二维混沌置乱矩阵构成置换群的理论和实验证明

Theoretical and Experimental Proof That 2D Chaotic Arrays Are Permutation Groups

群论是研究对称性问题的强有力的数学工具。混沌指的是确定性非线性动力系统表现出来的内在随机性,具有有界、非周期、对初始条件和参数极度敏感等特点,由其产生的离散序列可用来对数字图像等数据进行加密。目前已有的文献中对二者之间的关联现象鲜有研究。基于群和混沌的基本理论,结合置换群的概念,从理论和实验两方面证明了二维混沌置乱矩阵对置换变换构成置换群的结论,并由此指明了试图用不同初值,经不同混沌系统产生多个混沌二维置乱矩阵对数字图像、视频等多媒体数据进行多重置乱加密以加强安全性的做法的无效性。

Group theory is a sort of strong mathematics tool for the researches of the symmetry property. Chaos is the internal randomicity put up by the definite non-linear dynamical system. It has several properties, including the limitary, the nonperiodic and the dependence on initial condition and parameters. The discrete sequences produced by chaos system are often used to enerypt data such as digital pictures. In former papers, the relationship of Group theory and Chaotic system has seldom been studied. In this paper, it proves the result that the scrambling transform of two-dimensional chaotic scrambling arrays will form permutation groups. It is propesed on the basic theory of Group and Chaos and is proved in theoretical and experiment ways. According to the result, it is demonstrated invalid to use two-dimensional chaotic scrambling arrays created by different chaotic system and different initial values to encrypt multimedia data such as digital images and videos.