New constructions by composition of families of multidimensional periodic arrays for video watermarking

Digital watermarking is a technique for embedding secure information into digital media in order to address issues of illegal distribution and content authentication. Families of periodic arrays have recently been applied to video watermarking, where they are embedded into video frames in a way that remains imperceptible to users but can be securely and unambiguously detected by the owner without requiring the original media. Detection relies on cross-correlation between the watermark and the watermarked frames, where a high ratio between the correlation peak and the surrounding values ensures reliable detection. Suitable watermark families must satisfy several properties: a large number of members, a high peak auto-correlation to cross-correlation ratio (with a high unique peak auto-correlation value, low non-peak auto- and cross-correlation values), and high cryptographic security measured by the linear complexity parameter. These conditions determine the effectiveness of a family of periodic arrays as watermark candidates. In this work we present four types of families of multidimensional periodic arrays, types A, B, D and E. Families A and B are constructed by composing a Legendre array with sequences derived from cyclic groups of points on an elliptic curve over a prime field. Family A was first introduced in previous work, but contains a miscaracterization of cross-correlation. We correct this error and we propose Family B as a slightly different alternative. Families A and B have high linear complexity, but low correlation ratios when compared to other known constructions. To address this limitation, we introduce families of types D and E. Families D and E are defined by two versions of composition: the plane version and the column version. In both constructions, the cyclic shifts are defined by composing logarithmic quadratic and fractional functions, respectively, over the direct product of cyclic groups of integers modulo coprime integers. The column version uses these cyclic shifts applied to a Sidelnikov ternary sequence, while the plane version applies cyclic shifts to a plane obtained by a folded version of the same Sidelnikov ternary sequence. The peak auto-correlation to non-peak auto-correlation and cross-correlation ratios are on the order of p^l where p is an odd prime and l is a positive integer, whereas the best ratio for other known constructions is on the order of p^2. Moreover, these families maintain high average linear complexity, and exhibit optimal low cross-correlation values with respect to the Welch bound. Finally, an embedding and detection scheme for the proposed watermark families in video is presented. Experimental results demonstrate that the families remain robust and imperceptible under distinct levels of H.264 compression, a widely used video coding format in streaming platforms and online services.