Efficient computation of universal elliptic Gau{\ss} sums

TL;DR

This paper presents efficient algorithms for computing elliptic Gauss sums using modular functions, with implementations and improvements, also extending methods to elliptic Jacobi sums, and analyzing their computational complexity.

Contribution

It provides detailed algorithms and implementations for computing elliptic Gauss sums and Jacobi sums, with significant speed improvements and complexity analysis.

Findings

Algorithms are faster and more efficient.

Implementations in C demonstrate practical applicability.

Complexity analysis guides future optimizations.

Abstract

In a former paper it has been shown that the elliptic Gau{\ss} sums, whose use has been proposed in the context of counting points on elliptic curves and primality tests, can be computed by using modular functions. In this work we give detailed algorithms for the necessary computations, all of which have been implemented in C. We analyse the relatively straightforward algorithms derived from the theory and provide several improvements speeding up computations considerably. In addition, slightly generalizing former results we describe how (elliptic) Jacobi sums may be determined in a very similar way and show how this can be used. We conclude by an analysis of space and run-time requirements of the algorithms.