A fast encryption algorithm of color image based on four-dimensional chaotic system

https://doi.org/10.1016/j.jvcir.2015.09.014Get rights and content

Highlights

  • A new four-dimensional hyper-chaotic system is proposed.

  • A new pseudo-random sequence generator is proposed.

  • Row-major and column-major are proposed to diffuse the image.

  • The parameters of cat map are related to the original image pixels.

Abstract

As the low complexity of low-dimensional chaotic system and the slow speed of image encryption, this paper proposes a fast encryption algorithm of color image based on four-dimensional chaotic system. Firstly, we propose a new method of designing four-dimensional chaotic system based on the classical equations of three-dimensional chaotic system, to increase the complexity and key space of the encryption algorithm. Secondly, according to the nature of color images’ pixels channel, we design a new pseudo-random sequence generator and reuse the random sequence, to improve the speed of image encryption. Finally, the methods of row-major and column-major are used to diffuse the original image and the cat map with parameter is used to scramble the image pixels, respectively, to achieve the effect of encryption. The results of simulation and security analysis show that the proposed encryption algorithm is of good performance on security, robustness and high encryption speed.

Introduction

Network information exchanging platform provides a great convenience for people’s life. Multimedia communication has become an important means of communication. At the same time, the information security is becoming more and more important. Image as a carrier of information dissemination, its security is becoming very important. The main method to protect the image information is image encryption. Due to the large amount information of the image and the high correlation between pixels, traditional encryption methods are not suitable in image encryption. Because the dynamic properties of chaotic system are very similar with traditional cryptography, the chaotic cryptography has been widely studied in image encryption. The common encryption process is generating pseudo-random sequence by chaotic sequences, then using the pseudo-random sequence to scramble and diffuse the image pixels [1], [2], [3], [4]. By increasing the scrambling and diffusing rounds to increase the complexity of the encryption algorithm or using complex chaotic equations to generate chaotic sequence [5], [6] to increase the periodicity of the original chaotic system can well satisfy the performance requirements of image encryption, such as sensitivity to initial value, the ability of resisting differential attack and statistical analysis. By the principle of cryptanalysis, the security of an encryption algorithm is decided by its key space, the key space of encryption algorithm needs to be large enough to resist the brute force attack at the existing computing capacity. Common method of expanding the key space is designing a high-dimensional differential equations. Most of the methods of designing high-dimensional differential equation are adding a non-linear fourth-dimension based on the classic three-dimensional differential equations, then adding a linear feedback to the previous three dimensions [7], [8], [9], [10]. Jia et al. [11] proposed a four-dimensional differential equation which exists chaotic state in a large range. In this way, they set the parameter of the chaos equation to be un-known as a key for the system to expand the key space. With the introduction of four-dimensional differential equation in image encryption, the method of generating pseudo-random sequence has a great influence on the speed and security of image encryption. Zhu et al. [12] proposed a method to generate a pseudo-random sequence which needs iterate the chaotic equation 3×M×N/4 times for diffusing a M×N color image one round. Wang and Jiang [13] proposed an improved HIE [14] algorithm which need iterate the chaotic equation at least M×N times for a M×N color image encryption. Using the chaotic equation to generate array for scrambling is the common method of scrambling image pixels, it will take extra time to generate the scrambling array. Guan et al. [15], [16] used the cat map with parameters to scramble image pixels which can solve the extra time of generate scrambling array.

In order to improve the security and encryption speed of encryption system, this paper proposes a fast algorithm of color image encryption. The core is proposing a new method to design four-dimensional differential equation on the basis of the classic three-dimensional chaotic system. By validating dynamics characteristic under different parameters, we select the chaotic system in a better state. Then we use one round of row-major and column-major to diffuse the image, and make a reusing of key stream in the diffusion process which can greatly reduce the time of encryption. Finally, the cat map with parameters is used to scramble the image pixels, and the parameters are related to the original plaintext image, it can increase the complexity of the encryption algorithm.

In this paper, the chapters are organized as follows. Section 2 introduces the method of designing a new four-dimensional chaotic equation in detail and describes the dynamic characteristics of chaotic equation. Section 3 describes the algorithm of pseudo-random sequence generator. Section 4 introduces the specific encryption algorithm of color image based on hyper-chaotic system. Section 5 is the implementation of the encryption algorithm and security analysis of the encrypting algorithm. The last part makes a summary for this paper.

Section snippets

The design of new four-dimensional chaotic equation

Because of the special nature of chaotic systems, researchers are constantly researching the new equations with chaotic dynamics on the basis of the existing chaotic equation. In this paper, based on the classic three-dimensional chaotic system, we design a method to generate a new four-dimensional chaotic equation. Eq. (1) is the model of three-dimensional chaotic system used in this paper.dy1dt=a1(y2-y1)dy2dt=b1y1+c1y2-y1y3dy3dt=d1y3+y1y2

When a1 = 16, b1 = 45, c1 = −1, d1 = −2, Eq. (1) is the

Design of pseudo-random sequence generator

Image encryption is similar with the traditional stream cipher encryption, the length of the key stream which is used to encrypt depends on the size of the image. For a gray image with size of M×N, we should generate M×N bytes binary key stream at least, and for a color image with size of M×N we should generate M×N×3 bytes binary key stream at least. The key stream is produced by the chaotic sequence, so reducing the length of chaotic sequence is an important method to reduce the time in image

The encryption algorithm of color image

According to the nature of color image’s channel, this paper processes the three channels of image pixel respectively. The specific structure of image encryption is shown in Fig. 9:

On the basis of Section 2, the chaotic sequences are generated by the hyper-chaotic system when the parameters of this system are set to be a=16,b=45,c=-2,d=48,f=-4,g=-1,e=1. Then use the pseudo-random sequence generator proposed in Section 3 to generate three key streams key1, key2 and key3. The concrete steps of

Encryption results

Set the parameters of chaotic systems to be a=16,b=45,c=-2,d=48,f=-4,g=-1,e=1. Use matlab to simulate the encryption and decryption algorithm on personal computer with 2G memory, 2.8 GHz CPU. Taking an example of 1024 × 1024 color image Lena, we set the initial values to be [0, 1, 0, 2], the results of image encryption and decryption with right keys and error keys are shown in Fig. 12.

Security analysis

  • (1)

    Key space analysis.

    On the basic of hyper-chaotic system, the key space depends on the initial value of the chaotic

Conclusions

This paper proposes a method to generate a new four-dimensional chaotic equation by combining the three-dimensional equations, and chooses the hyper-chaotic system to generate the chaotic sequence by analyzing the complexity of the four-dimensional chaotic system. A new pseudo-random sequence generator is designed in this paper. It can make full use of the chaotic sequence to generate key stream, and reuses the key stream in diffusion, which can reduce the chaotic sequence length greatly. We

Acknowledgements

This work was supported by the National Natural Science Foundation of China (60973162), the Natural Science Foundation of Shandong Province of China (ZR2014FM026 and ZR2009GM037), the Science and Technology of Shandong Province, China (2013GGX10129, 2010GGX10132 and 2012GGX10110), the National Cryptology Development Foundation of China (No. MMJJ201301006), Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-14-005) and the Engineering Technology and Research Center

References (29)

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This paper has been recommended for acceptance by M.T. Sun.

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