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Rates of reliable transmission of hidden information are derived for watermarking problems involving parallel Gaussian sources, which are often used to model host images and audio signals. Constraints are imposed on the average squared-error distortion that can be introduced by the information hider and by the attacker. When distortions are measured with respect to the original host data, the optimal covert and attack channels are two banks of Gaussian test channels. The solution to the watermarking game involves an optimal allocation of distortions by the information hider and by the attacker to the different channels. A fast algorithm is given for computing the optimal solution based on duality theory. For each channel, we derive analytical expressions for two asymptotic regimes: weak and strong host signals. Finally, we extend these results to the class of stationary Gaussian host signals with bounded, continuous spectral density. The analysis also provides an upper bound on watermarking capacity for nonGaussian host signals.