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arXiv · 1605.07749

Computing cardinalities of Q-curve reductions over finite fields

François Morain·Charlotte Scribot·Benjamin Smith
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Abstract

We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

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François Morain, Charlotte Scribot, Benjamin Smith. 2016-05-25. Computing cardinalities of Q-curve reductions over finite fields. https://doi.org/10.1112/s1461157016000267

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