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The multiple number field sieve for medium- and high-characteristic finite fields

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

Published online by Cambridge University Press:  01 August 2014

Razvan Barbulescu  and
Cécile Pierrot
Razvan Barbulescu
Affiliation:
Université de Lorraine, France email [email protected]
Cécile Pierrot
Affiliation:
Laboratoire d’Informatique de Paris 6, UPMC, France email [email protected]

Abstract

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In this paper we study the discrete logarithm problem in medium- and high-characteristic finite fields. We propose a variant of the number field sieve (NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_{p^n}$ for the whole range of applicability of the NFS and lowers the asymptotic complexity from $L_{p^n}({1/3},({128/9})^{1/3})$ to $L_{p^n}({1/3},(2^{13}/3^6)^{1/3})$ in the medium-characteristic case, and from $L_{p^n}({1/3},({64/9})^{1/3})$ to $L_{p^n}({1/3},((92 + 26 \sqrt{13})/27)^{1/3})$ in the high-characteristic case.

MSC classification

Secondary: 11T99: None of the above, but in this section 11Y16: Algorithms; complexity
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 230 - 246
Copyright
© The Author(s) 2014 

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