Computing discrete logarithms in an interval

Galbraith, Steven; Pollard, John; Ruprai, Raminder

doi:10.1090/S0025-5718-2012-02641-X

Computing discrete logarithms in an interval
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by Steven D. Galbraith, John M. Pollard and Raminder S. Ruprai;

Math. Comp. 82 (2013), 1181-1195

DOI: https://doi.org/10.1090/S0025-5718-2012-02641-X

Published electronically: August 17, 2012

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Abstract:

The discrete logarithm problem in an interval of size $N$ in a group $G$ is: Given $g, h \in G$ and an integer $N$ to find an integer $0 \le n \le N$, if it exists, such that $h = g^n$. 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 $(2 + o(1)) \sqrt {N}$ 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 $(1.715 + o(1)) \sqrt {N}$ 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 $(1.661 + o(1)) \sqrt {N}$ group operations. We give experimental results that show that the methods do work close to that predicted by the theoretical analysis.

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