  This work addresses private communication with distributed systems in mind.
We consider how to best use secret key resources and communication to transmit
signals across a system so that an eavesdropper is least capable to act on the
signals. One of the key assumptions is that the private signals are publicly
available with a delay---in this case a delay of one. We find that even if the
source signal (information source) is memoryless, the design and performance of
the optimal system has a strong dependence on which signals are assumed to be
available to the eavesdropper with delay.
  Specifically, we consider a distributed system with two components where
information is known to only one component and communication resources are
limited. Instead of measuring secrecy by "equivocation," we define a value
function for the system, based on the actions of the system and the adversary,
and characterize the optimal performance of the system, as measured by the
average value obtained against the worst adversary. The resulting optimal
rate-payoff region is expressed with information theoretic inequalities, and
the optimal communication methods are not standard source coding techniques but
instead are methods that stem from synthesizing a memoryless channel.
