Summary: In this paper, we present a classification of elliptic curves defined over a cubic extension of a finite field with odd characteristic which have coverings over the finite field therefore are subjected to the GHS attack. The densities of these weak curves, with hyperelliptic and non-hyperelliptic coverings, then are analyzed respectively. In particular, we show, for elliptic curves defined by Legendre forms, at least half of them are weak. We also give an algorithm to determine if an elliptic curve belongs to one of two classes of weak curves.