Improving NFS for the discrete logarithm problem in non-prime finite fields

Abstract

The aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(p n) where n is a small integer greater than 1. Though less studied than the small characteristic case or the prime field case, the difficulty of this problem is at the heart of security evaluations for torus-based and pairing-based cryptography. The best known method for solving this problem is the Number Field Sieve (NFS). A key ingredient in this algorithm is the ability to find good polynomials that define the extension fields used in NFS. We design two new methods for this task, modifying the asymptotic complexity and paving the way for record-breaking computations. We exemplify these results with the computation of discrete logarithms over a field GF(p 2) whose cardinality is 180 digits (595 bits) long.