Elliptic Gau{\ss} sums and Schoof's algorithm

TL;DR

This paper introduces a novel method for improving Schoof's algorithm by leveraging polynomially cyclic algebras to efficiently handle Atkin primes, resulting in reduced computational runtime.

Contribution

The paper presents a new approach using polynomially cyclic algebras to optimize Schoof's algorithm for elliptic curve point counting, especially for Atkin primes.

Findings

Reduced runtime in Schoof's algorithm for Atkin primes

Effective transfer of computations to isomorphic algebraic structures

Improved efficiency over classical methods

Abstract

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.