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

In this paper, we present a classification of classes of elliptic curves defined over cubic extension of finite fields with odd characteristics, which have coverings over the finite fields therefore can be attacked by the GHS attack. We then show the density of these weak curves with hyperelliptic and non-hyperelliptic coverings respectively. In particular, we shown for elliptic curves defined in Legendre forms, about half of them are weak.