Relation Collection for the Function Field Sieve

Abstract

In this paper, we focus on the relation collection step of the Function Field Sieve (FFS), which is to date the best algorithm known for computing discrete logarithms in small-characteristic finite fields of cryptographic sizes. Denoting such a finite field by Fpn, where p is much smaller than n, the main idea behind this step is to find polynomials of the form a(t)-b(t)x in Fp[t][x] which, when considered as principal ideals in carefully selected function fields, can be factored into products of low-degree prime ideals. Such polynomials are called "relations", and current record-sized discrete-logarithm computations need billions of those. Collecting relations is therefore a crucial and extremely expensive step in FFS, and a practical implementation thereof requires heavy use of cache-aware sieving algorithms, along with efficient polynomial arithmetic over Fp[t]. This paper presents the algorithmic and arithmetic techniques which were put together as part of a new public implementation of FFS, aimed at medium-to record-sized computations.