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The conjugacy search problem in public key cryptography: unnecessary and insufficient

The conjugacy search problem in a group G is the problem of recovering an x in G from given g in G and h=x^{-1}gx. This problem is in the core of several recently suggested public key exchange protocols, most notably the one due to Anshel, Anshel, and Goldfeld, and the one due to Ko, Lee at al. In this note, we make two observations that seem to have eluded most people's attention. The first observation is that solving the conjugacy search problem is not necessary for an adversary to get the common secret key in the Ko-Lee protocol. It is sufficient to solve an apparently easier problem of finding x, y in G such that h=ygx for given g, h in G. Another observation is that solving the conjugacy search problem is not sufficient for an adversary to get the common secret key in the Anshel-Anshel-Goldfeld protocol.