Universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm

TL;DR

This paper introduces universal elliptic Gauss sums for Atkin primes within Schoof's algorithm, providing a new theoretical framework and computational approach, but with limited practical efficiency improvements.

Contribution

It defines universal elliptic Gauss sums for Atkin primes and derives an efficient representation in terms of modular functions, expanding the theoretical understanding of Schoof's algorithm.

Findings

Provides a new representation of elliptic Gauss sums in terms of modular functions.

Analyzes computational aspects and proposes an alternative Frobenius trace computation method.

Concludes that the new method is not practically competitive with existing approaches.

Abstract

This work builds on earlier results. We define universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable representation in terms of the $j$-invariant and two other modular functions. We analyse the necessary computations in detail and derive an alternative approach for determining the trace of the Frobenius homomorphism for Atkin primes using these pre-computations. A rough run-time analysis shows, however, that this new method is not competitive with existing ones.