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

TL;DR

This work addresses information leakage in stochastic cyber-physical systems by introducing a quantitative notion of opacity based on conditional entropy, $H(W_T|O_{0:T}; \pi)$, where $W_T$ indicates whether the final state is secret. It develops a budget-constrained dynamic masking framework and a primal-dual policy-gradient algorithm, with a novel gradient computation for the conditional entropy through observable operators in hidden Markov models. The method is formulated as an augmented-state MD P and solved via $L(\theta,\lambda)$, updating $\theta$ to maximize opacity while respecting a masking-cost budget $\epsilon$. Experimental validation on a small illustrative example and a stochastic gridworld demonstrates that the synthesized dynamic masks significantly increase observer uncertainty about secrets under cost constraints, outperforming baseline masking policies. This approach enables principled, cost-aware design of information-release policies for CPS and lays groundwork for extending to other opacity notions and transparency-privacy trade-offs.

Abstract

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.

Figures (6)

• Figure 1: hmm for the illustrative example. The colored boxes represent the sensors in the environment. Red - sensor $R$, Green - sensor $G$, Purple - sensor $P$, and Brown - sensor $B$.
• Figure 2: A fragment of the mdp for the hmm in Ex. \ref{['ex: 1']}.
• Figure 3: Experimental validation of the illustrative example.
• Figure 4: Gridworld depiction of a pharmaceutical research facility with cells $\{12, 30\}$ as the initial states for the robot.
• Figure 5: The sensor setup with sensors $A$ (Green), $B$ (Yellow), $C$ (Grey), and $D$ (Brown), and the robot's goal policy (Blue arrows) in the grid world environment.

Theorems & Definitions (9)

• Remark 1
• definition 1: State-based final state opacity
• Remark 2
• definition 2
• definition 3: Observable operator(jaeger2000observable)
• proposition 1
• proof
• proposition 2
• proof