An efficient double-image encryption and hiding algorithm using a newly designed chaotic system and parallel compressive sensing
Introduction
Currently, the perfection of information technology brings convenience to people, but at the same time, there are many security problems. Images, as more intuitive information carriers, have been widely applied in the fields of information transmission, medicine, military and so on. Considering that personal privacy and confidential documents cannot be leaked, how to design a safe and efficient image encryption algorithm is a subject that requires in-depth research.
The advanced encryption standard (AES) algorithm, which is relatively simple, is a well-known encryption algorithm. However, inherent image characteristics cause inconsistency in the traditional encryption algorithm, and its implementation effect is not outstanding. In addition, the simple AES algorithm has difficulty resisting some means of attack. We can find new means of attack to break this advanced encryption standard at any time. Therefore, the security of AES in image encryption is poor. At present, image encryption schemes using completely different technologies, such as chaotic systems [1], [2], [3], compressive sensing [4], [5], [6], cellular automata [7], [8], DNA sequences [9], [10], and Boolean networks [11], [12], have been researched. Among them, chaotic systems have been widely applied to image encryption owing to their high computational efficiency. The unpredictability, initial value sensitivity and pseudorandomness of chaos theory have been gradually explored. Meanwhile, different chaotic image encryption methods have been proposed. Chaotic systems can be roughly divided into three categories: low-dimensional chaotic systems, high-dimensional chaotic systems and spatiotemporal chaotic systems. Although the chaotic behavior of the low-dimensional chaotic system is not complicated, it is easy to implement and has merits in running time and cost [13]. The high-dimensional chaotic system has at least two control parameters and a large key space. Although it has a better chaotic effect, its implementation cost is higher [14]. A spatiotemporal chaotic system varies with time and space [15]. Considering the characteristics of each type of system, it is of practical significance to design a system with a simple nonlinear form under the condition that the number of control parameters is as few as possible and the chaotic trajectory is complex. Inspired by previous researchers, a new one-dimensional piecewise chaotic map is proposed in this paper, and it is applied to the field of image encryption. The designed one-dimensional chaotic system ICTM combines two traditional 1D chaotic systems, maintaining the advantages of the simple implementation of the original one-dimensional chaotic system but also improving the disadvantage of the small key space of the 1D chaotic system. In particular, ICTM has a wider range of control parameters and can exhibit complex chaotic trajectory phenomena represented by high-dimensional systems. At the same time, compared with a high-dimensional system, its implementation is relatively simple, and its real-time transmission is also excellent. It is worth noting that the high-dimensional chaotic system can produce multiple chaotic sequences under a limited number of iterations, while the proposed system can only obtain multiple chaotic sequences by increasing the number of iterations or truncating the generated chaotic sequences many times according to the experimental requirements. Nevertheless, the computational complexity of this system is much less than that of a high-dimensional system under the condition of meeting the same requirements. In summary, the proposal of the new system is necessary.
Reviewing the existing image encryption techniques, most of them convert plain images into noise-like images so that important information of the image cannot be captured during the transmission of the image, and the security of the digital image in the content is protected. However, it is undeniable that noise-like images are hardly targeted by other attackers during transmission, which increases the risk of encryption algorithms being cracked. Therefore, some visually meaningful image encryptions combining cryptographic technology and hiding technology [16], [17], [18] were introduced. Bao et al. [19] first introduced a visually secure cryptographic system. First, the plain image was transformed into a noise-like image by a conventional image encryption algorithm, and then the integer wavelet transform was used to split the ciphertext data, and finally, it was embedded in another carrier image. It is worth noting that the carrier image selected by this scheme is four times larger than the original image. To solve this problem, a series of visual image encryption algorithms were proposed based on compressive sensing [20], [21], [22], [23], [24]. For instance, Chai et al. [21] introduced an image encryption and hiding algorithm combining compressive sensing and dynamic least significant bit (LSB) embedding, making the size of the cipher image equal to that of the ordinary image. This approach saves a considerable amount of transmission bandwidth and storage space compared with that in Ref. [19]. It is worth mentioning that this method is completely reversible, and there is no energy loss in image embedding and extraction, so the image decryption quality will not be affected. Nevertheless, another drawback of LSB embedding is that it is vulnerable to channel noise in the transmission process, and the robustness of the algorithm is difficult to guarantee. Therefore, Zhu et al. [22] proposed a robust meaningful image encryption algorithm based on block compressive sensing and singular value decomposition embedding, which process plain images in a block manner, achieving better parallel processing of medium- and large-scale images while pursuing high security. It is not difficult to find that the majority of encryption algorithms proposed at present are single image encryption, and there are relatively few studies on visually meaningful image encryption schemes for multiple images [25], [26]. More importantly, it is difficult to obtain impartial metrics for the algorithm. Considering that different embedding methods [27], [28], [29], [30] have their own characteristics, it is meaningful to design and implement a multi-image encryption and hiding algorithm by using the characteristics of certain embedding methods.
