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In the recently emerging field of group-based cryptography, the ConjugacySearch Problem (CSP) has gained traction as a non-commutative replacement ofthe Discrete Log Problem (DLP). The problem of finding a secure class ofnonabelian groups for use as platforms is open and a subject of activeresearch. This paper demonstrates a polynomial time solution of the CSP in animportant class of nonabelian groups, the extraspecial $p$-groups. For thispurpose, and as a further result, we provide a reduction of the CSP in certaintypes of central products. The consequences of our results are practicallyrelevant for ruling out several groups as platforms, since several nonabeliangroups are constructed by combining smaller groups by taking direct and centralproducts.

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