A New Image Encryption Scheme Based on 6D Hyperchaotic System and Random Signal Insertion

With the development of communication techniques and the Internet, how to protect private image transmission on the Internet has been a research highlight. Scholars have proposed different kinds of image encryption algorithms. Some of them are not secure enough. A novel cryptosystem is presented which is based on a 6D hyperchaotic system and random signal insertion. To enhance the dynamic performance of the hyperchaotic system, insert some random signals into the system variables during iteration. The sum value of all plaintext pixels is used to produce the initial values of the system. Thus, the proposed cryptosystem is closely related to the plain image. Split a pixel into two equal parts and form a larger matrix. Scrambling, cycle shift, and diffusion are operated on the new matrix. Finally, we obtain the encrypted image. The simulation results reveal that the proposed method has a very large key space, high key sensitivity and robustness, and is able to resist many different attacks. It is more secure than many other image encryption schemes.


I. INTRODUCTION
This image is one of the most important information carriers over the Internet. It is famous for high correlation and redundancy. It is also well known for large data capacity. With the rapid development of science and technology, how to transmit a private image securely over an open channel has become a very important research hotspot. There are two popular methods to protect private images: encryption and steganography [1]. Shannon in 1949 first proposed the confusion-diffusion structure in secure communication [2]. In 1989, Matthews designed the first chaotic encryption method based on a logistic map [3]. Fridrich used the confusion and diffusion structure for image encryption in 1998 [4]. Scholars have put forward many different image The associate editor coordinating the review of this manuscript and approving it for publication was Shuangqing Wei . encryption schemes in the past 20 years. These schemes were based on DNA techniques [5], [6], [7], [8], quantum techniques [9], [10], [11], [12], transform domain techniques [13], [14], [15], [16], compressive sensing techniques [17], [18], [19], [20], fractal techniques [21], [22], [23], [24], multiple image techniques [25], [26], [27], [28] and others [29], [30], [31], [32]. Some encryption schemes are not secure enough and have been broken [33], [34], [35], [36], [37], [38]. Hyperchaotic systems have many state variables and system parameters, a complicated structure, and good dynamic characteristics [39]. Yang et al. discussed the characteristic of the fractional-order hyperchaotic system and applied it for image encryption [22]. Wang and Yang designed a hyperchaotic system based on fractional order CNN in [23]. In [28], the authors proposed a method to encrypt multiple images based on a 3D scrambling and hyperchaotic system. A 3D image was produced by multiple plain images. The authors in [33] designed a color image encryption algorithm using a fractional order hyperchaotic system.
The proposed scheme uses the 6D hyperchaotic system to produce pseudorandom sequences [40]. For the sake of increasing the security of the cryptographic system, we insert some random signals into state variables during iteration. They will change the orbit of the system effectively. Since the sum value of the plaintext pixel is applied to obtain the initial values of the cryptographic system, the proposed method is closely associated with a plain image. Split an 8-bit pixel into two equivalent parts and form a new matrix. Confusion, circle shift, and diffusion are operated on the new matrix and finally we obtain the encrypted image.
The remainder of the manuscript is described as follows. Section II generates the chaotic sequences. In Section III, the proposed method is given. The simulation results are provided in Section IV. The security tests are listed in Section V, and the conclusion is presented in Section VI.

B. RANDOM SIGNAL INSERTION
For the sake of enhancing the dynamic effects of system (1), some random signals are inserted into the system variables during iteration. It will effectively change the chaotic orbit.

