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Title:
The Q-curve Construction for Endomorphism-Accelerated Elliptic Curves
Authors: Benjamin Smith
Abstract:We give a detailed account of the use of (\mathbb{Q})-curve reductions to construct elliptic curves over (\mathbb{F}{p^2}) 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) 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}) equipped with efficient endomorphisms for every (p > 3), and exhibit examples of twist-secure curves over (\mathbb{F}_{p^2}) for the efficient Mersenne prime (p = 2^{127}-1).
ePrint: https://eprint.iacr.org/2014/727
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