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

*Gaetan Bisson and Andrew V. Sutherland*

**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*log2(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.

**Category / Keywords: **

**Date: **received 3 Jan 2011

**Contact author: **gaetan bisson at loria fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20110105:023628 (All versions of this report)

**Short URL: **ia.cr/2011/004

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