Diffie-Hellman type key exchange protocols based on isogenies

In this paper, we propose some Diffie-Hellman type key exchange protocols using isogenies of elliptic curves. The first method which uses the endomorphism ring of an ordinary elliptic curve E, is a straightforward generalization of elliptic curve Diffie-Hellman key exchange. The method uses commutativity of the endomorphism ring End(E). Then using dual isogenies, we propose a second method. This case uses the endomorphism ring of an elliptic curve E, which can be ordinary or supersingular. We extend this method using isogenies between two elliptic curves E and E′. Our methods have the security level of that of [D. Jao and L. De Feo, Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies, J. Math. Cryptol. 8 (2014), no. 3, 209--247], with the advantage of transmitting less information between two parties.