The purpose of the paper is to give new key agreement protocols
(a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization
of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key
cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups
of a given group. To support these and other known cryptographic schemes we present a
general technique to produce a family of instances being matrix groups (over finite commutative
rings) which play a role for these schemes similar to the groups Z_n^* in the existing
cryptographic constructions like RSA or discrete logarithm.