Local Negative Base Transform and Image Scrambling

Scrambling transform is an important tool for image encryption and hiding. A new class of scrambling algorithms is obtained by exploiting negative integer as the base of number representation to express the natural numbers. Unlike Arnold transform, the proposed scrambling transform is one-dimensional and nonlinear, and an image can be shuffled by using the proposed transform to rearrange the rows and columns of the image separately or to permute the pixels of the image after scanned into a sequence of pixels; it can be also applied to shuffle certain part region of an image. Firstly, the transformation algorithm for converting nonnegative integers in base to the corresponding integers in base is given in this paper, which is the computational core of scrambling transform and the basis of studying scrambling transform. Then, the three kinds of transforms are introduced, that is, negative base transform (abbreviated as NBT), modular negative base transform (MNBT), and local negative base transform (LNBT) with three parameters, where NBT is an injection and MNBT a surjection and LNBT a bijection. The minimum transform periods of LNBT are calculated for some different values of the three parameters, and the algorithm for calculating the inverse transform of LNBT is given. The image scrambled by LBNT can be recovered by the transform period or the inverse transform. Numerical experiments show that LNBT is an efficient scrambling transform and a strong operation of confusing gray values of pixels in the application of image encryption. Therefore, the proposed transform is a novel tool for information hiding and encryption of two-dimensional image and one-dimensional audio.