Non-commutative Key Exchange Protocol Based on Morphisms of Supersingular Curves

In cryptography, key-exchange protocols are of central interest and their main goal is securely exchanging a secret key between two parties over a public channel. Th e security of these protocols is based on the hardness of some famous mathematical problems. Diffie-Hellman key exchange algorithm is one of the most practical protocols in this area which is based on the discrete logarithm problem on a group. Th e commutativity of the underlying group is an essential property in the original Diffie-Hellman protocol. In this paper, we focus on non-commutative groups and some of their properties that seem to be useful in constructing needed one-way functions in noncommutative key exchange protocols based on conjugacy problem. We particularly consider the non-commutative group of endomorphism ring of the supersingular elliptic curves and study the transmitted form of conjugacy problem on these rings in order to construct a key exchange algorithm based on the hardness of finding a trivialpoint-preserve morphism between these curves over a finite field. Finally, we will discuss the efficiency and security features of the proposed scheme.