  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.
