Cryptology ePrint Archive: Report 2013/312

Families of fast elliptic curves from Q-curves

Benjamin Smith

Abstract: We construct new families of elliptic curves over \(\FF_{p^2}\) 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. Our construction is based on reducing \(\QQ\)-curves---curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates---modulo inert primes. As a first application of the general theory we construct, for every \(p > 3\), two one-parameter families of elliptic curves over \(\FF_{p^2}\) equipped with endomorphisms that are faster than doubling. Like GLS (which appears as 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. Unlike GLS, we also offer the possibility of constructing twist-secure curves. Among our examples are prime-order curves equipped with fast endomorphisms, with almost-prime-order twists, over \(\FF_{p^2}\) for \(p = 2^{127}-1\) and \(p = 2^{255}-19\).

Category / Keywords: Elliptic curve cryptography, endomorphisms, GLV, GLS, exponentiation, scalar multiplication, Q-curves

Date: received 23 May 2013

Contact author: smith at lix polytechnique fr

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Version: 20130528:212430 (All versions of this report)

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