Root Extraction in Certain Finite Abelian $p$-Groups

TL;DR

This paper introduces the Generalized Root Extraction problem in finite Abelian groups, especially torsion subgroups of elliptic curves, providing conditions for solutions and algorithms for root extraction and its variants.

Contribution

It formulates a new root extraction problem in finite Abelian groups, offers necessary and sufficient conditions, and develops algorithms applicable to groups of prime power order.

Findings

Necessary and sufficient conditions for root existence

Algorithms for root extraction in specific Abelian groups

Extension to simultaneous root extraction problems

Abstract

We formulate a problem called \emph{Generalized Root Extraction} in finite Abelian groups that have more than one generator. We then study this problem for the specific case of the torsion subgroups of elliptic curves. We give a necessary and sufficient condition for the existence of a solution. We also present an algorithm to find a solution. Our algorithm easily generalizes to Abelian groups of prime power order that have a specific structure. We then discuss a variant of this problem called \emph{Simultaneous Root Extraction} and present an algorithm for solving it.