  We present a new approach to handling the case of Atkin primes in Schoof's
algorithm for counting points on elliptic curves over finite fields. Our
approach is based on the theory of polynomially cyclic algebras, which we
recall as far as necessary. We then proceed to describe our method, which
essentially relies on transferring costly computations in extensions of
$\mathbb{F}_p$ to isomorphic ones endowed with a special structure allowing to
reduce run-time. We analyse the new run-time and conclude this procedure yields
some improvement as compared to the classical approaches.
