On the Conjugacy Problem in Groups and its Variants

TL;DR

This thesis investigates the conjugacy problem and its variants in groups, analyzing recent theoretical results, developing algorithms, and exploring applications such as cryptographic schemes, with implementation in GAP.

Contribution

It extends understanding of twisted conjugacy problems, introduces new conditions affecting subgroup structures, and applies these concepts to cryptography and computational group theory.

Findings

Analysis of recent results on twisted conjugacy in free groups

Development of algorithms for doubly-twisted conjugacy relations

Application to cryptographic schemes based on matrix semigroups

Abstract

This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy problem in free-by-cyclic groups and some further group extensions. We also consider the doubly-twisted conjugacy problem in free groups. Staecker has developed an algorithm for deciding doubly-twisted conjugacy relations in the case where the involved homomorphisms satisfy a certain remnant inequality. We show how a similar condition affects the equalizer subgroup and raise new questions regarding this subgroup. As an application we discuss the Shpilrain-Ushakov authentication scheme based on the doubly-twisted conjugacy search problem in matrix semigroups over truncated polynomials over finite fields. Part of this thesis is devoted to the implementation and testing of some of the previously mentioned concepts in the GAP programming language.