Some baby-step giant-step algorithms for the low hamming weight discrete logarithm problem

Abstract

In this paper, we present several baby-step giant-step algorithms for the low hamming weight discrete logarithm problem. In this version of the discrete log problem, we are required to find a discrete logarithm in a finite group of order approximately $2^m$, given that the unknown logarithm has a specified number of 1’s, say $t$, in its binary representation. Heiman and Odlyzko presented the first algorithms for this problem. Unpublished improvements by Coppersmith include a deterministic algorithm with complexity $O \left ( m \left (\begin {smallmatrix}m / 2 t / 2\end {smallmatrix}\right ) \right )$, and a Las Vegas algorithm with complexity $O \left ( \sqrt {t} \left (\begin {smallmatrix}m / 2 t / 2\end {smallmatrix} \right )\right )$. We perform an average-case analysis of Coppersmith’s deterministic algorithm. The average-case complexity achieves only a constant factor speed-up over the worst-case. Therefore, we present a generalized version of Coppersmith’s algorithm, utilizing a combinatorial set system that we call a