Enforcement for infinite-step opacity and K-step opacity via insertion mechanism

Opacity is an important information-flow property concerning the security and privacy of cyber–physical systems. We investigate the synthesis problem of infinite-step opacity and K-step opacity by using insertion function. An insertion function is a monitoring interface placed between the system and the outside observer that inserts fictitious events to the system output if necessary. To successfully enforce infinite-step opacity and K-step opacity, in this paper, we first review the insertion mechanism without considering the opacity enforcement issue, and propose two new automata for recognizing the safe languages for infinite-step opacity and K-step opacity respectively. Then, we enforce the infinite-step opacity and the K-step opacity with the reviewed insertion mechanism. Computational complexity issues are also discussed. Our results extend the prior results about the synthesis of insertion functions from the notion of current-state opacity to the notions of infinite-step opacity and K-step opacity.