The Opacity of Real-Time Automata

Opacity is an important property on information flow to guarantee that a system under attack keeps its “secrets”, possibly subsets of traces (language-based opacity) or subsets of states (state-based opacity), opaque to the outside intruder with partial observability. In this paper, we investigate the opacity problems of real-time automata (RTA), which is a popular model for real-time systems. In order to prove that the language-opacity problem of RTA is decidable, we introduce the notion of trace-equivalence and then translate RTA into finite-state automata (FA) with timed alphabets. Besides, we also introduce the notions of partitioned timed alphabet and language to guarantee trace equivalence is preserved by complementation and product operations over FA with timed alphabets. Thus, our decision procedure can be sketched as follows: first, translate the RTA to model a system under attack and the RTA to specify the secret behavior of the system into FA, respectively; then, compute another FA, which accepts all traces accepted by the first FA, but not by the second one; afterwards, project these FA onto the given observable set; finally, unify the alphabets of these FA such that for any two timed actions with the same event, their time parts do not have any overlap. Thus, whether the original system is language-opaque with respect to the secret RTA and the observable set is reduced to the inclusion problem of regular languages. Similarly, we can show decidability of initial-opacity of RTA.