International Association for Cryptologic Research (IACR)
Abstract
The discrete logarithm problem in an interval of size N in a group G is: Given g,h∈G and an integer N to find an integer 0≤n≤N, if it exists, such that h=gn. Previously the best low-storage algorithm to solve this problem was the van Oorschot and Wiener version of the Pollard kangaroo method. The heuristic average case running time of this method is (2+o(1))N group operations.
We present two new low-storage algorithms for the discrete logarithm problem in an interval of size N. The first algorithm is based on the Pollard kangaroo method, but uses 4 kangaroos instead of the usual two. We explain why this algorithm has heuristic average case expected running time of (1.715+o(1))N group operations. The second algorithm is based on the Gaudry-Schost algorithm and the ideas of our first algorithm. We explain why this algorithm has heuristic average case expected running time of (1.661+o(1))N group operations. We give experimental results that show that the methods do work close to that predicted by the theoretical analysis.
This is a revised version since the published paper that contains a corrected proof of Theorem 6 (the statement of Theorem 6 is unchanged). We thank Ravi Montenegro for pointing out the errors