  The secrecy of a communication system in which both the legitimate receiver
and an eavesdropper are allowed some distortion is investigated. The secrecy
metric considered is the exponent of the probability that the eavesdropper
estimates the source sequence successfully within an acceptable distortion
level. The problem is first studied when the transmitter and the legitimate
receiver do not share any key and the transmitter is not subject to a rate
constraint, which corresponds to a stylized model of a side channel and reveals
connections to source coding with side information. The setting is then
generalized to include a shared secret key between the transmitter and the
legitimate receiver and a rate constraint on the transmitter, which corresponds
to the Shannon cipher system. A single-letter characterization of the highest
achievable exponent is provided, and asymptotically-optimal strategies for both
the primary user and the eavesdropper are demonstrated.
