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Computing cardinalities of $\mathbb{Q}$ -curve reductions over finite fields

Part of: Computational number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 August 2016

François Morain ,
Charlotte Scribot  and
Benjamin Smith
François Morain
Affiliation:
École Polytechnique/LIX, and Centre national de la recherche scientifique (CNRS), and Institut national de recherche en informatique et en automatique (INRIA), France email morain@lix.polytechnique.fr
Charlotte Scribot
Affiliation:
Ministère de l’Éducation Nationale, France
Benjamin Smith
Affiliation:
Institut national de recherche en informatique et en automatique (INRIA), and École Polytechnique/LIX, and Centre national de la recherche scientifique (CNRS), France email smith@lix.polytechnique.fr

Abstract

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We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$ -curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ 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.

MSC classification

Primary: 11Y16: Algorithms; complexity
Secondary: 11G20: Curves over finite and local fields 11G05: Elliptic curves over global fields

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