Diffie-Hellman type key exchange protocols based on isogenies

Abstract

‎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.