Enforcing opacity by insertion functions under multiple energy constraints

This paper investigates the enforcement of opacity by insertion functions subject to multiple quantitative constraints capturing resource or energy limitations. There is a malicious intruder attempting to infer secrets of the system from its observations. To prevent the disclosure of secrets, the insertion function inserts fictitious events to the output of the system to obfuscate the intruder. The system is initialized with several types of resources, referred to as energy. The energy is consumed or replenished with event occurrences while always consumed with event insertions. The insertion function must enforce opacity while ensuring that each type of resource is never depleted. This problem is then reduced to a two-player game between the insertion function and the system (environment), with properly defined objectives. A game structure called the Energy Insertion Structure, denoted by EIS is proposed, which provably contains solutions to the energy constrained opacity enforcement problem. Then we further study the bounded cost rate insertion problem on the insertion function’s winning region of EIS, which requires that the long run average rate of insertion cost be bounded. This problem is formulated as a multidimensional mean payoff game and a special method called hyperplane separation technique is applied to efficiently solve it.