The Theoretical Limits of Watermark Spread Spectrum Sequence

At present, the spread spectrum (SS) sequences used in watermark include i.i.d. random sequences and the sequences used in SS communications. They appear earlier than digital watermark. Almost no researchers pay attention to whether they are really fit for watermark. In this paper, we compare the SS watermark channel and the traditional SS communication channel. We find out that their correlation property is different. Considering cropping and translation attacks, we define watermark auto- and cross-correlation and propose Loose Autocorrelation and Tight Cross-Correlation (LAC&TCC) properties for SS watermark. The LAC&TCC properties are that, whether or not synchronized, the autocorrelation is equal or close to 1 and the cross-correlation is equal or close to 0. Accordingly, the peak correlation is divided into the peak autocorrelation R (a)(l) and the peak cross-correlation R (c)(l). We establish the lower bound of R (c)(l) and the higher bound of R (a)(l), respectively. The two bounds indicate that, no matter how small the cover is reserved, the extractor can always find a threshold to distinguish auto- and cross-correlation in theory.