C. GENERATION CHAOTIC SEQUENCES
1. Compute the initial values of the system as follows: where sm and t i are secret keys, i = 1, 2,. . . , 6; mod is a modular operator. 2. Discard the first 900 values to avoid the transient behavior.
3. Continue to iterate system (1) 2MN times and generate pseudorandom sequence x i , i = 1, 2,. . . , 6. 4. Obtain the new sequences a 1 , a 2 , and a 3 . where abs represents the absolute value and floor obtains the closest integer to zero. Here, the three sequences a 1 , a 2 , and a 3 are non-negative integers. Especially

Split the 8-bit pixel of the original image into two 4-bit parts. It forms a new matrix
Here, i ′ and j ′ could be determined by Equations 8 and 9.
4. Convert sequences S and a 2 into respective binary sequences.
where CST [b, c, d] means the cyclic shift d bits on the binary sequence b. MIB represents the most important bit. The shift to the left or right will be determined by c = 0 or 1. 6. Union of two adjacent 4-bit into one 8-bit in sequence S 2 .
7. Convert binary sequence S 2 to decimal sequence P 2 . 8. Encrypt the sequence P 2 as in Equation (11), as shown at the bottom of the next page, where C(i), C(i-1), sm, a 3 (i), and P 2 (i), respectively, denote the encrypted pixel, the previous encrypted pixel, the sum value of the plain text pixels, the new sequence value, and the decimal sequence value. 9. Change the sequence C into an EC matrix of size M ×N . Finally, we obtain the encrypted image EC.
The block diagram of the designed scheme is shown in Figure 3. The proposed scheme is a reversible method. The decryption process is reversible from the encryption process and is not introduced again.

IV. SIMULATION RESULTS
The simulation is performed in MATLAB. Four images named Lena, Clock, Resolution, and Peppers are used for the experiment. The dimensions of the test images are 256 × 256. The encrypted result of Lena is used to compare with other similar algorithms. The simulation results are shown in Figure 4.
In Figure 4 it is revealed that the simulation result is noise-like and illegible. The decrypted image is the same as the plain image. It also shows that the distribution of the plain pixel value is quite concentrated, whereas the decrypted pixel value is flatter. Attackers will not find useful information from the histogram of the encrypted image.

V. SECURITY ANALYSIS A. KEY SPACE
If the key space of the cryptosystem is more than 2 128 , then it will be able to resist brute-force attacks [41]. The keys of the cryptosystem consist of: (1) the sum of the plain pixel sm; (2) the system parameter h; (3) the initial values t i , i = 1, 2, . . . , 6. The operational precision in this scheme is 10 −15 , and the key space of the designed scheme will be almost equal to (10 15 ) 7 ≈ 2 348 . It means that the cryptosystem will be tolerant to brute-force attacks and secure.

B. KEY SENSITIVITY
A cryptosystem should be very sensitive to the key. The key sm comes from the sum value of the plaintext pixels, and therefore, the proposed scheme is sensitive to the plaintext image. Alter one of the keys insignificantly (β =10 −15 ) and keep the others unchanged. Apply the new keys to encrypt and decrypt the Lena image.
In Figure 5, it is concluded that even if two keys have a small difference, the test results will be very different. The difference between two encrypted images will be 99.64% with almost the same keys.

C. HISTOGRAM
The Chi-square test [39] is used for histogram analysis and can be computed as formula (12): where f i is the real frequency of the grayscale level. If the significance level is 0.05 and the result is smaller than χ 2 0.05 (255) = 293.25 [39], then the encrypted image histogram will be recognized as uniform. The results of the Chi-square test are listed in Table 1.
From Table 1, it reveals that all the results of the Chi-square test are less than 293.25. The histogram of the encrypted image is fairly consistent, and the proposed algorithm could resist the attack on histogram analysis.

D. ADJACENT PIXELS CORRELATION
The adjacent correlation coefficient (ACC) between pixels u and v can be obtained as: 66012 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   Choose 6000 pairs of neighboring pixels in horizontal, vertical, and diagonal directions from the plain and encrypted images. Figure 6 reveals the distribution of adjacent pixels in plain and corresponding encrypted images. Table 2 shows three directional coefficients of encrypted images. The ACC values are listed in Table 3. Table 2 reveals the ACC value of the adjacent pixels in three directions. All of these values are approaching zero. This means that the proposed scheme could obviously decrease the correlation of neighboring pixels. The attackers could not find anything by correlation analysis. As shown in Table 3, the performance of the proposed scheme is superior to other methods.

