The $$\mathbb {Q}$$ Q -curve Construction for Endomorphism-Accelerated Elliptic Curves

Abstract

We give a detailed account of the use of $$\mathbb {Q}$$Q-curve reductions to construct elliptic curves over $$\mathbb {F}_{p^2}$$Fp2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant---Lambert---Vanstone (GLV) and Galbraith---Lin---Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves and thus finding secure group orders when $$p$$p is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $$\mathbb {F}_{p^2}$$Fp2 equipped with efficient endomorphisms for every $$p > 3$$p>3, and exhibit examples of twist-secure curves over $$\mathbb {F}_{p^2}$$Fp2 for the efficient Mersenne prime $$p = 2^{127}-1$$p=2127-1.