On the Conjugacy Problem in Certain Metabelian Groups

TL;DR

This paper investigates the computational complexity of the conjugacy search problem in specific metabelian groups, showing it is generally exponential but polynomial in some cases, and reducible to discrete logarithm in others.

Contribution

It provides a detailed complexity analysis of the conjugacy search problem in certain metabelian groups, identifying cases of polynomial solvability and reductions to known problems.

Findings

Conjugacy search problem is at most exponential in general.

In some subfamilies, the problem is solvable in polynomial time.

In other subfamilies, the problem reduces to discrete logarithm.

Abstract

Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups we prove that the conjugacy search problem is polynomial. We also show that for a different subfamily the conjugacy search problem reduces to the discrete logarithm problem.