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1 GRACE - Geometry, arithmetic, algorithms, codes and encryption Inria Saclay - Ile de France 2 LIX - Laboratoire d-informatique de l-École polytechnique Palaiseau

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\.

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

Author: Benjamin Smith -

Source: https://hal.archives-ouvertes.fr/


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