Public key cryptography based on some extensions of group

TL;DR

This paper demonstrates that certain group extensions with unsolvable conjugacy problems can be constructed to have solvable word problems, providing a new approach for cryptographic group design.

Contribution

It proves that extensions with unsolvable conjugacy problems always have solvable word problems, enabling systematic creation of cryptographically relevant groups.

Findings

Extensions have solvable word problems despite unsolvable conjugacy problems

Explicit construction of an extension of Thompson group F

Proposed group as a basis for public key cryptography

Abstract

Bogopolski, Martino and Ventura in [BMV10] introduced a general criteria to construct groups extensions with unsolvable conjugacy problem using short exact sequences. We prove that such extensions have always solvable word problem. This makes the proposed construction a systematic way to obtain finitely presented groups with solvable word problem and unsolvable conjugacy problem. It is believed that such groups are important in cryptography. For this, and as an example, we provide an explicit construction of an extension of Thompson group F and we propose it as a base for a public key cryptography protocol.