An image encryption scheme based on double chaotic cyclic shift and Josephus problem☆
Introduction
Nowadays, with the rapid development of Internet technology and social media, more and more digital information, especially digital images, is transmitted frequently in the network. In order to prevent these digital images containing a large amount of privacy information from being obtained and exploited by unauthorized adversaries, image encryption as an effective means has attracted great attention. Compared with traditional text data, digital images have some inherent properties, such as strong correlation of adjacent pixels, massive data and high redundancy. Therefore, if classical text encryption methods such as DES [1] and AES [2] are directly applied to digital image encryption, it is proved that the encryption effect is not ideal.
To solve these problems, in recent years, so as to improve the security and efficiency of image transmission, many methods in various fields have been used in the encryption and decryption of images. In particular, various image encryption algorithms have been proposed based on compressed sensing [3], [4], DNA coding [5], [6], [7], [8], magic square transformations [9], cellular automata [10], [11], [12], [13], and chaos theory [14], [15], [16], [17], [18], [19]. Among them, due to the characteristics of pseudo-randomicity, ergodicity and high sensitivity to initial values and initial conditions of chaotic systems, they are very consistent with the basic requirements of cryptography. In 1998, Fridrich first introduced chaos theory into Shannon’s classic image encryption structure based on permutation and diffusion [14]. And this method greatly improves the efficiency under the premise of ensuring security. Accordingly, chaos theory is widely used to design image encryption algorithms. Wang et al. [20] proposed a fast image encryption algorithm based on Logistic map, which is used to generate the keys. Khan et al. [18] presented a novel Intertwining and NCA-based image encryption scheme. The Intertwining logistic map applied in this algorithm can overcome the shortcomings of Logistic map. Hua et al. [21] constructed a new 2D-LSCM map and further proposed a 2D-LSCM-based image encryption algorithm. In [22], a 5D multiwing hyper-chaotic system was utilized in the generation of the key stream associated with the original image. Specifically, in the scrambling process, not only pixel-level permutation but also bit-level permutation is used to enhance the security of the cryptosystem. Moreover, by dividing the image into blocks, the 2D image is transformed into 3D space, and the 2D and 3D reversible modular chaotic maps are used to permute the pixels in [23].
Although high-dimensional chaotic maps with more complex structures have more variables and parameters, the cost of generating sequences and computations increases accordingly. In addition, many chaotic image encryption schemes confuse pixels by sorting chaotic sequences, which greatly increases the computational complexity. The authors in [24] presented a new hybrid digital chaotic system by combining two chaotic maps as part of a new chaotification method. Further, the new chaotic system is used to design a novel image cipher. Liu et al. [25] developed a new 2D-SIMM map and chaotic shift transform (CST) to encrypt digital images. However, CST needs at least two rounds of encryption to achieve the desired visual effect.
To sum up, we know that when designing image encryption algorithm based on chaos, high computational cost and hardware consumption should be avoided due to chaotic sequences that must be generated by the algorithm. Due to the huge amount of data in digital images, we should pay attention to these problems when applying chaotic sequences. If the number of sequences generated is equal to or more than the number of plaintext pixels, then pixel confuse and diffusion operations are performed on this basis. Even multiple rounds of encryption are needed to achieve the desired effect. Even if the effect of encryption can be qualified, it is against the original intention of light-weight algorithms. Therefore, when using chaotic sequences, how to reduce computational complexity becomes the focus of attention. In [19], chaotic sequence is used to sort the row and column vectors of the image, so as to achieve the effect of scrambling the image and reduce the complexity from to . In addition, we can also use other technical means combined with chaotic map to design image encryption algorithm, to achieve the purpose of compensating for the shortcomings of chaotic map itself. Recently, a DNA key based chaotic meaningful image encryption scheme was given by Khan et al. [8]. In the proposed scheme, the authors take advantage of the good properties of DNA computing, such as wide parallelism, ultra-low power consumption and large storage space. Khedmati et al. [13] presented an image hidden method based on a 2D hybrid chaotic system and the framelet transform. And the cellular automata was also used in encryption system to improve the security of the algorithm. Based on the above reasons, we apply Josephus problem to design an image encryption scheme based on chaos.
