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Secret key can be used to provide privacy guarantees in a communication channel.  Our approach is to measure the level of privacy by the ability (or, preferably, inability) of an adversary to produce a correlated signal based on the intercepted message but without the secret key.  If a distortion function is relevant, then we wish to inflict maximal distortion on the adversary's reconstruction.

One key assumption for our setup and results is a causally informed adversary.  Not only does the adversary observe the public communication, he also has access to the protected information itself strictly causally.  For instance, the information sequence might be an actuator signal in a distributed control setting.  An adversary who is trying to disturb the system can observe the behavior of the system to gather the past part of the information signal, which can potentially be used along with the public communication to make inferences about the future actions of the system.

We provide an information theoretic region that fully characterizes the optimal level of protection that secret key can provide against an adversary.  Our results include a more general setting where the intended receiver of the communication is not required to reconstruct the information sequence but instead produces actions that are correlated with the information.

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