We formulate notions of opacity for cyberphysical systems modeled as discrete-time linear time-invariant systems. A set of secret states is k-ISO with respect to a set of nonsecret states if, starting from these sets at time 0, the outputs at time k are indistinguishable to an adversarial observer. Necessary and sufficient conditions to ensure that a secret specification is k-ISO are established in terms of sets of reachable states. We also show how to adapt techniques for computing underapproximations and overapproximations of the set of reachable states of dynamical systems in order to soundly approximate k-ISO. Furthermore, we provide a condition for output controllability, if k-ISO holds, and show that the converse holds under an additional assumption. We extend the theory of opacity for single-adversary systems to the case of multiple adversaries and develop several notions of decentralized opacity. We study the following scenarios: first, the presence or lack of a centralized coordinator, and, second, the presence or absence of collusion among adversaries. In the case of colluding adversaries, we derive a condition for nonopacity that depends on the structure of the directed graph representing the communication between adversaries. Finally, we relax the condition that the outputs be indistinguishable and define a notion of _-opacity, and also provide an extension to the case of nonlinear systems.

中文翻译：

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线性系统中集中和分散不透明度的概念


我们为建模为离散时间线性时不变系统的网络物理系统制定不透明的概念。如果从时间 0 处的这些集合开始，时间 k 处的输出对于敌对观察者来说是无法区分的，则一组秘密状态相对于一组非秘密状态而言是 k-ISO。确保秘密规范是 k-ISO 的必要和充分条件是根据可达状态集来建立的。我们还展示了如何采用计算动力系统可达状态集的欠近似和过近似的技术，以便可靠地近似 k-ISO。此外，如果 k-ISO 成立，我们提供了输出可控性的条件，并表明在附加假设下反之亦成立。我们将单对手系统的不透明理论扩展到多个对手的情况，并发展了几种去中心化不透明的概念。我们研究以下场景：首先，是否存在集中协调员；其次，对手之间是否存在共谋。在对手共谋的情况下，我们得出了一个不透明的条件，该条件取决于表示对手之间通信的有向图的结构。最后，我们放宽了输出不可区分的条件并定义了_-不透明度的概念，并且还提供了对非线性系统情况的扩展。