Extended supersingular isogeny Diffie–Hellman key exchange protocol: Revenge of the SIDH

Abstract

The supersingular isogeny Diffie-Hellman key exchange protocol (SIDH) was introduced by Jao and De Feo in 2011. SIDH operates on supersingular elliptic curves defined over F p 2 , where p is a large prime number of the form p ¼ 4 e A 3 e B − 1 and e A and e B are positive integers such that 4 e A ≈ 3 e B . A variant of the SIDH protocol, dubbed extended SIDH (eSIDH), is presented. The eSIDH makes use of primes of the formHere ℓ B and ℓ C are two small prime numbers; f is a cofactor; and e A , e B , and e C are positive integers such that 4 e A ≈ ℓ e B B ℓ e C C . It is shown that for many relevant instantiations of the SIDH protocol, this new family of primes enjoys faster field arithmetic than the one associated with traditional SIDH primes. Furthermore, its richer opportunities for parallelism yield a noticeable speed-up factor when implemented on multicore platforms. A supersingular isogeny key encapsulation (SIKE) instantiation using the prime eSIDH-p765 yields an acceleration factor of 1.06, 1.15 and 1.14 over a SIKE instantiation with the prime SIKE-p757 when implemented on k = {1, 2, 3}-core processors. To the authors' knowledge, this work reports the first multicore implementation of SIDH and SIKE.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Faster Key Generation of Supersingular Isogeny Diffie-Hellman

“…For each of them, the three-point ladder [3] and isogeny computation (including isogeny construction and isogeny evaluation at points) are dominant. In the SIDH implementation, they take 17%∼19% and 81%∼83% of the overall computational cost, respectively [5]. Although the latter one requires more computational resources, the optimization of the three-point ladder is still meaningful to improve the performance of SIDH and SIKE.…”