Research on k-Step Active Opacity of Stochastic Discrete Event Systems

In recent years, the opacity of discrete event systems has achieved remarkable application results in information security mechanisms, attracting increasing attention from the researchers of the world. With the aim to address the security issue of leakage paths in the system, this paper proposes a method to enhance the $k$-step opacity of the system through active control based on the framework of stochastic discrete event systems, ensuring that all secret states in the system are not disclosed. First, the $k$-step active opacity of stochastic discrete event systems is formalized. Second, by constructing a $k$-step active opacity verifier, a controllable event string with the lowest leakage probability is found for each secret state in the system, allowing the system to enter the secret state through the safest path by enforcing controllable events. In this way, the $k$-step opacity can be enhanced, and additionally a necessary and sufficient condition for the $k$-step active opacity of stochastic discrete event systems is obtained, which can be used to verify the $k$-step active opacity of stochastic discrete event systems.