Elliptic curves with weak coverings over cubic extensions of finite fields with odd characteristic

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 subjected to the GHS attack. The densities of these weak curves, with hyperelliptic and non-hyperelliptic coverings, are then 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. keywords Elliptic curves, Hyperelliptic curves, Non-hyperelliptic curves, Index calculus, GHS attack, Cover attack