# A New One-Dimensional Compound Chaotic System and Its Application in High-Speed Image Encryption
The compound map has a very wide range of chaotic parameters and the system structure is simple, which can realize fast calculation and is more suitable for real-time image encryption processing. It can also generate a uniformly distributed sequence filled with the whole state value space, which validates its application in cryptography with a large enough secret key space and can resist the violent attack of key exhaustion. The uniformly distributed state values can better resist the attack of statistical analysis. Through a series of famous chaos theory tools, such as theoretical proof of state boundedness, Lyapunov exponent, bifurcation diagram, approximate entropy, time-series diagram and cobweb graph, we prove the highly chaotic behavior of the proposed compound chaotic map. Based on the new chaotic map system, we also designed a new high-speed image encryption scheme. In this new image encryption scheme, we adopt an improved permutation-substitution strategy which completes the permutation of image pixel positions and the substitution of pixel values simultaneously and takes the image row or column as the cyclic processing unit instead of a single pixel as the cyclic processing unit, which greatly improves the security and speed of the scheme. In addition, the new image encryption scheme can achieve satisfactory encryption effects with a small number of encryption rounds. The number of encryption rounds can be adjusted according to the actual requirement. We use histogram analysis, information entropy, key analysis, and ciphertext sensitivity to plaintext and key to evaluate the efficiency and security of the scheme. In the same experimental environment, we also compare this scheme with some latest encryption schemes. The comparison results show that the new scheme is superior to other schemes in safety and speed. The main contributions of this work are as follows:  
(1) A new model of one-dimensional discrete compound chaotic map is proposed, which has two system parameters and unlimited parameter range. The parameter interval of this system, which can generate chaotic phenomena, is much wider than the existing 1D maps, and the generated chaotic signals are more evenly distributed in the range of (0, 1). Therefore, the system used for information encryption can obtain larger key space and better security. (2) Based on the new chaotic system, a new image encryption algorithm is proposed, which combines the traditional permutation and substitution operations and simultaneously realizes the pixel position permutation and pixel value substitution. It can not only improve the encryption speed, but also increase the difficulty of deciphering the key. In addition, the encryption operation cycle process takes rows (columns) as processing units instead of pixels as processing units, which can significantly improve the encryption speed. (3) The intermediate key is related to the content of the image to be encrypted. By introducing this mechanism, the ability of the cryptosystem to resist chosen-plaintext attack is improved. In addition, the sensitivity of ciphertext to plaintext is also enhanced, which leads to the improvement of the performance of the algorithm against differential cryptanalysis. The rest of this paper is arranged as follows: Section 2 introduces the proposed new chaotic map system and proves its strong chaotic behavior with common chaos theory tools. Section 3 introduces the proposed new image encryption scheme based on this chaotic map. Section 4 tests the performance and security analysis of the proposed new image encryption scheme. The last section is the summary of the whole paper.