Embedded Insertion Functions for Opacity Enforcement

We investigate the enforcement of opacity, an information-flow privacy property, using insertion sequences that modify the output of the system by event insertions. Previous work considered the problem of enforcing the opacity under the assumption that the insertion functions were based on the observed system strings. Now, we investigate the more powerful method of insertion sequences based on the exact system states and events. In this case, the insertion function would be embedded into the system itself, rather than being an output interface. In this article, we develop methods that verify if a valid insertion function exists in this setting; if one exists, synthesize one using a computationally effective algorithm; and investigate a special case where it is possible to verify and synthesize a valid embedded insertion function with polynomial complexity in the size of the system.