In this paper, we propose two variants of the Number Field Sieve (NFS) to compute discrete logarithms in medium characteristic finite fields. We consider algorithms that combine two ideas, namely the Multiple variant of the Number Field Sieve (MNFS) taking advantage of a large number of number fields in the sieving phase, and two recent polynomial selections for the classical Number Field Sieve. Combining MNFS with the Conjugation Method, we design the best asymptotic algorithm to compute discrete logarithms in the medium characteric case. The asymptotic complexity of our improved algorithm is L<inf>p</inf><sup>n</sup>(1/3, (8(9 + 4 √ 6)/15)<sup>1/3</sup>) ≈ Lpn(1/3, 2.156), where (image found)<inf>p</inf><sup>n</sup> is the target finite field. This has to be compared with the complexity of the previous state-of-the-art algorithm for medium characteristic finite fields, NFS with Conjugation Method, that has a complexity of approximately L<inf>p</inf><sup>n</sup>(1/3, 2.201). Similarly, combining MNFS with the Generalized Joux-Lercier method leads to an improvement on the asymptotic complexities in the boundary case between medium and high characteristic finite fields.
Annual International Conference on the Theory and Applications of Cryptographic Techniques

Pierrot, C.A. (2015). The multiple number field sieve with conjugation and generalized Joux-Lercier methods. In EUROCRYPT (pp. 156–170). doi:10.1007/978-3-662-46800-5_7