Research On Digital Image Scrambling Algorithm

The digital image scrambling algorithm and its applications are discussed in thispaper, it include the methods of constructing the image scrambling matrix, scramblingmatrix periodicity, and application of scrambling matrix in image scrambling. The maincontributions are as follows.（1） Based on Euclid algorithm, two methods of constructing two-dimensionalgeneralized Arnold matrix were proposed. One is based on the Fibonacci series and theother is based on the Dirichlet series. Its advantage is that the users can choose a largertwo-dimensional generalized period matrix Arnold and enter encryption key bythemselves, One-time Pad Cipher is realized, which resolve the problem of only foursituation about the generalized two-dimensional Arnold matrix and greatly increase theimage encryption system security.（2）Three methods for construction n-dimensional generalized Arnold matrix areproposed. The first is constructing n-dimensional generalized Arnold Based on sequencematrices, the second is on chaotic sequence and three is on Chebyshev Chaotic NeuralNetworks. Its advantages are include that each method only determining by theencryption key is simple and can be made public. Each algorithm has larger key spaceand can produce “One-time Pad Cipher” encryption matrix with very high security. Theencryption results have good chaos and autocorrelation with the natural frequency of theplaintext being hidden and homogenized, which is favorable for resisting statisticalanalysis attacks. Transformation matrix and inverse transform matrix produced by keydo not involve complicated matrix operations in the algorithm, whose time complexityis lower, so high-dimensional transformation matrix will not beyond computing power.The image is scrambled by transformation matrix and restored by the inversetransformation matrix.（3） The inherent relationship between the two-dimensional Arnold map periodicand Fibonacci module series periodic is described, that the smallest period of theFibonacci module sequence is twice of that of two-dimensional Arnold transformationis also proved, The upper bound for the smallest module period of Arnoldtransformation is obtained, which is3N, which greatly promote the existing conclusions（N²/2）.（4） The twin Fibonacci sequence is first proposed, and its properties and theoremsare given. The theorem of the least period of the twin Fibonacci sequence mod p^r is proved by means of the mathematical induction. The paper describes the inherentrelationship between the periodic of3-dimensional Arnold map and that of the twinFibonacci sequence, the upper bound for the smallest module period of the3-dimensional Arnold transformation is obtained, which is3.14N². It is a great advancecompared with the existing best upper bound N³in the literatures.（5） The theorem of the least period of n-dimensional invertible generalized Arnoldmatrix mod p^r is proved by means of the mathematical induction, i.e., for any primenumber p, positive integer r，and n-dimensional invertible matrix A on the Galois fieldFP, if the least positive period of A（mod p） is T, then the least positive period of A（modp^r） is p^r-1T. The upper bound for the smallest module period of the n-dimensionalArnold transformation is obtained, which is Nⁿ/2. These conclusions resolve theperennial problem for us to find the model period of the transformation matrix, whichprovide a much solider theoretical foundation for image scrambling.（6） The definition for the best scrambling period of Arnold transformation is firstproposed, the number of the best scrambling transformation using Arnoldtransformation is given.（7）The several rounds encryption algorithm based on n-dimensional generalizedArnold transformation matrix with One-time Pad Cipher is proposed, which isscrambling both in image position space and color space. Its features are included longperiod, the algorithm completely open, the effectively prevention to many attacks.Experimental results show that the scrambling transformation method is efficient andhas strong security.