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Finding roots in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_{p^n}$ with the successive resultants algorithm

Part of: Finite fields and commutative rings (number-theoretic aspects) Number theory

Published online by Cambridge University Press:  01 August 2014

Christophe Petit
Christophe Petit*
Affiliation:
UCL Crypto Group, Université catholique de Louvain, Place du Levant 3 , B1348 Louvain-la-Neuve, Belgium email christophe.petit@uclouvain.be

Abstract

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The problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.

Supplementary materials are available with this article.

MSC classification

Secondary: 11-04: Explicit machine computation and programs (not the theory of computation or programming) 11T06: Polynomials 11T55: Arithmetic theory of polynomial rings over finite fields 11T71: Algebraic coding theory; cryptography

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