Public key exchange using extensions by endomorphisms and matrices over a Galois field

In this paper, we describe a public key exchange protocol based on an extension of a semigroup by automorphisms (more generally, by endomorphisms). One of its special cases is the standard DiffieHellman protocol, which is based on a cyclic group. However, when our protocol is used with a non-commutative (semi)group, it acquires several useful features that make it compare favorably to the Diffie-Hellman protocol. Here we suggest a couple of instantiations of our general protocol, with a non-commutative semigroup of matrices over a Galois field as the platform and show that security of the relevant protocols is based on quite different assumptions compared to that of the standard DiffieHellman protocol. Our key exchange protocols with this platform are quite efficient, too: with private keys of size 127 bits and public keys of size 1016 bits, the run time is 0.03 s on a typical desktop computer.