Enhancing opacity of stochastic discrete event systems using insertion functions

This paper extends prior works on the enforcement of opacity for systems modeled as finite-state automata using insertion functions. Insertion functions insert fictitious events at the output of the system, thereby obfuscating its secret behaviors. In this paper, we use insertion functions to enhance opacity for systems modeled as probabilistic finite-state automata. Specifically, we quantify each insertion function based on the opacity level that it achieves, defined as the probability that the system remains opaque. We then develop an algorithmic procedure that synthesizes an optimal insertion function that maximizes the opacity level, when the system's observable output is finite-length.