  This paper addresses the problem of infinite-step opacity and K-step opacity
of discrete event systems modeled with Petri nets. A Petri net system is said
to be infinite-step/K-step opaque if all its secret states remains opaque to an
intruder for any instant within infinite/K steps. In other words, the intruder
is never able to ascertain that the system used to be in a secrete state within
infinite/K steps based on its observation of the systems evolution. Based on
the notion of basis reachability and the twoway observer, an efficient approach
to verify infinite-step opacity and K-step opacity is proposed.
