Download Algebraic Geometry and Its Applications: Dedicated to Gilles by Jean Chaumine, James Hirschfeld, Robert Rolland PDF

By Jean Chaumine, James Hirschfeld, Robert Rolland

This quantity covers many issues together with quantity idea, Boolean services, combinatorial geometry, and algorithms over finite fields. This ebook comprises many attention-grabbing theoretical and applicated new effects and surveys awarded through the simplest experts in those parts, akin to new effects on Serre's questions, answering a query in his letter to most sensible; new effects on cryptographic functions of the discrete logarithm challenge relating to elliptic curves and hyperellyptic curves, together with computation of the discrete logarithm; new effects on functionality box towers; the development of latest sessions of Boolean cryptographic services; and algorithmic functions of algebraic geometry.

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E. ℓ | [OK : Z[π]] and ℓ2 ∤ [OK : Z[π]]), k the element nℓ α can be replaced by an element of the form π ℓ−1 . This replacement is useful since by [11, Fact 10], determining whether an element k of the form π ℓ−1 is an endomorphism is equivalent to testing the field of definition of the ℓ-torsion. 4. Let A ⊂ B ⊂ C be abelian groups, with [C : A] finite. Let ℓ be a prime, and suppose ℓ divides [C : A] and ℓ2 does not divide [C : A]. Suppose there is some β ∈ B such that β ∈ A and ℓβ ∈ A. Then for any α ∈ C such that ℓα ∈ A, α ∈ B.

Flon and R. Oyono. Fast arithmetic on Jacobians of Picard curves. In Public Key Cryptography - PKC 2004, volume 2947 of LNCS, pages 55–68. Springer, 2004. 9. S. Flon, R. Oyono and C. Ritzenthaler. Rationality of the intersection points of a line with a plane quartic. In progress, 2007. 10. G. Frey and M. M¨ uller. Arithmetic of modular curves and applications. In Algorithmic Algebra and Number Theory, pages 11–48. Ed. , Springer-Verlag, Berlin, 1999. 11. S. D. Galbraith. Equations for modular curves.

1 we just have to replace f3 by f4 . Furthermore, if char(k) = 3, we let Y = y + h1 (x)/3 and we can assume that C is of the following form: Y 3 + h2 Y = f4 , with h2 and f4 as above. If in addition char(k) = 2, then we can assume that f4 has no x3 term. 4. Comments on implementation We deal in this part with an optimized implementation in the case of the existence of a rational flex. To make the algorithm more efficient, we use the following well known methods: (1) In order to reduce the number of field inversions, we use Montgomery’s trick to compute simultaneous inversions.

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