This paper revisits the problem of watermarking a Gaussian host, where the embedder and attacker are subject to mean-squared distortion constraints. The worst (nonadditive) attack and unconstrained capacity have been identified in previous work. Here we constrain the encoding function to lie in a given family of encoding functions — such as spread-spectrum or fixed-dimensional Quantization Index Modulation (QIM), with or without time-sharing, with or without external dithering. This gives rise to the notion of constrained capacity. Several such families are considered in this paper, and the one that is best under the worst attack is identified for each admissible value of the watermark-to-noise ratio (WNR) and the noise-to-host ratio (NHR). With suitable improvements, even scalar QIM can outperform any (improved) spread-spectrum scheme, for any value of WNR and NHR. The remaining gap to unconstrained capacity can be bridged using higher-dimensional lattice QIM.
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