Galois invariant smoothness basis

TL;DR

This paper constructs Galois-invariant smoothness bases for finite fields to improve discrete logarithm computations, addressing a question by Joux and the second author, and applies these methods to linear and function field sieves.

Contribution

It introduces models for finite fields that enable Galois-invariant smoothness bases, enhancing the efficiency of discrete logarithm algorithms.

Findings

Constructed models for broad classes of finite fields.

Enabled Galois-invariant smoothness bases for these fields.

Applied to linear and function field sieves for discrete log computations.

Abstract

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\bK$, one looks for a smoothness basis for $\bK^*$ that is left invariant by automorphisms of $\bK$. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve).