A low-memory algorithm for finding short product representations in finite groups

Details A low-memory algorithm for finding short product representations in finite groups A low-memory algorithm for finding short product representations in finite groups

2011-01-03

arxiv.org/abs/1101.0564v1

Mathematics

Number Theory

12 pages

Scientific paper

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-rho approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log_2 n, where n=#G and d >= 2 is a constant, we find that its expected running time is O(sqrt(n) log n) group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.

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