By Jean Chaumine, James Hirschfeld, Robert Rolland

ISBN-10: 9812793429

ISBN-13: 9789812793423

ISBN-10: 9812793437

ISBN-13: 9789812793430

This quantity covers many themes together with quantity idea, Boolean capabilities, combinatorial geometry, and algorithms over finite fields. This booklet includes many fascinating theoretical and applicated new effects and surveys provided via the simplest experts in those components, reminiscent of new effects on Serre's questions, answering a query in his letter to best; new effects on cryptographic purposes of the discrete logarithm challenge concerning elliptic curves and hyperellyptic curves, together with computation of the discrete logarithm; new effects on functionality box towers; the development of latest periods of Boolean cryptographic features; and algorithmic functions of algebraic geometry.

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Additional info for Algebraic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications)

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Lange and R. Oyono. On using expansions to the base of −2. Inter. J. of. Comp. , 81(4):403–406, 2004. 3. E. Reinaldo Barreiro, J. Estrada Sarlabous and J. P. Cherdieu. Efficient reduction on the Jacobian variety of Picard curves. In Coding theory, cryptography and related areas (Guanajuato, 1998), pages 13–28. Springer, Berlin, 2000. 4. A. Basiri, A. Enge, J-C. Faug`ere and N. G¨ urel. Implementing the Arithmetic of C3,4 Curves. In Algorithmic Number Theory Symposium - ANTS-VI, volume 3076 of LNCS, pages 87–101.

1. Let K be a primitive quartic CM field and K ∗ the reflex of K. The following algorithm takes as input the field K, a prime p that splits completely in K and splits completely into principal ideals in K ∗ , and a curve C defined over the finite field Fp . The algorithm returns true or false according to whether End(J) is an order in OK , where J = Jac(C). If the answer is true, the algorithm also outputs a set S ⊂ OK that consists of the Aut(K/Q)-orbit of the Frobenius endomorphism of J. g. [6, Alg.

This computation is outlined in [11, Section 5] and described in more detail in Section 2 below. If End(J) is an order in OK , we compute a set of possible elements π ∈ OK that could represent the Frobenius endomorphism of J. If π represents the Frobenius endomorphism, then its complex conjugate π represents the Verschiebung endomorphism. We next determine a set {αi } of elements of OK such that Z[π, π, {αi }] = OK . It follows that End(J) = OK if and only if each αi is an endomorphism of J. We show in Section 3 that we can take each αi k to have one of two forms: either αi = π ℓ−1 for some positive integer k and prime ℓ, or αi = hiℓ(π) for some cubic polynomial hi with integer coeffid cients and some prime power ℓd .

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Algebraic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications) by Jean Chaumine, James Hirschfeld, Robert Rolland


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