An image encryption algorithm based on cascade chaotic system

At present, chaotic encryption has become one of the hot topics in cryptography research, but most of the existing chaotic encryption algorithms are based on a single chaotic system. Facts show that some chaotic maps can be accurately predicted by phase space reconstruction methods [1]. In addition, due to the limitation of computer precision, the time series output by a single chaotic system cannot achieve theoretically complete randomness, but this defect can be improved by cascading multiple chaotic systems [2]. To this end, this paper proposes an image encryption algorithm based on the cascade of multiple chaotic systems. Both theoretical analysis and numerical experiments show that this algorithm can achieve the purpose of obfuscation and diffusion required by cryptography, and can effectively prevent differential attacks.

1 Generation of chaotic sequences

1.1 Logistic Mapping

The logistic map was proposed by mathematical ecologist May in 1976. It is a nonlinear iterative equation and the most widely studied dynamical system. The definition of the logistic map is:

When 3.569 945 6<μ≤4, the Logistic map works in a chaotic state, that is, the sequence {xk} generated by the initial condition x0 under the action of the Logistic map is non-periodic, non-convergent, and very sensitive to the initial value; when μ=4, the map is surjective, and the generated chaotic sequence is ergodic on the interval (0, 1). Since the Logistic map has characteristics similar to white noise, simplicity, and initial value sensitivity, many chaotic image encryption algorithms are based on the Logistic map.

1.2 Spatiotemporal Chaos Mapping

A spatiotemporal chaotic system is an extended system in space[3], which exhibits chaos in time and space. Coupled map lattices (CMLs) are usually used as spatiotemporal chaotic systems. This system is a dynamic system with discrete time, discrete space and continuous state. It consists of nonlinear mappings called local mappings located at lattice sites, and each local mapping is coupled with other local mappings according to certain rules. Due to the inherent nonlinear dynamic characteristics of each local mapping and the divergence caused by the mutual coupling, CML can exhibit spatiotemporal chaos. Therefore, different forms of CML can be constructed by using different local mappings and coupling methods[4]. The two-dimensional CML constructed by this algorithm is:

2 Implementation of encryption and decryption

The chaotic systems selected in this algorithm are spatiotemporal chaotic systems and one-dimensional Logistic mapping. First, the spatiotemporal chaotic system is used to generate a random sequence, and then the sequence values ​​are used as the initial values ​​of the Logistic mapping of formula (1). After a specific number of iterations, the final required chaotic sequence is obtained. This specific number is determined by the result of the encryption of the previous image pixel.

(4) Starting from the last two pixels of image c1, operate the pixel values ​​according to step (3) in the opposite direction to obtain image c, which is the encrypted ciphertext image.
2.2 Decryption process

The decryption process is the opposite of the encryption process, that is, the number of iterations mentioned in step (2) is changed to be determined by the first two pixel values ​​of the ciphertext image, and the order of steps (3) and (4) is reversed to complete the decryption of the ciphertext image.

3 Security Analysis

This algorithm has high security, a larger key space, and can resist most common attacks.

3.3 Statistical analysis

The correlation between adjacent pixels in an image is very large. In order to defend against statistical attacks during the encryption process, the correlation between adjacent pixels must be reduced [5]. This paper randomly selects 2,008 pairs of pixels from the image to be encrypted and the encrypted image, tests the pixel correlation in the horizontal, vertical, and diagonal directions, and calculates the correlation coefficient using formula (8):

3.4 Differential Attack Analysis

By making a small change to the encrypted image and then observing the result of the change, the attacker can obtain the correlation between the encrypted image and the original image. If an encryption algorithm can make a small change to the original image, so that the encrypted results before and after change greatly, then the algorithm can effectively prevent differential attacks.

The number of pixels change rate (NPCR) refers to the rate of change of the number of pixels in the encrypted image when the image to be encrypted changes one pixel. The larger the NPCR, the more sensitive the encryption algorithm is to the changes in the image to be encrypted, and the stronger the encryption algorithm is in resisting plaintext attacks; the average intensity change rate (UACI) refers to the rate of change of the average intensity of the corresponding pixels of the image to be encrypted and the encrypted image. The larger the index, the greater the change in the average intensity of the encrypted image compared to the image to be encrypted, and the stronger the encryption algorithm is in resisting differential attacks. Suppose the two encrypted images are c1 and c2, then:

This paper proposes an image encryption algorithm based on cascade chaotic system. It uses one-dimensional CML composed of Logistic mapping as the spatiotemporal chaotic system, and then uses its output sequence as Logistic to obtain the final key sequence after a certain number of iterations from a certain initial value. Security analysis shows that the key space of this algorithm is large enough to make brute force attacks impossible. Simulation experimental results also show that this algorithm has high performance and has certain potential application value in image encryption and image transmission.