Universal elliptic Gau{\ss} sums and applications

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.