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We study the number of effective divisors of a given degree of an algebraic function field defined over a finite field. We first give somme lower bounds and upper bounds when the function field, the degree and the underlying finite field are fixed. Then we study the behavior of the number of effective divisors when some of the parameters, namely the underlying finite field, the degree of the effective divisors, the algebraic function field can be variable.

Thanks to a new construction of the so-called Chudnovsky- Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation and they allow computations to be parallelized while maintaining a low number of bilinear multiplications. We give an example with the finite field F1613\\mathbb {F}_{16^{13}}.

Nous donnons des bornes inf\\'erieures sur le nombre de diviseurs effectifs de degr\\'e $\\leq g-1$ par rapport au nombre de places d'un certain degr\\'e d'un corps de fonctions alg\\'ebriques de genre $g$ d\\'efini sur un corps fini. Nous d\\'eduisons des bornes inf\\'erieures du nombre de classes qui am\\'eliorent les bornes de Lachaud - Martin-Deschamps et des bornes inf\\'erieures asymptotiques atteignant celles de Tsfasman-Vladut. Nous donnons des exemples de tours de corps de fonctions alg\\'ebriques ayant un grand nombre de classes. We give lower bounds on the number of effective divisors of degree $\\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.

Nous donnons des bornes inférieures sur le nombre de diviseurs effectifs de degré ≤ g — 1 par rapport au nombre de places d'un certain degre d'un corps de fonctions algebriques de genre g défini sur un corps fini. Nous déduisons des bornes inférieures du nombre de classes qui améliorent les bornes de Lachaud-Martin-Deschamps et des bornes inférieures asymptotiques atteignant celles de Tsfasman-Vladut. Nous donnons des exemples de tours de corps de fonctions algébriques ayant un grand nombre de classes. We give lower bounds on the number of effective divisors of degree ≤ g — 1 with respect to the number of places of certain degrees of an algebraic function field of genus g defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.

This paper focuses on parallel hash functions based on tree modes of operation for an inner Variable-Input-Length function. This inner function can be either a single-block-length (SBL) and prefix-free MD hash function, or a sponge-based hash function. We discuss the various forms of optimality that can be obtained when designing parallel hash functions based on trees where all leaves have the same depth. The first result is a scheme which optimizes the tree topology in order to decrease the running time. Then, without affecting the optimal running time we show that we can slightly change the corresponding tree topology so as to minimize the number of required processors as well. Consequently, the resulting scheme decreases in the first place the running time and in the second place the number of required processors.

For any finite field \\({\\mathbb F}_q\\) with \\(q\\) elements, we study the set \\({\\mathcal F}_{(q,m)}\\) of functions from \\({\\mathbb F}_q^m\\) into \\({\\mathbb F}^q\\). We introduce a transformation that allows us to determine a linear system of \\(q^{m+1}\\) equations and \\(q^{m+1}\\) unknowns, which has for solution the Hamming distances of a function in \\({\\mathcal F}_{(q,m)}\\) to all the affine functions.

In this paper we propose a signature scheme based on two intractable problems, namely the integer factorization problem and the discrete logarithm problem for elliptic curves. It is suitable for applications requiring long-term security and provides a more efficient solution than the existing ones.

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case. Nous déterminons des majorations du nombre de points d'un ensemble algébrique affine ou projectif, défini sur une extension d'un corps fini par un système d'équations polynomiales, y compris dans le cas où l'ensemble algébrique n'est pas défini sur le corps fini lui-même. Une attention particulière est portée aux ensemble algébriques irréductibles mais non absolument irréductibles, pour lesquels nous obtenons de meilleures bornes. Nous étudions le cas des intersections complètes, pour lesquelles nous construisons une décomposition moins fine que la décomposition en composantes irréductibles, mais plus directement liée aux polynômes qui définissent l'ensemble algébrique. Enfin, nous construisons des familles d'ensembles algébriques atteignant le nombre maximum de points rationnels dans le cas affine, et comportant de nombreux points dans le cas projectifs.