Families of fast elliptic curves from Q-curves
Benjamin Smith (INRIA Saclay - Ile de France, LIX)

TL;DR
This paper introduces new families of elliptic curves over _{p^2} with efficient endomorphisms, enabling faster cryptographic computations and broader curve selection for enhanced security and performance.
Contribution
It constructs novel families of elliptic curves from -curves with efficient endomorphisms, expanding options for secure and fast cryptographic implementations.
Findings
Constructed two one-parameter families over _{p^2} with endomorphisms faster than doubling
Provided examples of prime-order and twist-secure curves over _{p^2} for specific primes
Demonstrated wider curve selection compared to GLS, with applications to cryptography.
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…
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