Information-theoretic analysis of spherical fingerprinting
Pierre Moulin, Ying Wang · 2009
Information-theoretic performance limits of digital fingerprinting systems subject to almost-sure squared-error distortion constraints on the fingerprint embedder and the colluders are derived in this paper. The rate of the fingerprinting code is R = 1/N log M where N is codelength and M is the number of users. No assumption is made on the host signal statistics, but the collusion channel is also subject to a location-invariant condition. The receiver knows neither the collusion channel nor even the number of colluders. Capacity is the supremum of achievable rates and is shown to be equal to 1/2K log(1 + Df/KDc) where K is the number of colluders, and Dfand Dcare the L2-distortion tolerance levels for the fingerprint embedder and the colluders, respectively. The worst collusion is shown to consist of uniform linear averaging of the coalition's marked copies followed by addition of independent spherical noise. Positive error exponents are achieved at all rates below capacity using random spherical fingerprinting codes and a new universal decoding criterion based on empirical Gaussian mutual information. It is also shown that minimum-distance decoding fails for this problem, and that a simple single-user decoder is almost as good as the universal decoder for large K. Geometric interpretations for all the results are given.