Structure computation and discrete logarithms in finite abelian p-groups

TL;DR

This paper introduces improved algorithms for computing discrete logarithms and bases in finite abelian p-groups, enhancing efficiency and providing new methods for structure determination and pth root extraction.

Contribution

It presents a generic algorithm surpassing Pohlig-Hellman, a direct basis computation method, and a Monte Carlo approach for structure determination of finite abelian groups.

Findings

Enhanced discrete logarithm algorithm for abelian p-groups

Direct basis computation method without relation matrices

Monte Carlo algorithm for group structure with O(|G|^0.5) complexity

Abstract

We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G.