Combinatorial group theory and public key cryptography

After some excitement generated by recently suggested public key exchange protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether choosing a different group, or a class of groups, can remedy the situation; (2) whether some other "hard" problem from combinatorial group theory can be used, instead of the conjugacy search problem, in a public key exchange protocol. Another question that we address here, although somewhat vague, is likely to become a focus of the future research in public key cryptography based on symbolic computation: (3) whether one can efficiently disguise an element of a given group (or a semigroup) by using defining relations.