Computing cardinalities of Q-curve reductions over finite fields

TL;DR

This paper introduces a faster point-counting algorithm for certain elliptic curves over finite fields, leveraging a new endomorphism to improve practical efficiency while maintaining theoretical complexity.

Contribution

It develops a specialized SEA variant using a low-degree endomorphism, enhancing practical performance for cryptographically relevant curves.

Findings

Algorithm is significantly faster in practice than traditional SEA.

Maintains the same asymptotic complexity as SEA.

Applicable to elliptic curves with low-degree isogenies to their Galois conjugates.

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.