In content fingerprinting, the same media covertext - image, video, audio, or text - is distributed to many users.
A fingerprint, a mark unique to each user, is embedded into each copy of the distributed covertext. In a collusion
attack, two or more users may combine their copies in an attempt to "remove" their fingerprints and forge a
pirated copy. To trace the forgery back to members of the coalition, we need fingerprinting codes that can reliably
identify the fingerprints of those members. Researchers have been focusing on designing or testing fingerprints
for Gaussian host signals and the mean square error (MSE) distortion under some classes of collusion attacks,
in terms of the detector's error probability in detecting collusion members. For example, under the assumptions
of Gaussian fingerprints and Gaussian attacks (the fingerprinted signals are averaged and then the result is
passed through a Gaussian test channel), Moulin and Briassouli1 derived optimal strategies in a game-theoretic
framework that uses the detector's error probability as the performance measure for a binary decision problem
(whether a user participates in the collusion attack or not); Stone2 and Zhao et al.3 studied average and other
non-linear collusion attacks for Gaussian-like fingerprints; Wang et al.4 stated that the average collusion attack
is the most efficient one for orthogonal fingerprints; Kiyavash and Moulin5 derived a mathematical proof of the
optimality of the average collusion attack under some assumptions.
In this paper, we also consider Gaussian cover signals, the MSE distortion, and memoryless collusion attacks.
We do not make any assumption about the fingerprinting codes used other than an embedding distortion
constraint. Also, our only assumptions about the attack channel are an expected distortion constraint, a memoryless
constraint, and a fairness constraint. That is, the colluders are allowed to use any arbitrary nonlinear
strategy subject to the above constraints. Under those constraints on the fingerprint embedder and the colluders,
fingerprinting capacity is obtained as the solution of a mutual-information game involving probability density
functions (pdf's) designed by the embedder and the colluders. We show that the optimal fingerprinting strategy
is a Gaussian test channel where the fingerprinted signal is the sum of an attenuated version of the cover signal
plus a Gaussian information-bearing noise, and the optimal collusion strategy is to average fingerprinted signals
possessed by all the colluders and pass the averaged copy through a Gaussian test channel. The capacity result
and the optimal strategies are the same for both the private and public games. In the former scenario, the original
covertext is available to the decoder, while in the latter setup, the original covertext is available to the encoder
but not to the decoder.