Towards Quantitative Analysis of Opacity

Opacity is a general approach for describing and unifying security properties expressed as predicates. A predicate is opaque if an observer of the system is unable to determine the satisfaction of the predicate in a given run of the system. The meaning of opacity is straightforward when considering the standard (qualitative) operational semantics, but there are a number of possible interpretations in a context where quantitative information about system evolutions is available. We propose four variants of quantitative opacity defined for probabilistic labelled transition systems, with each variant capturing a different aspect of quantifying the opacity of a predicate. Moreover, we present results showing how these four properties can be checked or approximated for specific classes of probabilistic labelled transition systems, observation functions, and system predicates.