On the error exponent and capacity games of private watermarking systems

Watermarking systems are analyzed as a game between an information hider, a decoder, and an attacker. The information hider is allowed to cause some tolerable level of distortion to the original data within which the message is hidden, and the resulting distorted data can suffer some additional amount of distortion caused by an attacker who aims at erasing the message. Two games are investigated: the error exponent game and the coding capacity game. Motivated by a worst case approach, we assume that the attacker is informed of the hiding strategy taken by the information hider and the decoder, which are uninformed of the attacking scheme. This approach leads to the maximin error exponent and maximin coding capacity as objective functions. It is assumed that the host data is drawn from a finite-alphabet memoryless stationary source, and its realization (side information) is available at the encoder and the decoder. A single-letter expression for the maximin error exponent is found under large deviations distortion constraints. Moreover, we find an asymptotically optimal random coding distribution, a universal decoder, and a worst case attack channel. It is proved that there is a saddle point in the asymptotic exponent and that the minimax and the maximin error exponents are equal. Finally, a single letter expression for the coding capacity, i.e., the maximin reliable information rate, is found.