Arnold cat map encryption in image watermarking

In the proposed scheme, the selected scrambling method is Arnold Transform Map. Although the transform is simple in Arnold Transform Map, but it not only scrambles the pixel position by encoding the number of iterations, but it also reduces the key storage and transfer spaces.

Assume that on the one hand, the original watermark image pixels are (x, y), if these pixels are scrambled N times, it results in the scrambled pixels (x’, y’). On the other hand, Arnold transforming is reversible and in order to obtain the original state of the watermark image, a fixed time of the transforming should be applied on the scrambled watermark image. Thus, the watermark image scrambling time can be regarded as a watermark key. Suppose the watermark image is with size of M×M; in addition, x and y are the coordinates of the watermark image’s pixel; moreover, x’ and y’ are the coordinates after transforming.