On the Conjugacy Search Problem in Extraspecial $p$-Groups

TL;DR

This paper presents a polynomial time algorithm for solving the Conjugacy Search Problem in extraspecial p-groups, impacting the selection of secure groups in cryptography by identifying insecure group classes.

Contribution

It provides the first polynomial time solution for CSP in extraspecial p-groups and reduces CSP in certain central product groups, informing cryptographic group choices.

Findings

Polynomial time solution for CSP in extraspecial p-groups

Reduction of CSP in certain central product groups

Implications for cryptographic group security

Abstract

In the recently emerging field of group-based cryptography, the Conjugacy Search Problem (CSP) has gained traction as a non-commutative replacement of the Discrete Log Problem (DLP). The problem of finding a secure class of nonabelian groups for use as platforms is open and a subject of active research. This paper demonstrates a polynomial time solution of the CSP in an important class of nonabelian groups, the extraspecial $p$-groups. For this purpose, and as a further result, we provide a reduction of the CSP in certain types of central products. The consequences of our results are practically relevant for ruling out several groups as platforms, since several nonabelian groups are constructed by combining smaller groups by taking direct and central products.