A framework for opacity in linear systems

We present a framework for opacity in 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 a passive adversarial observer. Necessary and sufficient conditions for k-ISO are given in terms of reachable sets of the system. Properties of k-ISO under unions and intersections are verified. It is seen that while unions of opaque sets preserve opacity, this is not necessarily true for intersections. We show that under certain conditions, k-ISO is equivalent to output controllability. Finally, we present an algorithm to compute a k-ISO set of states, given candidate secret and nonsecret sets of initial states.