The special number field sieve in double-struck Fpn: Application to pairing-friendly constructions

Abstract

In this paper, we study the discrete logarithm problem in finite fields related to pairing-based curves. We start with a precise analysis of the state-of-the-art algorithms for computing discrete logarithms that are suitable for finite fields related to pairing-friendly constructions. To improve upon these algorithms, we extend the Special Number Field Sieve to compute discrete logarithms in double-struck Fpn, where p has an adequate sparse representation. Our improved algorithm works for the whole range of applicability of the Number Field Sieve. © 2014 Springer International Publishing.