Synthesis of Dynamic Masks for Information-Theoretic Opacity in Stochastic Systems

In this work, we investigate the synthesis of dynamic information releasing mechanisms, referred to as ''masks'', to minimize information leakage from a stochastic system to an external observer. Specifically, for a stochastic system, an observer aims to infer whether the final state of the system trajectory belongs to a set of secret states. The dynamic mask seeks to regulate sensor information in order to maximize the observer's uncertainty about the final state, a property known as final-state opacity. While existing supervisory control literature on dynamic masks primarily addresses qualitative opacity, we propose quantifying opacity in stochastic systems by conditional entropy, which is a measure of information leakage in information security. We then formulate a constrained optimization problem to synthesize a dynamic mask that maximizes final-state opacity under a total cost constraint on masking. To solve this constrained optimal dynamic mask synthesis problem, we develop a novel primal-dual policy gradient method. Additionally, we present a technique for computing the gradient of conditional entropy with respect to the masking policy parameters, leveraging observable operators in hidden Markov models. To demonstrate the effectiveness of our approach, we apply our method to an illustrative example and a stochastic grid world scenario, showing how our algorithm optimally enforces final-state opacity under cost constraints.