Extended Insertion Functions for Opacity Enforcement in Discrete-Event Systems

Opacity is a confidentiality property that holds if certain secret behavior of a system, typically represented by a predicate, cannot be revealed under any system evolution. Among other proposed methodologies, when opacity is violated, it can be enforced using insertion mechanisms, i.e., by inserting symbols before an actual system output (in real time as the system evolves) in order to replace observation sequences that lead to opacity violations with observation sequences that can be generated by system behavior that does not violate opacity. This article focuses on opacity enforcement in discrete-event systems modeled with finite-state automata and proposes an extended insertion mechanism that can enforce opacity in a practical manner to a wide class of systems by inserting symbols before and after an actual system output. This article also introduces event insertion constraints that require only certain specific symbols to be inserted before and after an actual system output. For each case, we obtain a necessary and sufficient condition (based on the construction of an appropriate verifier) for opacity enforceability using the proposed extended insertion mechanism and devise a pertinent extended insertion strategy.