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

作者：Bisson Gaetan^*; Sutherland Andrew V

来源：Designs, Codes and Cryptography, 2012, 63(1): 1-13.

DOI：10.1007/s10623-011-9527-8

• Short product
• Generic group algorithm
• Pollard-rho
• Isogeny search

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 %26gt;= 2 is a constant, we find that its expected running time is O(root n log n) group operations (we give a rigorous proof for d %26gt; 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.

• 出版日期2012-4