Exposure and Revelation Times as a Measure of Opacity in Timed Stochastic Discrete Event Systems

Opacity is a security notion that focuses on determining whether a given system's behavior is kept secret to intruders. Various notions of opacity have received significant attention during the last decade including current state opacity and initial state opacity, which have been studied for deterministic and probabilistic systems in untimed contexts. In timed systems, opacity requirements may vary with time and one could also be interested in knowing the time duration for which opacity requirements are violated or preserved. The main contribution of this article is to introduce and analyze opacity exposure and opacity revelation times as measures of vulnerability in timed discrete event system (DES) that behave according to Markovian dynamics (i.e., at any given time, all enabled events are independent and distributed in time with exponential probability density functions). Labeled stochastic Petri nets (LSPNs) are used to model timed stochastic DESs, and appropriate constructions (involving current and initial state observers) are used to evaluate exposure and revelation times for a given LSPN.