Measuring secrecy by the probability of a successful guess

A secrecy system in which both the legitimate receiver and an eavesdropper are allowed some distortion is studied. The eavesdropper is allowed to make one guess, and the considered secrecy metric is the exponent of the probability that the guess is within an acceptable distortion level. A subsequent notion of information leakage is suggested. In the absence of any shared key between the transmitter and the legitimate receiver, a single-letter characterization of the highest achievable exponent is provided. Moreover, asymptotically-optimal universal strategies for both the primary user and the eavesdropper are demonstrated, where universality means independence of the source statistics. When a secret key is shared between the transmitter and the legitimate receiver, and a rate constraint is imposed, upper and lower bounds on the optimal exponent are given. Certain sufficient conditions for the bounds to meet are provided, and examples where they apply are discussed.