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. . 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 yields a (nonuniform) deterministic algorithm with complexity . We also present some explicit constructions for splitting systems that make use of perfect hash families.
It is proved in REF that the average-case complexity achieves only a constant factor speed-up over the worst case.
Some baby-step giant-step algorithms for the low hamming weight discrete logarithm problem
Mathematics of Computation
Mathematics Computer Science