A low-memory algorithm for finding short product representations in finite groups
Number Theory
2012-06-26 v1 Cryptography and Security
Group Theory
Abstract
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.
Cite
@article{arxiv.1101.0564,
title = {A low-memory algorithm for finding short product representations in finite groups},
author = {Gaetan Bisson and Andrew V. Sutherland},
journal= {arXiv preprint arXiv:1101.0564},
year = {2012}
}
Comments
12 pages