E. INFORMATION ENTROPY
Integral information entropy (IE) and local information entropy (LE) are often used to measure uncertainty. They can be computed from formulas (16) and (17): where e l means the l-th signal and p(e l ) represents the probability of e l . R i is a data block with W pixels. The data block is randomly chosen and does not overlap. H(R i ) means the IE of the R i data block and (n, W ) = (30, 1936) [42]. The maximum value of IE is 8 for the 256-level grayscale image. Thus, the IE value is closer to 8, the security of an encryption scheme is higher. LE is a more strict measure to test randomness than IE. If the LE value is located in the interval [h l * α left , h l * α right ], then the ciphered result will pass the LE test and the security of the cryptosystem is extremely high [43]. The IE and LE values of the encrypted images are listed in Table 4.
It concludes from Table 4 that the average value of IE is about 7.9974 and approaches the theoretical value 8. All LE values pass the test except one. It represents the proposed scheme could resist an entropy attack.

F. DIFFERENTIAL ATTACK TEST
NPCR and UACI are often adopted to measure the effect of defending against differential attack [27].
where EC 1 and EC 2 are two images with size M × N . The ideal value of UACI is 33.4635% and that of NPCR is 99.6094% [43]. It reveals from Table 5 that the values of UACI and NPCR are approaching the desired values. It also concludes that the proposed scheme is superior to other methods and could resist differential attack.

G. ENCRYPTION QUALITY
The deviation from the uniform histogram (DUH) and the irregular deviation (ID) are adopted to measure encryption quality [45]. ID tests the extent of deviation to approach the average statistical distribution. DUH tests the deviation of histograms between the encrypted image and the theoretical one. An excellent encryption algorithm will have small values of ID and DUH. ID and DUH can be computed as where H p and H EC signify the histogram of the plaintext image P and the encrypted image EC. The ID and DUH values are shown in Table 6. It concludes that the proposed method has smaller ID and DUH values than other methods. It also shows that the encryption quality of the proposed method is very high.

H. TEXTURE TEST
Homogeneity, contrast, and energy are three characteristics of texture. Homogeneity is employed to test the distribution of co-occurrence matrices (CM) [48]. Contrast is adopted to measure the difference between adjacent pixels. Energy is used to test the sum of squared elements in the GLCM. they can be obtained as formulas (23)(24)(25) : The values of homogeneity and energy are as small as possible. The higher value of contrast will be much better. The texture test results are listed in Table 7.

I. ROBUSTNESS ANALYSIS
Robustness is very important in the image encryption algorithm. It means that the plain image could be recovered under different conditions.

1) NOISE ATTACK TEST
The encrypted image is added with Salt & Peppers noise which intensity is from 0.001 to 0.3. The decrypted result is shown in Figure 7.   (b1) decrypted image of (a1); (b2) decrypted image of (a2); (b3) decrypted image of (a3); (b4) decrypted image of (a4).  Figure 7 that the recovered image is recognized. It means that the proposed method could defend itself against noise attack.

J. COMPEXITY ANALYSIS
Complexity is one of the important indexes for the cryptosystem. The computational complexity of different schemes is compared in Table 8.
It shows that the proposed algorithm is the second fastest in four methods. It also reveals that the proposed method is efficient.

VI. CONCLUSION
In this paper a new plain-related image encryption scheme based on a 6D hyperchaotic system and random signal insertion is proposed. The 6D hyperchaotic system has good dynamic performance and is adopted to produce chaotic sequences. The sum of all pixels of the plain image is applied to strengthen the connection to the original image. Random signal injection will effectively enhance the dynamic performance of the cryptosystem. Split the pixel into two equal parts and form a new larger matrix. Scrambling, circle shift, and diffusion operate on the new matrix. The encrypted image is obtained. The simulation results denote the performance of the designed scheme. Secure analysis demonstrates that the proposed scheme has a large key space, a tiny pixel correlation, a high sensitivity and encryption quality, and an ideal information entropy. Meanwhile, the cryptosystem could effectively resist histogram analysis, differential attack, texture analysis, and robustness attacks.