Block QIM watermarking games

While binning is a fundamental approach to blind data embedding and watermarking, an attacker may devise various strategies to reduce the effectiveness of practical binning schemes. The problem analyzed in this paper is design of worst-case noise distributions against L-dimensional lattice quantization index modulation (QIM) watermarking codes. The cost functions considered are 1) probability of error of the maximum-likelihood decoder, and 2) the more tractable Bhattacharyya upper bound on error probability, which is tight at low embedding rates. Both problems are addressed under the following constraints on the attacker's strategy: the noise is independent of the marked signal, blockwise memoryless with block length L, and may not exceed a specified quadratic-distortion level. The embedder's quadratic distortion is limited as well. Three strategies are considered for the embedder: optimization of the lattice inflation parameter (also known as Costa parameter), dithering, and randomized lattice rotation. Critical in this analysis are the symmetry properties of QIM nested lattices and convexity properties of probability of error and related functional of the noise distribution. We derive the minmax optimal embedding and attack strategies and obtain explicit solutions as well as numerical solutions for the worst-case noise. The role of the attacker's memory is investigated; in particular, we demonstrate the remarkable effectiveness of impulsive-noise attacks as L increases. The formulation proposed in this paper is also used to evaluate the capacity of lattice QIM under worst-noise conditions.