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Robert Rolland

Publications and source records attributed to Robert Rolland.

16 recordsLinked to original sources
arXiv2025

On the Weil descent of Artin-Schreier algebraic function fields over finite fields

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is algebraically closed in $F$. We give general results on the descent over the fields $k= \F_{p^t}$ for $t$ dividing $n$. Then, we completely handle the bi-cyclic case of the descent over the fields $k_1=\F_{p}$ and $k_2= \F_{p^2}$ of all the sub-extensions of $F$ defined over $\F_{p^4}$. We give explicit examples with small prime numbers $p$.

math.NT↗
arXiv2024

Chaining Multiplications in Finite Fields with Chudnovsky-type Algorithms and Tensor Rank of the k-multiplication

We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension of order n a finite field K. We prove that these algorithms give a tensor decomposition of the k-multiplication for which the rank is linear in n uniformly in $q$. We give uniform upper bounds of the rank of k-multiplication in finite fields. They use interpolation on algebraic curves which transforms the problem in computing the Hadamard product of $k$ vectors with components in K. This generalization of the widely studied case of $k=2$ is based on a modification of the Riemann-Roch spaces involved and the use of towers of function fields having a lot of places of high degree.

math.NT↗
arXiv2020

On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative understanding of the domain. In particular, we identify and clarify certain results not completely proved and we emphasis the link with open problems in number theory, algebraic geometry, and coding theory.

math.AG↗
arXiv2017

Optimization of Tree Modes for Parallel Hash Functions: A Case Study

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.

cs.DC↗
arXiv2015

Hamming distances from a function to all codewords of a Generalized Reed-Muller code of order one

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.

cs.IT↗
arXiv2015

On Chudnovsky-Based Arithmetic Algorithms in Finite Fields

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 ${\mathbb F}_{16^{13}}$.

cs.DM↗
arXiv2015

A Digital Signature Scheme for Long-Term Security

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.

cs.CR↗
arXiv2014

On the Number of Points of Algebraic Sets over Finite Fields

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.

math.AG↗
arXiv2013

Remarks on low weight codewords of generalized affine and projective Reed-Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.

cs.IT↗
arXiv2013

Lower bounds on the number of rational points of Jacobians over finite fields and application to algebraic function fields in towers

We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are especially useful for estimating the class number of function fields in towers of function fields over finite fields. We give examples in the case of asymptotically good towers. In particular we estimate the class number of function fields which are steps of towers having one or several positive Tsfasman-Vladut invariants. Note that the study is not done asymptotically, but for each individual step of the towers for which we determine precise parameters.

math.AG↗
arXiv2011

Lower bounds on the class number of algebraic function fields defined over any finite field

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 and asymptotics for the class number, depending mainly on the number of places of a certain degree. We give examples of towers of algebraic function fields having a large class number.

math.AG↗
arXiv2009

The second weight of generalized Reed-Muller codes in most cases

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d (n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.

cs.IT↗
arXiv2009

Families of curves over any finite field with a class number greater than the Lachaud - Martin-Deschamps bounds

We study and explicitly construct some families of asymptotically exact sequences of algebraic function fields. It turns out that these families have an asymptotical class number widely greater than the general Lachaud - Martin-Deschamps bounds. We emphasize that we obtain asymptotically exact sequences of algebraic function fields over any finite field $\F_q$, in particular when $q$ is not a square and that these sequences are dense towers.

math.NT↗
arXiv2009

On the existence of dimension zero divisors in algebraic function fields defined over F_q

Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree γ-1 where γis the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.

math.NT↗
arXiv2004

Descent of the Definition Field of a Tower of Function Fields and Applications

Let us consider an algebraic function field defined over a finite Galois extension $K$ of a perfect field $k$. We give some conditions allowing the descent of the definition field of the algebraic function field from $K$ to $k$. We apply these results to the descent of the definition field of a tower of function fields.We give explicitly the equations of the intermediate steps of an Artin-Schreier type extension reduced from $\F_{q^2}$ to $\F_q$. By applying these results to a completed Garcia-Stichtenoth's tower we improve the upper bounds and the upper asymptotic bounds of the bilinear complexity of the multiplication in finite fields.

math.NT↗