Selecting polynomials for the Function Field Sieve

TL;DR

This paper introduces methods for selecting optimal polynomials for the Function Field Sieve, improving the efficiency of discrete logarithm computations in finite fields by analyzing polynomial properties and providing a rapid testing algorithm.

Contribution

It presents a new algorithm for quickly testing large sets of polynomials and explains the behavior of inseparable polynomials, including their relation to the Coppersmith algorithm.

Findings

Developed an efficient polynomial testing algorithm

Analyzed the size, root, and cancellation properties of polynomials

Connected inseparable polynomials to the Coppersmith algorithm

Abstract

The Function Field Sieve algorithm is dedicated to computing discrete logarithms in a finite field GF(q^n), where q is small an prime power. The scope of this article is to select good polynomials for this algorithm by defining and measuring the size property and the so-called root and cancellation properties. In particular we present an algorithm for rapidly testing a large set of polynomials. Our study also explains the behaviour of inseparable polynomials, in particular we give an easy way to see that the algorithm encompass the Coppersmith algorithm as a particular case.