Families of fast elliptic curves from Q-curves
This work addresses the need for faster and more secure elliptic curve cryptography, offering incremental improvements over existing methods like GLV and GLS by providing a wider range of curves and twist-secure options.
The paper tackles the problem of accelerating elliptic curve-based cryptosystems by constructing new families of elliptic curves over finite fields with efficiently computable endomorphisms, resulting in faster operations than doubling and enabling secure group orders and twist-secure curves, with examples including prime-order curves for specific primes like p = 2^127-1 and p = 2^255-19.
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 \(p = 2^{255}-19\).