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The Q-curve construction for endomorphism-accelerated elliptic curves

Benjamin Smith 1, 2
1 GRACE - Geometry, arithmetic, algorithms, codes and encryption
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}_{p^2}$ 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. Like GLS (which is 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 for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $\mathbb{F}_{p^2}$ equipped with efficient endomorphisms for every $p > 3$, and exhibit examples of twist-secure curves over $\mathbb{F}_{p^2}$ for the efficient Mersenne prime $p = 2^{127}-1$.
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Submitted on : Tuesday, March 24, 2015 - 10:44:46 AM
Last modification on : Thursday, March 5, 2020 - 6:31:06 PM
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Benjamin Smith. The Q-curve construction for endomorphism-accelerated elliptic curves. Journal of Cryptology, Springer Verlag, 2016, 29 (4), pp.27. ⟨10.1007/s00145-015-9210-8⟩. ⟨hal-01064255v2⟩



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