On the tensor rank of multiplication in any extension of \F_2

In this paper, we obtain new bounds for the tensor rank of multiplication in any extension of \F_2. In particular, it also enables us to obtain the best known asymptotic bound. In this aim, we use the generalized algorithm of type Chudnovsky with derivative evaluations on places of degree one, two and four applied on the descent over \F_2 of a Garcia-Stichtenoth tower of algebraic function fields defined over \F_{2^4}

A new algorithm for multiplication in finite fields

Cover title.Includes bibliographical references.by Antonio Pincin

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

International audienceWe indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in O(n(2q)^log * q (n)) where n corresponds to the extension degree, and log * q (n) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field F 3^57. Index Terms Multiplication algorithm, bilinear complexity, elliptic function field, interpolation on algebraic curve, finite field

Optimization of the scalar complexity of Chudnovsky2^2 multiplication algorithms in finite fields

We propose several constructions for the original multiplication algorithm of D.V. and G.V. Chudnovsky in order to improve its scalar complexity. We highlight the set of generic strategies who underlay the optimization of the scalar complexity, according to parameterizable criteria. As an example, we apply this analysis to the construction of type elliptic Chudnovsky$^2$ multiplication algorithms for small extensions. As a case study, we significantly improve the Baum-Shokrollahi construction for multiplication in $\mathbb F_{256}/\mathbb F_4$.Comment: 25 pages, 0 figur

CONSTRUCTION OF ASYMMETRIC CHUDNOVSKY ALGORITHMS WITHOUT DERIVATED EVALUATION FOR MULTIPLICATION IN FINITE FIELDS

The Chudnovsky and Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Ran-driambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. In this article, we first translate this generalization into the language of algebraic function fields. Then, we propose a strategy to effectively construct asymmetric algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in F 16 13 using only rational places, in F 4 5 using also places of degree two, and in F 2 5 using also places of degree four

Classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial, part I: The algebra G[u]<Q(u)l>, l>1

AbstractIn this paper we will classify all the minimal bilinear algorithms for computing the coefficients of(∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l where deg Q(u)=j,jl=n and Q(u) is irreducible.The case where l=1 was studied in [1]. For l>1 the main results are that we have to distinguish between two cases: j>1 and j=1. The first case is discussed here while the second is classified in [4]. For j>1 it is shown that up to equivalence every minimal (2n-1 multiplications) bilinear algorithm for computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l is done by first computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) and then reducing it modulo Q(u)l (similar to the case l = 1, [1])

New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

International audienceWe obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields $\mathbb{F}_q$ for any prime power $q$; moreover these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are substantially better by using a family of Shimura curves defined over $\mathbb{F}_q$, with an optimal ratio of $\mathbb{F}_{q^t}$-rational places to their genus, where $q^t$ is a square