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

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 2m, 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 Unpublished improvements by Coppersmith include a deterministic algorithm with complexity O m m/2 t/2 , and a Las Vegas algorithm with complexity O t m/2 t/2 . 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 splitting system. Using probabilistic methods, we prove a new existence result for these systems that yield...

• Type: Journal
• Year: 2002
• Where: MOC
• Authors: Douglas R. Stinson
• Keywords: Algorithm, Deterministic Algorithm, Logarithm, MOC 2002