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We present a specialized point-counting algorithm for a class of elliptic curves over
that includes reductions of quadratic
-curves modulo inert primes and, more generally, any elliptic curve over
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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