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Get Arithmetic, Geometry, Cryptography and Coding Theory 2009 PDF

By David Kohel, Robert Rolland (ed.)

ISBN-10: 0821849557

ISBN-13: 9780821849552

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Extra info for Arithmetic, Geometry, Cryptography and Coding Theory 2009

Example text

On a Xα = 22 q si et seulement si Tr ηu = 1. De plus, on a Tr u = 1. (2) Si e ∈ (k∗ )3 , alors Xα = 0 ou 22 q. Il existe y ∈ k tel que y 4 +y 3 +y 2 +y = e et Xα = 22 q si et seulement si Tr ηy = 0. D´ emonstration. Lorsque e = 1, d’une part, si Z est racine de P (x), alors Q(Z) = Tr(aZ 5 + cZ) et d’autre part, si Z est une racine de x5 P (x) + 1, alors Q(Z ) = Tr(aZ 5 + cZ ). Supposons que u soit un ´el´ement de k v´erifiant Tr u = 1 et (u2 + u)3 = e = On a alors a7 α7 (u5 + u) = u−1 + u−2 et −7 .

His algorithm is very efficient on a certain family of elliptic curves, called Montgomery’s curves. In this case, the differential addition costs 4M + 2S and the doubling 2M + 2S + 1D where 1D is a multiplication by a constant. Brier and Joye’s adaptation requires 9M + 2S for an addition and 6M + 3S for a doubling. The complexity of this general algorithm is then n(15M + 5S) + 3M + S + I for a n-bit scalar, where I is a modular inversion in the field Fp and 3M + S + I is the cost to recover the Y -coordinate at the end.

Lin, and M. Scott, Endomorphisms for faster elliptic curve cryptography on a large class of curves, EUROCRYPT 2009, LNCS 5479 (2009), 518–535. H. -E. Hamid, H. Choukri, D. Naccache, M. Tunstall, and C. org/2004/100. T. Izu and T. Takagi, A fast parallel elliptic curve multiplication resistant against side channel attacks, PKC 2002, LNCS 2274 (2002), 371–374. M. Joye and C. Tymen, Protections against differential analysis for elliptic curve cryptography, CHES 2001, LNCS 2162 (2001), 377–390. M. M.

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Arithmetic, Geometry, Cryptography and Coding Theory 2009 by David Kohel, Robert Rolland (ed.)


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