In general, a new chaotic system model is designed and applied on the basis of the existing one-dimensional chaotic system. Compared with previous studies, our new one-dimensional chaotic system has better chaotic performance. Based on the new system, two sparse plain images are encrypted and measured. Then, two compressed and encrypted noise-like images are embedded into the real part and the imaginary part of the carrier image processed by FFT. As a result, the efficiency of image encryption and transmission is improved. Overall, the main advantages of the proposed algorithm are as follows:
a) To enhance the unpredictability of secret code streams and the efficiency of encryption algorithms, a new chaotic system is improved on the basis of two classical one-dimensional chaotic maps.
b) To improve the transmission efficiency of the image, the carrier image is transformed into the form of a complex matrix by fast Fourier transform in the embedding process, and the two compressed and encrypted plain images are embedded into their real part and imaginary part, respectively, to realize double image encryption.
c) To improve the quality of the cipher image, the step of quantifying secret information is omitted in the encryption process to reduce the transmission of the key and the accumulation of truncation error.
d) In the decryption process, blind extraction is realized; that is, the extraction of secret data does not depend on the carrier image.
The remainder of this paper is arranged as follows. In the second section, the relevant knowledge of the proposed algorithm is introduced, including a detailed description of a new chaotic system, and its performance is analyzed by some tests. In the third section, the implementation process of the encryption and decryption algorithm is described in detail. Simulation experiments are presented in the fourth section. In the following section, the performance analysis of the proposed scheme is introduced. Then, a brief summary is given in the final section.
Section snippets
1D infinite collapse map
In general, the 1D iterative chaotic mappings discussed have finite collapses in the iterative region, which limits their Lyapunov exponent to some extent. The 1D infinite collapse map (ICM) has improved this deficiency, and its expression is defined as follows [31]:where u ≠ 0 is a control parameter. The bifurcation diagram of ICM in the range of (0,10) is shown in Fig. 1(a). The periodic window indicates that the chaos phenomenon does not exist everywhere within the parameter
The generation of parameters
Step 1. On the basis of Eq. (12), select two plain images P1 and P2 of size M × N, and the sum of some pixels of the two images are denoted as P. Convert P to a 64-bit hexadecimal number n1 through SHA-256.
Step 2. Through Eq. (13), calculate the average of the pixel values of the two plain images and generate an 8-bit hexadecimal number n2.
Step 3. Use n2 to disturb n1. Since the upper bits have a
Experimental results
To fully illustrate the feasibility of the proposed scheme, simulation experiments are conducted using MATLAB R2020a, which is on a desktop computer (2.90 GHz i5-9400F CPU, 8 GB RAM). The external secret keys are installed as follows: k1 = 0.6093, k2 = 0.7849, and k3 = 0.3168, and the remaining encryption parameters are set to CR = 0.25 and TS = 25.
Key space analysis
The key space is the scope of the encryption key size, and its size determines the ability to resist violent attacks. Therefore, when designing an encryption scheme, the key space should be as large as possible. Generally, at the current computing speed, a key space of 2100 can resist brute force attacks. In this paper, 256-bit hash values are used as key streams to obtain the initial values. In addition, we also introduce k1, k2, and k3 as external keys, each of which is a number composed of
Conclusions
In this paper, an efficient double-image visually meaningful encryption algorithm based on a new one-dimensional chaotic system ICTM, PCS and FFT embedding, is proposed. It not only provides double protection for digital image content and vision, but it also doubles the transmission efficiency. In the key generation process, SHA is employed to correlate the algorithm with plain images to enhance the capability of withstanding known- and chosen- plaintext attacks. In the encryption process, PCS
CRediT authorship contribution statement
Xingyuan Wang: Writing – review & editing, Funding acquisition, Project administration, Resources, Validation, Supervision. Cheng Liu: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft, Writing – review & editing, Data curation, Writing – original draft. Donghua Jiang: Methodology, Resources.
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
This research is supported by the National Natural Science Foundation of China (No. 61672124), the Password Theory Project of the 13th Five-Year Plan National Cryptography Development Fund (No. MMJJ20170203), the Liaoning Province Science and Technology Innovation Leading Talents Program Project (No. XLYC1802013), the Key R&D Projects of Liaoning Province (No. 2019020105-JH2/103), the Jinan City ‘20 Universities’ Funding Projects Introducing Innovation Team Program (No. 2019GXRC031), and the
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