International Association for Cryptologic Research

International Association
for Cryptologic Research

IACR News item: 28 May 2013

Benjamin Smith
ePrint Report ePrint Report
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\\).

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