Root Extraction in Finite Abelian Groups

TL;DR

This paper studies root extraction in finite Abelian groups, providing algorithms and complexity bounds, and shows it is comparable in difficulty to discrete logarithm problems once a basis is known.

Contribution

The paper introduces algorithms for root extraction in finite Abelian groups and analyzes their complexity, extending previous work to more general group structures.

Findings

Root extraction complexity is similar to discrete logarithm computation.

Algorithms are provided with bounds on group operations needed.

Root extraction becomes easier after basis computation and discrete log solving.

Abstract

We formulate the Root Extraction problem in finite Abelian $p$-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these algorithms. We observe that once a basis is computed and the discrete logarithm relative to the basis is solved, root extraction takes relatively fewer "bookkeeping" steps. Thus, we conclude that root extraction in finite Abelian groups is no harder than solving discrete logarithms and computing basis.