Universal elliptic Gau{\ss} sums and applications

TL;DR

This paper introduces a new method for computing elliptic Gauss sums using modular functions, enabling efficient point counting on elliptic curves and primality testing.

Contribution

It defines universal elliptic Gauss sums via modular functions and demonstrates their efficient computation and application to elliptic curve point counting and primality proofs.

Findings

Efficient representation of elliptic Gauss sums in terms of the $j$-invariant.

Application of these sums to point counting on elliptic curves.

Use in primality testing algorithms.

Abstract

We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By means of certain well-known modular functions we define the universal elliptic Gau{\ss} sums and prove they admit an efficiently computable representation in terms of the $j$-invariant and another modular function. After that, we show how this representation can be used for obtaining the elliptic Gau{\ss} sum associated to an elliptic curve over a finite field $\mathbb{F}_p$, which may then be employed for counting points or primality proving.