Periodicity and Application for a kind of n-dimensional Arnold-type Transformation

Abstract

Summary form only given. Along with the rapid development and wide application of Internet technique, the research about communication security becomes a very important topic in theory and the practice. For the explicitness and large information content of images, much attention has been paid to the security of image storing and transmission in communication. As one of the key technique of digital image security problem, the digital image scrambling technique can be seem as one of the method for digital image encryption and used for digital image hiding, digital watermark, precondition, and after-treatment process of numerical resume for image deposit. Most digital image scrambling technique are based on the Arnold transformation, magic square, and fractal etc. By the Arnold transformation, the image becomes more and more complicity, but the image is recovered in a certain moment. In this way, the hiding purpose for image in communication can be reached. The periodicity of the two-dimensional Arnold transformation was discussed in many literatures. The necessary and sufficient condition about the existence of periodicity for the general Arnold matrix transformation had been given in literatures. The periodicity of n-dimensional Arnold transformation had been discussed in literatures. Some new image scrambling method based on three-dimensional Arnold transformation had been given in literatures. In this paper, we extend the conclusion about the periodicity of n-dimensional Arnold transformation, apply it in digital image scrambling, and give the key-dependent digital image scrambling scheme. For the n-dimensional transformation, the necessary and sufficient condition about the existence of periodicity is gcd(|A|, N)=1, where |A|=det(A) and A is the transformation matrix, N is module. We define the transformation satisfied above character to n-dimensional Arnold-type transformation. Obviously, we can derive the conclusion that the n-dimensional Arnold-type transformation is periodic. Using properties of the similar matrixes, we can derive the following conclusion. For a given module N, if the n-dimensional Arnold-type transformation matrix A is similar to B, then the period of transformation A is equal to that of transformation B. There is a shortcoming for the application of the Arnold transformation in digital image hiding. The security of the algorithm is based on the assumption that the attacker doesn't know about the algorithm (i.e. the attacker don't know the transformation matrix). If the attacker know about the algorithm, it is easy to recover the image. So this algorithm doesn't fill the requirement of the modern cryptography. However, we can apply the above n-dimensional Arnold-type transformation to avoid this lack. It can ensure the security of the digital image scrambling by the secrete key even the algorithm is public. We select a n-dimensional invertible matrix as the secrete key, then derive transformation matrix. In this paper, we extend the conclusion which about the Arnold transformation in the former paper using the property of the similar matrix and a digital image scrambling scheme is proposed by a kind of Arnold-type transformation in order to raise the security of the image scrambling.