Stochastic encryption against stealthy attacks in CPSs: A zero-sum game approach

The zero-sum game-based stochastic encryption problem for cyber–physical systems (CPSs) under stealthy attacks is investigated in this paper. Firstly, a deterministic encryption strategy is presented to achieve the same estimation performance as the standard Kalman filter in the absence of attacks. Since a deterministic procedure may be deciphered, the encryption/attack strategy will be randomly employed to counter potential adversaries. The impact of stochastic strategies on the estimation performance is evaluated by calculating the expected error covariance. Based on the trade-off among the energy levels of encryption parameters, attackers, and defenders, the Nash equilibrium for the attacker–defender game is derived offline, providing guidance on when to implement encryption and attack strategies. In particular, we also provide a criterion for selecting encryption parameters to restrict the attacker’s payoff. The theoretical results are validated through simulations.