Faster Compact Diffie-Hellman: Endomorphisms on the x-line

Abstract

We describe an implementation of fast elliptic curve scalar multiplication optimized for Diffie-Hellman Key Exchange at the 128-bit security level. The algorithms are compact (using only x-coordinates) run in constant time with uniform execution patterns and do not distinguish between the curve and its quadratic twist, they thus have a built-in measure of side-channel resistance. (For comparison we also implement two faster but non-constant-time algorithms.) The core of our construction is a suite of two-dimensional differential addition chains driven by efficient endomorphism decompositions built on curves selected from a family of Q-curve reductions over F-p2 with p = 2(127) - 1. We include state-of-the-art experimental results for twist-secure constant-time x-coordinate-only scalar multiplication.