Arnold型置乱变换周期分析

Arnold变换是通信安全中数字图像置乱技术之一.在一定条件下,Arnold型置乱变换具有周期性,使得通信双方可以随机控制图像传输中变换的次数.为统一地分析各类Arnold型变换的周期性,首先建立整数矩阵模算术的基本法则,然后证明模N的Arnold型变换的周期等于以N的两两互素的因数为模的变换的周期之最小公倍数.问题于是归结为模是素数及其幂的情形.最后导出模取素数之不同的幂时相应变换周期间的关系,引入周期特征码的概念,获得对各类Arnold型置乱变换的周期的统一、规整而简洁的理解与把握.

On Periods of Arnold-type Transformations

Arnold transformation is one of the techniques for scrambling digital images for the sake of communication security. Under certain conditions Arnold transformation A modulo N possesses periodicity by which the encoder and decoder can control randomly the number of the transformations in the image transportation. To study periodicity of Arnold-type transformations, basic modular arithmetic properties of integer matrices are firstly established. A theorem is secondly obtained to show that the periodicity of A modulo N equals the least common multiple of all those moduloes mutually prime factors of N. Thus the periodicity analysis can be reduced to transformation modulo prime numbers (and their powers). Finally formulas for periodicity of transformation modulo powers of prime number are proved and the concept of period character code is introduced by which an elegant and concise method to calculate the period of Arnold-type scrambling transformation is obtained.