ACM and rectangular images: Overlapping partitions, implementation, and periodicity analysis

Abstract

The Arnold Cat Map (ACM) is a popular chaotic map used in image encryption. Chaotic maps are known for their sensitivity to initial conditions and their ability to permute, or rearrange, pixels. However, ACM is periodic, and its period is relatively short. This periodicity decreases the effective key-space and security of a cryptosystem using ACM. Further, ACM is typically only able to be performed on square images. To solve the low periodicity and typical limitation to square images, this paper proposes performing ACM on overlapping square partitions which cover the entirety of an image. The presence of overlap results in a greatly increased image period. The resulting system will be referred to as overlapping ACM or OACM. Several papers have already discussed systems involving overlapping ACM. However, they did not discuss the implementation or periodicity of such a system in detail. This paper does cover the implementation and periodicity analysis of OACM and proposes a simple symmetric encryption system which uses OACM. The proposed encryption system is not as sophisticated or secure as other modern encryption schemes, since it is mainly intended as an initial test of OACM’s utility. Histogram and sensitivity analyses did however indicate a level of security against various cryptographic attacks, and OACM performed reasonably in both the permutation and diffusion stages of the cryptosystem.