We first present a modified Hopfield network, the clipped Hopfield network, with synaptic weights assigned to three values {-1,0,+1}. We give the necessary conditions under which a set of 2n binary vectors can be stored as stable points of the network. We show that in the parallel updating mode, for most of the state vectors, the network will always converge to these 2n stable points. We further demonstrate that these 2n stable points can be divided into two groups, the alpha group and the beta group, each with n stable points. It is shown that the basins of attraction of the stable points in the alpha group are evenly distributed, and the basins of attraction of the stable points in the beta group are also evenly distributed. By ways of application, we show that this class of Hopfield network can be used to build a cryptographically secure keystream generator.