use any binary string(e.g. an email address) as a public key.A trusted third party, the Public Key Generator, publishes a master public key from whichall public keys are derived using only public data, and keeps a master secret key to computethe secret keys corresponding to the public data. This scheme allows to encrypt messages, orcheck digital signatures, without prior distribution of public keys as in the classical public keycryptography model. However, the users must highly trust the Public Key Generator, because itcan compute any private key, which is not the case in the classical infrastructure. Nonetheless,different variants exist where this pitfall is avoided.The first instantiation of such an encrypting scheme appeared only many years later: in 2001in the works of Cocks [52], using quadratic residues, and of Boneh and Franklin [20, 21], usingthe Weil pairing.Elliptic curves used in such schemes should not only be resistant to attacks against the discretelogarithm, but also have a small enough embedding degree so that the pairing is efficientlycomputable. Hence, supersingular curves are natural candidates, but their embedding degreesbeing always really small may not be sufficient and discards them for several applications. Testingrandom curves will definitely not yield suitable curves. Using the CM method it is howeverpossible to find ordinary curves which not only have a prime, or nearly prime, number of points,but also have a controlled embedding degree. More precisely, for a given discriminant ∆, wewant to find a couple (t, q) where q is a prime power, t 2 − 4q = ∆ and q + 1 − t has large primedivisor l which divides q k − 1 for a suitable k, but not q i − 1 for i < l. This last condition can berephrased as l dividing Φ k (q) where Φ k is the k-th cyclotomic polynomial. To summarize theabove discussion, the following conditions are needed:1. t 2 − 4q = ∆ where q is a prime power;2. l | q + 1 − t where l is a large prime;
5.4. Elliptic curves in cryptography 1573. l | Φ k (q) where Φ k is the k-th cyclotomic polynomial.Different methods and families of curves verifying such conditions have been proposed. Overviewsof these constructions can be found in the survey of Freeman, Scott and Teske [101], the Master’sthesis of Bisson [17], or the Ph.D. thesis of Naehrig [215].
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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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Part IIBent functions and pointcoun
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96 Chapter 3. Bent functions and al
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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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Page 185 and 186: Chapter 6Complex multiplication in
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Page 189 and 190: 6.1. Abelian varieties 163Theorem 6
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Page 216 and 217: Table A.8: Coefficients for d = 8,
Page 218 and 219: 192 Bibliography[13] Elwyn Ralph Be
Page 220 and 221: 194 Bibliography[42] Claude Carlet,
Page 222 and 223: 196 Bibliography[72] Cunsheng Ding,
Page 224 and 225: 198 Bibliography[102] David Freeman
Page 226 and 227: 200 Bibliography[132] Marc Hindry a
Page 228 and 229: 202 Bibliography[161] Philippe Lang
Page 230 and 231: 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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