We formulate notions of opacity for cyberphysical systems modeled as discrete-time linear time-invariant systems. A set of secret states is @math -ISO with respect to a set of nonsecret states if, starting from these sets at time @math , the outputs at time @math are indistinguishable to an adversarial observer. Necessary and sufficient conditions to ensure that a secret specification is @math -ISO are established in terms of sets of reachable states. We also show how to adapt techniques for computing under-approximations and over-approximations of the set of reachable states of dynamical systems in order to soundly approximate k-ISO. Further, we provide a condition for output controllability, if @math -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: i) the presence or lack of a centralized coordinator, and ii) 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 @math -opacity, and also provide an extension to the case of nonlinear systems.
 Opacity describes the inability for an external observer to know what happened in a system. Recently, opacity has been investigated in the framework of discrete event systems. In our previous paper, we define two types of opacities: strong opacity and weak opacity. Given a general observation mapping, a language is strongly opaque if all strings in the language are confused with some strings in another language and it is weakly opaque if some strings in the language are confused with some strings in another language. In this paper, we investigate properties of opacities. We show that opacities are closed under union, but may not be closed under intersection. Based on these properties, we discuss how to modify languages to satisfy the strong opacity, weak opacity, and no opacity by investigating the sublanguages and superlanguages that are strongly opaque, weakly opaque, and not opaque respectively. We find the largest sublanguages and smallest superlanguages. Examples are given to illustrate results. In this paper, we investigate opacity of discrete event systems in a decentralized framework with several agents, each of them performing its observation of the system. We consider two cases, one without coordination among agents and one with coordination. Both cases are useful because many systems used today are distributed over a network, some with agents coordinating among themselves and some without. We introduce general definitions of decentralized opacity for both cases. The definitions are based on languages. Therefore, they are flexible and can include other properties of discrete event systems as special cases. In particular, we show that co-observability used in supervisory control is a special case of decentralized opacity. We illustrate the usefulness of decentralized opacity by applying it in solving an interesting security problem in computer systems.