Chaos properties of the Chaotic Permutation generated by Multi Circular Shrinking and Expanding Movement

In the image encryption, chaotic map usually used for pixel permutation, due to it can generate chaotic permutations hence difficult to predict. However chaotic map which is widely used for image permutations like Arnold Cat Map (ACM) has a relatively low recurrence period so vulnerable to the known image attack. Besides, the key space Arnold cat map is also relatively low so vulnerable to brute-force attack and limited to N square matrix. ACM generally combined with other chaotic maps to produce the encryption that is more resistant to brute force attacks and the known image attacks. But there come with a cost, which is the additional process and the encryption can´t be purely based on permutations that known robust to noise. Therefore in this paper we propose a chaotic permutation method which has a very large recurrence period and key space, and can be applied to M × N matrix. The proposed method is to apply the Chaotic Permutation Multi Circular Shrinking and Expanding Movement (CPMCM) which is controlled by the expanded key. Based on the analysis of chaos properties, the chaotic method meets the characteristics of the chaos system, with a very large recurrence period which is equal to the Least Common Multiple (LCM) of all real numbers N where N is the size of the permuted block. And also has a very large key space that is equal to N factorial. The Spearman correlation test also showed that among the permutation sequences are not correlated or random.