Paper 2014/147 Original: eprint.iacr.org/2014/147 ↗

The Multiple Number Field Sieve for Medium and High Characteristic > Finite Fields

Abstract

In this paper, we study the discrete logarithm problem in medium and high characteristic finite fields. We propose a variant of the Number Field Sieve (NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in $\mathbb{F}_{p^n}$ for the whole range of applicability of NFS and lowers the asymptotic complexity from $L_{p^n}(1/3, (128/9)^{1/3})$ to $L_{p^n}(1/3, (2^{13} /3^6)^{1/3})$ in the medium characteristic case, and from $L_{p^n} (1/3, (64/9)^{1/3})$ to $L_{p^n}(1/3,((92 + 26\sqrt{13})/27))^{1/3})$ in the high characteristic case. Version 2 contains an erratum.