Notions of Centralized and Decentralized Opacity in Linear Systems

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 $\epsilon$ -opacity, and also provide an extension to the case of nonlinear systems.