Diffie-Hellman type key exchange protocols based on isogenies

Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi

Authors

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.

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History

• Receive Date: 13 May 2017
• Revise Date: 01 October 2017
• Accept Date: 19 May 2017