Reduced-Complexity Approach for Opacity Enforcement Based on Extended Insertion Functions with Limited Insertion Events

Opacity is a security and privacy property that guarantees predefined secret behavior of a cyber-physical system will not be inferred with certainty by a malicious entity. If a system is opacity-violating, extended insertion functions that insert fake symbols can be implemented to enforce opacity. By constricting the ways in which the fake symbols are inserted, this paper addresses opacity enforcement under the framework of finite state automata using extended insertion functions under event insertion constraints and limited insertion events constraints, i.e., only a limited number of symbols in a specified set can be inserted before and after an actual system output. The notion of (k b ,k a )-EIC-enforceability is proposed to describe how to enforce opacity via extended insertion functions in the presence of both event insertion constraints and limited insertion events constraints. A necessary and sufficient condition to check (k b ,k a )-EIC-enforceability is derived by constructing an appropriate verifier.