Current-state opacity modelling and verification in partially observed Petri nets

Abstract

System opacity is a widely studied security notion, implying that a secret behaviour of a given system cannot be seen or assessed by an external observer based on the system evolution. This work deals with the problem of current-state opacity formulation and verification in the context of discrete event systems modelled with partially observed Petri nets (POPNs) (i.e., Petri nets containing place sensors that measure the number of tokens in observable places and event sensors that indicate the firing of observable transitions). A Petri net system is recognized as current-state opaque if the current-state estimate is never entirely contained in the set of secret states. In this regard, we introduce the notion of discernible markings to design a reduced state estimator called a discernible reachability graph, and then come up with formal modelling of current-state opacity in POPN systems. The main idea of the proposed approach consists in proving that if a system is current-state opaque, its current-state estimate, possibly established by an intruder, contains at least one non-secret state. We exploit the mathematical feasibility to formulate this concept by defining and solving an integer linear programming problem with respect to a given secret and an observation sequence collected from sensors. In the light of the proposed modelling, necessary and sufficient conditions are proposed for opacity verification, and examples are given to expose the results.