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

被引：3

作者：

Bisson, Gaetan ^{[1
,2
]}

Sutherland, Andrew V. ^{[3
]}

机构：

[1] LORIA, Vandoeuvre Les Nancy, France

[2] Eindhoven Univ Technol, NL-5600 MB Eindhoven, Netherlands

[3] MIT, Cambridge, MA 02139 USA

来源：

DESIGNS CODES AND CRYPTOGRAPHY | 2012年 / 63卷 / 01期

关键词：

Short product; Generic group algorithm; Pollard-rho; Isogeny search; KNAPSACKS;

D O I：

10.1007/s10623-011-9527-8

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

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(root 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.

引用

页码：1 / 13

页数：13