Synthesis of Dynamic Masks for Infinite-Step Opacity

We investigate the problem of synthesizing dynamic masks that preserve the infinite-step opacity in the context of discrete-event systems. Dynamic mask is an information acquisition mechanism that controls the observability of the system's events dynamically online, e.g., by turning sensors on/off . A system equipped with a dynamic mask is said to be infinite-step opaque if an outside intruder that can access all acquired information can never infer that the system was at some secret state for any specific previous instant. Existing works on the dynamic mask synthesis problem can only preserve the current-state opacity. However, synthesizing dynamic masks for the infinite-step opacity, which is stronger than the current-state opacity, is much more challenging. The main reason is that the delayed information is involved in this problem and whether or not a current secret can be revealed depends on sensing decisions to be synthesized in the future. In this paper, a new type of information state is proposed to capture all the delayed information in the infinite-step opacity synthesis problem. An effective algorithm is then presented to solve the synthesis problem, which extends existing dynamic mask synthesis techniques from the current-state opacity to infinite-step opacity. Additionally, an information-state-reduction-based approach is proposed to further mitigate the computational complexity of the synthesis procedure. Finally, we discuss how to generalize our results to a class properties with delayed information including infinite-step K-anonymity and infinite-step indistinguishability.