Exposure Time as a Measure of Opacity in Timed Discrete Event Systems

The reliance of many emerging applications on shared cyber-infrastructures has prompted the study of various notions for security and privacy, including notions for deterministic, non-deterministic, and probabilistic systems. The main contribution of this work is to extend these ideas to timed stochastic systems, by introducing and analyzing opacity exposure as a measure of vulnerability to security violations. More specifically, we consider partially observed stochastic Petri net (POSPN) models that behave according to Markovian dynamics (i.e., the time delays of all events are independent random variables and have exponential probability density functions). We assume that certain events and part of the marking trajectories are observable to an outside observer (intruder) who is interested in determining whether current-state opacity is violated (i.e., whether the intruder can infer with certainty that the marking of the system necessarily lies within a given secret set of states). In a timed system, as the POSPN considered in this paper, we are also interested in knowing how much time the intruder has to react in case current-state opacity is violated. We consider and analyze two critical aspects: (i) the frequency with which the system enters a time interval where an opacity violation occurs, and (ii) the amount of time the intruder has to take any malicious action.