The Function Field Sieve Is Quite Special

Abstract

In this paper, we describe improvements to the function field sieve (FFS) for the discrete logarithm problem in $\mathbb{F}p^n$ , when p is small. Our main contribution is a new way to build the algebraic function fields needed in the algorithm. With this new construction, the heuristic complexity is as good as the complexity of the construction proposed by Adleman and Huang [2], i.e L _p ⁿ[1/3,c] = exp((c+ o(1))log(p ⁿ)^1/3 log(log(p ⁿ))^2/3) where c = (32/9)^1/3. With either of these constructions the FFS becomes an equivalent of the special number field sieve used to factor integers of the form A ^N ± B. From an asymptotic point of view, this is faster than older algorithm such as Coppersmith’s algorithm and Adleman’s original FFS. From a practical viewpoint, we argue that our construction has better properties than the construction of Adleman and Huang. We demonstrate the efficiency of the algorithm by successfully computing discrete logarithms in a large finite field of characteristic two, namely $\mathbb{F}2^{521}$ .