Complexity of Conjugacy Search in some Polycyclic and Matrix Groups

The most prominent algorithmic problem employed in the recently emerging field of nonabelian group-based cryptography is the Conjugacy Search Problem (CSP). While several methods of attacks on nonabelian protocols have been devised, many of these are heuristic, protocol-specific, and focus on retrieving the shared keys without solving the underlying CSP in the group. So far, the true complexity of the CSP in different platform groups has not been sufficiently investigated. In this paper, we study the complexity of various versions of the CSP in polycyclic groups and matrix groups over finite fields. In particular we show that in $\GL_n(\fq)$ and in polycyclic groups with two generators, a CSP where conjugators are restricted to a cyclic subgroup is reducible to a set of $\mathcal{O}(n^2)$ DLPs. As a consequence of our results we also demonstrate the cryptanalysis of a few independently proposed cryptosystems.