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

Gaetan Bisson; Andrew V.  Sutherland

Paper 2011/004 Original: eprint.iacr.org/2011/004 ↗

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

Gaetan Bisson

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.

Metadata

Available format(s): PDF
Publication info: Preprint.
History: 2011-01-05: last updated; 2011-01-05: published
Short URL: https://ia.cr/2011/004
License: https://creativecommons.org/licenses/by/4.0/

BibTeX

@misc{cryptoeprint:2011/004,
      author = {Gaetan Bisson and Andrew V.  Sutherland},
      title = {A low-memory algorithm for finding short product representations in finite groups},
      howpublished = {Cryptology {ePrint} Archive, Paper 2011/004},
      year = {2011},
      url = {https://eprint.iacr.org/2011/004}
}