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.

中文翻译：

---

针对 CPS 中隐蔽攻击的随机加密：一种零和博弈方法


该文研究了隐蔽攻击下信息物理系统 （CPS） 的基于零和博弈的随机加密问题。首先，提出了一种确定性加密策略，以便在没有攻击的情况下实现与标准卡尔曼滤波器相同的估计性能。由于可以破译确定性程序，因此将随机采用加密/攻击策略来对抗潜在对手。随机策略对估计性能的影响是通过计算预期误差协方差来评估的。基于加密参数、攻击者和防御者之间的能量水平之间的权衡，攻击者-防御者博弈的纳什均衡是离线推导出的，为何时实施加密和攻击策略提供了指导。特别地，我们还提供了一个标准，用于选择加密参数以限制攻击者的收益。理论结果通过仿真进行了验证。