The Application of a Novel Fractional-order Hyper-chaotic System in Image Encryption

In this paper, a novel fractional-order hyper-chaotic system is proposed. By drawing the phase trajectory and Lyapunov exponent spectrum, its dynamic characteristics are analyzed. The simulation results show that the fractional-order hyper-chaotic system has hyper-chaotic characteristic. Then, image encryption implementation based on the fractional-order hyper-chaotic system is investigated. And, a three-color separation and scrambling the image pixel location, the histogram, key space, pixel distribution, correlation coefficient and key sensitivity of cipher-text in color pictures are tested and analyzed. The results show that the algorithm has good security and practicability.


Introduction
In 1963, Lorenz E N delivered a paper on "deterministic a periodic flow", which reveals a series of properties of chaotic motion, such as deterministic aperiodicity, extreme sensitivity to initial values, long-term behavior unpredictability, and so on. 1 His work guides the direction for chaos theory research and development in chaotic systems, such as Chen system, 2 Lü system, 3 Liu system, 4 Qi system, 5 and so on, which make chaotic dynamics theory and applied research get rapid development.In recent years, because of the unpredictability of the chaotic system and the extreme sensitivity of the system parameters, 6 coupled with the rapid development of fractional calculus theory, the fractional-order calculus and chaotic system are integrated by scholars. 7The calculus operator which added to the chaotic system is studied. 8It is found that the fractional-order system can still exhibit the chaotic state sometime, and the application of the fractionalorder chaotic system has a better practical value. 9herefore, the fractional-order chaotic systems have aroused great enthusiasm of scholars.
With the rapid development of Internet technology, much of information is released and transmitted quickly through internet.Therefore, information security issues become a focus of people's attention.Image encryption based on chaotic system processes digital image, which makes the cipher-text image disorder to cover the plaintext information and to achieve the effect of image encryption. 10In early 1990, Matthews applied the chaotic system to the image encryption algorithm. 11In the early stage, the low dimensional chaotic system is applied to image encryption.Although the lowdimensional chaotic system was designed easily, the random sequence acquired was fast.Because of low security and low complexity, it is easy to be deciphered.The hyper-chaotic system has four state variables so that the system complexity is relatively high, the key space is larger than chaotic system.The initial conditions are extremely sensitive, and the overall system security is high.With further studies of fractional-order chaotic systems, it is found that fractional-order chaotic system not only has characteristics of sensitivity to initial values and pseudo-randomness, but also can reflect the historical information of the system, which has the strong historical memory. 12In addition, the fractionalorder chaotic system enhances nonlinearity and complexity of system researched, which increases key space of encryption algorithm. 13The existing integerorder chaotic system can not predict and enhance the security of communication.Based on the literature, 14 a novel fractional-order hyper-chaotic system is proposed in this paper, and the application of this fractional-order chaotic system in digital image encryption is studied by numerical simulation.

Numerical Analysis of a Novel Fractionalorder Hyper-chaotic System
In this paper, a novel fractional-order hyper-chaotic system is constructed on the basis of literature. 14The mathematical model as follows: The system parameters are a = 33.2,b = 10, c = 18, d = 15, e = 26.The calculous order of the system (1) is q , and ) 1 , 0 ( q , where q = 0.95.The phase trajectory of the novel fractional-order hyper-chaotic system is shown in Fig. 1: The Lyapunov exponent can characterize the motion characteristics of the system.Its positive and negative values and size in one direction, denotes the degree of divergence or convergence of the adjacent orbits in the attractors for a long time. 15For the hyper-chaotic system, there must be two or more positive Lyapunov exponents.The Lyapunov exponent spectrum of system (1) is shown in Fig. 2. When a =33.

Image encryption and decryption process
Each color image is composed of R, G, B, which belongs to three primary colors.Different tricolor distributions make up different images.Image encryption is to scramble each color dot.The novel fractional-order hyper-chaotic system proposed in this paper adopts one-dimensional time-domain out-of-order at first, and then the other three-dimensional sequence matrix is used to realize the spatial encryption algorithm by performing XOR operation, to achieve the purpose of disorder.Decryption process is the inverse of the encryption process, the specific flow chart is shown in Fig. 3: Fig. 3 Flow chart of encryption and decryption The specific steps of the encryption scheme as follows: 1) The original image of R, G, B three primary colors is separated from the corresponding gray-scale image, the results are shown in Fig. 4:

P -178
The application of a The original RGB image R component

Histogram analysis
Fig. 9 is a gray histogram of the B component before and after encryption, which describes the number of pixels in each gray level.It can be seen from Fig. 9 that the number of pixels on each gray level before encryption is not uniform and the fluctuation is relatively large.The difference in the number of pixels between wave crests and troughs is relatively large, and the image distribution is obvious.After the encryption, the number of pixels in each gray level is relatively uniform, almost no fluctuations, a large degree of hiding the pixel distribution of the original image, which can be a good resistance to the statistical analysis of the decipher.Fig. 9 Histograms before and after image encryption

Adjacent pixel correlation analysis
One row and one column of pixels are selected from the original image and the encrypted image.The correlation P -179 coefficients of its adjacent pixels are calculated in Tab.
1. Observation of the data in Tab.1 can be drawn: the correlation coefficients of the original image adjacent pixels are close to 1, the correlative performance is relatively high.In addition, the correlation coefficients of the adjacent pixels of the encrypted image is close to zero, and the adjacent pixels are almost irrelevant.Therefore, the encryption algorithm based on system (1) has higher resistance to attack.

Conclusion
In this paper, a novel fractional-order hyper-chaotic system is proposed and its dynamical properties are analyzed.Firstly, the phase trajectories and Lyapunov exponent spectrum are gained, which verify that the fractional-order system has hyper-chaotic attractors.Then, this system is applied to the digital image encryption.Algorithm based on the one-dimensional time-domain chaotic sequence, and three-dimensional spatial re-encryption is used to encrypt and decrypt the color image.Finally, the key sensitivity, histogram characteristics and correlation coefficients of adjacent pixels are analyzed.The analysis results show that the encryption algorithm based on a novel fractional-order hyper-chaotic system proposed has advantages of high sensitivity and reliability. Tab.

Fig. 1
Fig.1 Phase portraits of system (1)The Lyapunov exponent can characterize the motion characteristics of the system.Its positive and negative values and size in one direction, denotes the degree of divergence or convergence of the adjacent orbits in the attractors for a long time.15For the hyper-chaotic system, there must be two or more positive Lyapunov exponents.The Lyapunov exponent spectrum of system

Fig. 4 5 :Fig. 5 Fig. 6 Fig. 7 3 . 2 . Algorithm security performance analysis 3 . 2 . 1 Fig. 8
Fig.4 The image of original and each component 2) Firstly, the chaotic sequences of w series is used to scramble the components.And then use x, y, z sequence respectively on the R, G, B matrix to perform XOR operation to achieve spatial domain encryption.The results of the encryption of each component are shown in Fig.5: This work is supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No.11202148).