In this paper, we propose an image encryption scheme based on double chaotic cyclic shift (DCCS) and Josephus problem, which we denote as JP-DCCS. JP-DCCS is further improved while following the classical obfuscation and diffusion structure. In order to further improve CST proposed in [25], we proposed DCCS to permute pixels. As a result, only a round of random row shift and column shift is achieved so that the plain-image does not have a specific texture. In the diffusion stage, we first divided the target image into blocks and selected the pixels in different sub-blocks through the Josephus problem [26], so as to perform modular addition and XOR operation to change pixel values. The goal is to further break the correlation between the pixels so that both permutation and diffusion operations can be achieved in one step. In addition, the suggested scheme is highly sensitive to small changes in pixel values, and if a pixel value changes, it will propagate to subsequent pixels. Simulation results and security analysis demonstrated that JP-DCCS is efficient and able to encrypt a meaningful plain-image into a high quality cipher-image that can resist all kinds of attacks. Additionally, compared with other image encryption algorithms, our algorithm is proved to be excellent and performance.
The rest of this paper is organized as follows: Section 2 shows the performance analysis of chaotic map used in this paper and the specific description and improvement of Josephus problem. Section 3 presents JP-DCCS Image encryption scheme, and proposes a new secret key generation algorithm. Section 4 provides the simulation experiment and the security analysis for our proposed algorithm. Finally, the conclusion is given in Section 5.
Section snippets
Preliminaries
In this section, we will introduce the two chaotic maps used in this paper, which are Logistic maps [27] and Standard maps [28], [29], and further elaborate the mathematical explanation of Josephus problem. In order to reduce the computational complexity of the encryption algorithm, we use one-dimensional and two-dimensional chaotic maps with simple structure. And to enhance security, we introduce chaotic sequence into Josephus permutation. The meanings of some symbols used in this paper are
JP-DCCS image encryption scheme
This section presents the full details of a new image encryption scheme based on JP-DCCS. The algorithm can be partitioned into two phases: key generation and image encryption. In the first phase, we design the key generation algorithm with a large key space and high sensitivity. And then in the second phase, we adopt the classical permutation–diffusion structure to encrypt images. In the permutation process, we improved the CST method proposed in [25]. We not only control the cyclic shift of
Simulation results and security analysis
In this section, we present the simulation results and security analysis of the proposed JP-DCCS algorithm to verify its effectiveness and superiority. A qualified image encryption algorithm needs to have the following basic properties : (1) The encrypted image has the ability to obtain no information about the original image from the human vision system; (2) The key space should be large enough to prevent violent cracking attacks; (3) The process of encryption and decryption is reversible, and
Conclusion
In this paper, we propose a new image encryption algorithm based on double chaotic cycle shift and Josephus problem. The chaotic maps used in this paper are analyzed and the optimal parameter range is selected to be applied to the encryption system. The whole image encryption scheme is divided into two main parts, including permutation phase and diffusion phase. In the permutation phase, we adopted the improved CST algorithm to improve the encryption strength, while in the diffusion phase, we
CRediT authorship contribution statement
Rui Wang: Conception and design of study, Software, Analysis of data, Writing - original draft. Guo-Qiang Deng: Conception and design of study, Writing - review & editing. Xue-Feng Duan: Methodology, Writing - review & editing, Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors wish thank Professor D. S. Suganya and the anonymous referee for providing very useful suggestions.
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The work was supported by the National Natural Science Foundation of China (No.11561015; 11761024), the Natural Science Foundation of Guangxi Province, China (No.2016GXNSFFA380009; 2017GXNSFBA198082; 2016GXNSFAA380074),Guangxi Science and Technology Project, China (Guike AD18281024).