A General Polynomial Selection Method and New Asymptotic Complexities for the Tower Number Field Sieve Algorithm

Abstract

In a recent work, Kim and Barbulescu had extended the tower number field sieve algorithm to obtain improved asymptotic complexities in the medium prime case for the discrete logarithm problem on $\mathbb{F}_{p^n}$ where $n$ is not a prime power. Their method does not work when $n$ is a composite prime power. For this case, we obtain new asymptotic complexities, e.g., $L_{p^n}(1/3,(64/9)^{1/3})$ (resp. $L_{p^n}(1/3,1.88)$ for the multiple number field variation) when $n$ is composite and a power of 2; the previously best known complexity for this case is $L_{p^n}(1/3,(96/9)^{1/3})$ (resp. $L_{p^n}(1/3,2.12)$). These complexities may have consequences to the selection of key sizes for pairing based cryptography. The new complexities are achieved through a general polynomial selection method. This method, which we call Algorithm-$\mathcal{C}$, extends a previous polynomial selection method proposed at Eurocrypt 2016 to the tower number field case. As special cases, it is possible to obtain the generalised Joux-Lercier and the Conjugation method of polynomial selection proposed at Eurocrypt 2015 and the extension of these methods to the tower number field scenario by Kim and Barbulescu. A thorough analysis of the new algorithm is carried out in both concrete and asymptotic terms