On the capacity game of private fingerprinting systems under collusion attacks

The problem of fingerprinting in the presence of collusive attacks is considered. It is modeled as a game between a fingerprinter and a decoder on the one hand, and a coalition of two or more attackers on the other. The fingerprinter distributes, to different users, different fingerprinted copies of a host data (covertext) embedded with different fingerprints. The coalition members create a forgery of the data while aiming at erasing the fingerprints in order not to be detected. Their action is modeled by a multiple-access channel (MAC). The decoder, who has access to the original covertext data, observes the forgery and decodes one of the messages in order to identify one of the members of the coalition. Motivated by a worst case approach, we assume that the coalition of attackers is informed of the hiding strategy taken by the fingerprinter and the decoder, while they are uninformed of the attacking scheme. A single-letter expression for the capacity is derived under the assumption that the host data is drawn from a memoryless; stationary source and some mild assumptions on the operation of the encoder. It is shown that for a coalition consisting of L < infinity members, the capacity scales with O(1/L), and whenever L grows with the length of the covertext, the capacity is essentially zero. Also, a lower bound on the error exponent is derived as a by-product of the achievability part, and asymptotically optimum strategies of the parties involved are characterized.