Constructions in public-key cryptography over matrix groups

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. ∗Partially supported by RFFI, grants, 03-01-00349, NSH-2251.2003.1. The paper was done during the stay of the author at the Mathematical Institute of the University of Rennes.