IACR News item: 28 May 2013
Benjamin Smith
ePrint ReportOur 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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