Double color image encryption based on fractional order discrete improved Henon map and Rubik’s cube transform
Introduction
With the rapid development of the Internet of Things (IoT) and Information and Communication Technology (ICT), the human society is increasingly oriented towards communications using the Internet. As text, images and videos are important carriers of information, the security in the transmission process is a key research issue. In the past 10 years, with the advances of technology, the proportion of color image information in multimedia became popular. Differently from text information, images have the characteristics of containing data capacity and strong correlation between pixels. However, traditional encryption methods are not suitable for image encryption and, therefore, various image-oriented algorithms have been advanced. We find in the literature image encryption schemes based on compressive sensing [1], [2], DNA computing [3], [4], [5] and chaos [6], [7], [8], [9], with the algorithms satisfying confusion and diffusion requirements [10], [11].
Chaotic systems have attracted the attention due to their sensitivity to initial values and the complex dynamics behavior. A number of image encryption algorithms based on chaotic systems were proposed [12], [13], [14], [15], [16]. In this case the chaotic system is used as a pseudo-random sequence generator to improve the security performance. However, traditional low-dimensional integer-order chaotic systems lead to a reduction in the key space of the algorithm. To overcome this shortcoming, fractional-order discrete chaotic systems were tested in image encryption because of their larger parameter selection space and dynamic characteristics [8], [9], [17]. Recent results show that the image encryption based merely on a chaotic system reveals loopholes [18], [19], [20]. Therefore, combining other methods to improve the security of encryption algorithm is currently an on-going research topic.
The DNA technology has high information density and high parallelism [21], [22]. Therefore, several researchers combined the DNA technique and chaos to design image encryption algorithms [23], [24], [25], [26]. The core of this technology is the DNA encoding, decoding and operation, so that the binary pattern of image information is transformed into the DNA sequence through coding rules. However, the DNA encryption algorithms still have shortcomings, such as the use of the same encoding rule for all pixels [27], [28]. This means that the encoding rule can be discovered by hackers, resulting in a limited security. On the other hand, the algorithm given by the DNA encoding is not sensitive to the plain image and secret key. On this issue, Guesmi et al. [29] proposed an encryption scheme based on DNA coding and SHA-2, by combining the initial value of the chaotic system with the HASH value of the plain image. This approach solved the problem of the algorithm’s insensitivity to the plain image.
During the transmission of digital image, multiple images need often to be encrypted. If we encrypt each image separately, then the efficiency is limited since it does not take advantage of the multiple images. Situ and Zhang proposed the multiple-image encryption algorithm [30]. As a special case of multiple-images encryption, double-image encryption is also of interest. The existing double-image encryption algorithms are mostly aimed at gray images. Zhou and Jiang [31] proposed a double-image compression-encryption scheme based on a co-sparse analysis model and random pixel exchanging. Mohammed and Saadon [32] developed a optical double-image encryption based on sparse phase information. Other relevant encryption algorithms can be found in [33], [34], [35]. Compared to the gray images, the colored ones are more common in information transmission. Nonetheless, due to the strong correlation between the three color channels, the above double-image encryption algorithm may not be able to be used directly.
Based on these ideas, a new double color image encryption algorithm is developed. First, a FODIHM is used as a pseudo-random number generator. The fractional-order discrete chaotic system has a parameter selection range far exceeding the one of the integer-order chaotic system, which greatly improves the key space of the encryption. Second, a Rubik’s cube transform method is proposed. The six color components of the double color image are used to compose a spatial structure similar to the Rubik’s cube. Based on this idea, the scrambling operation is performed to break the strong correlation between pixels while allowing both parts to save information of each other. Therefore, even if a certain image is completely lost, the plain image can be partially restored. Third, a CAT transform based on the discrete fractional Logistic map is implemented. The choice of parameters is related to the hash value of the plain image, which can effectively improve the security performance and better resist the chosen and known plain text attacks.
The rest of the paper is organized as follows. In Section 2, some important definitions and lemmas are introduced, and the novel FODIHM is formulated. In Section 3, the double color image encryption scheme based on the FODIHM is proposed. In Section 4, several simulation results and security tests are discussed. Finally, in Section 5 the main conclusions are given.
Section snippets
Fractional-order discrete improved Henon map
In this section, the FODIHM is derived by means of a three-dimensional generalization of the two-dimensional Henon map formulated by Hitzel [36]. In recent years, Manoj [37] proposed a new discrete fractional version of the generalized hyperchaotic Henon map. The dynamic behavior was analyzed and a control strategy was discussed. Jouini [38] designed a fractional three-dimensional generalized Hénon map. The adoption of tools such as bifurcation diagrams, phase portraits and Lyapunov exponents
The proposed image encryption technique
In this section, we introduce the new technique for encrypting two images simultaneously. The complete encryption system is composed of chaotic secret code stream generator, Rubik’s cube transform, Random DNA coding and CAT transform based on the fractional order discrete Logistic map and classic XOR. The corresponding flow chart is shown in Fig. 6.
Experimental platform
To verify the performance and effectiveness of the proposed encryption algorithm, we carried out several simulation experiments, using MATLAB on a laptop with intel(R) Core(TM) i5-8300H(2.3 GHz) and 8 GB RAM.
Security analysis
Four pixels color images of Lena and baboon, all black and all white are chosen to test the algorithm (https://download.csdn.net/download/qq_41987955/14882553), where Lena is as main plain image and the other three are auxiliary encrypted images. The five-parameter user-supplied
Conclusion
An improved 3-D Henon map and its fractional-order form were proposed, and their chaotic behaviors were analyzed under given initial conditions. Based on the proposed system, a new encryption algorithm for double color images was presented. Firstly, the hash value of the two images and the key entered are used as the initial value of fractional-order improved 3-D Henon map. This step greatly extends the key space of the encryption algorithm and leads to a richer system dynamics. Secondly, a new
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 would like to express their deep gratitude to the Editors and the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. This work was supported by the National Natural Science Funds of China (Nos. 62073114; 11971032), the Fundamental Research Funds for the Central Universities (No. JZ2019HGTB0090) and the Science and Technology Program of Guangzhou (No. 201707010031).
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2023, OptikCitation Excerpt :During the past several decades, the theory in fractional calculus, especially fractional difference has been developed [1–9]. More and more chaos behaviors of fractional-order systems are observed and applied in image encryption recently, such as variable-order fractional Chen’s systems [9], fractional-order double-ring erbium-doped fiber laser chaotic system [10], fractional hybrid chaos system with framelet transform [11], fractional-order memristive chaotic system with time delay [12], fractional order Chua’s system applied in phase masks and scrambling the interim result [13], fractional-order hyperchaotic system [14,15] and its encryption application with DNA approach [16], fractional order discrete improved Hénon map with Rubik’s cube transform [17], fractional-order multi-scroll Chen chaotic system with DNA Mutation [18], dynamically rotating fractional-order chaotic systems [19] and fractional-order Hénon chaotic map companied with a two-dimensional Discrete Wavelet Transform and a four-dimensional hyperchaotic system [20]. Most of the above systems are based on fractional differential equations rather than fractional difference equations.
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2022, OptikCitation Excerpt :The efficiency of the proposed QDFrKT based color image encryption techniques are reported in experimental results. Based on a new chaotic fractional-order(FO) discrete improved Hénon map(FODIHM) as a pseudo-random number generator and DNA(deoxyribonucleic acid) computation, a double color image encryption method is proposed [11]. Furthermore, using the hash value of the color image (SHA-256) and three additional user’s keys, the initial value of FODIHM is computed to increase the sensitivity of the encryption algorithm.