  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.
