On the conjugacy search problem and left conjugacy closed loops

The conjugacy search problem (CSP) is used as a primitive in several braid group-based public key encryption schemes. It has been pointed out that, in braid groups, it is unlikely to provide adequate security. Therefore, new structures need to be found. In this paper, we give a formulation of the CSP for left conjugacy closed loops. In order to construct a generalization of the Anshel–Anshel–Goldfeld key establishment method, we also define a partial conjugacy search problem PCSP and show it to be equivalent to the CSP, if the underlying structure is a group. We also study more closely the PCSP in a class of conjugacy closed loops of order p2, where p is a prime.