Image Encryption Based on High-Dimensional Manifold Computing and Block Dividing Algorithm

. The image encryption schemes developing more sensitive and more chaotic maps are used as a key sequence generator, such as cascade chaotic maps and high-dimension maps. High-dimension chaotic maps can generate sequences with little correlation after hundred times of iteration. In fact, the sequence is just a ﬂow of the manifold of the chaotic system. A fast way to select sensitive ﬂows of the dynamic systems is introduced in the paper. Combining with the divided blocks diﬀusion algorithm, the novel scheme yields the cipher image more randomly. Experimental results show that the sensitive ﬂows of the high-dimension system can generate a series with better ergodicity and less correlation compared with the common ﬂows of the same system. The encryption eﬃciency is enhanced by choosing the sensitive ﬂows of the high-dimension system. The analysis proved that the novel image encryption scheme can resist all common kinds of attacks.


Introduction
Image encryption has been widely used in information privacy, national safety, medical image protection, and other fields where only the right person gets the exactly true message. In the recent years, many encryption theories have been used in image encryption, such as elliptic curve approach [1,2], cellular automata theory [3], DNA encryption [4][5][6][7], secure hash algorithm [8], image compress method [9], biological operations [10], and image fusion technology, watermarking, and chaos encryption [11][12][13][14][15]. Among the theories mentioned above, the chaos encryption methods have been proved to be effective for its sensitivity, diffusion ability, and confusion ability. e chaos encryption approach does bifurcation analysis to the chosen nonlinear dynamic system, and then it takes advantage of the chaos nonlinear dynamic system to generate pseudorandom sequences which are used for image encryption [16,17]. In the other way, the encryption attack methods are also developed.
ere are brute-force attack [18], known-plaintext attack [19], chosen-plaintext attack [20], chosen ciphertext attack [21], differential attack [22,23], meet-in-middle attack [24][25][26], and so on. So more complex systems, bigger key space, and higher sensitivity methods are needed to resist different kinds of attacks. Wu et al. introduce an encryption method based on the chaos system and the elliptic curve [1]. is method uses elliptic theory to compress the original image first and then takes advantage of the 4-dimension system to do diffusion and confusion. Although the results show that both the pixel correlation and structural similarity index metrics are good, it has poor ability for noise attack. Lan et al. combine the sharing matrix and image encryption for lossless (k, n)secret image sharing [27]. is approach introduces the (k, n) sharing matrix producing algorithm, and it combines the sharing matrix with chaos encryption to do image encryption. e results show that it is sensitive for the initial value and has a high key space and can detect fake share. e most important is that the encrypted image has n shares, and only k shares are enough for decryption, where k< n. is means that even some information get lost in transmission, and we can still make the encrypted image decrypt without information loss. e disadvantages are that the sharing matrix computing costs too much time, and it cannot resist noise attack. Lee et al. proposed a parallel computing method for chaos encryption [12]. It divided the original image into blocks first and then did chaos encryption synchronously with multithreading. is method improved the computing speed greatly. Although it made the encryption methods with the chaos system more executable, not all the software supports multithreading operation. An approach combining DNA algorithms and chaos maps was presented in reference [8]. In that paper, the Watson-Crick complementary rule was used for bit-level changing, and chaotic maps generated a sequence to choose different biological and algebraic operations. It also introduces SHA-256 to the encryption process which increases the complexity. is method can resist noise attack, but key sensitivity is not good enough. Cellular automata (CA) and chaotic map to do encryption were employed in reference [8].
e CA rule made the "0" and "1" distribution in equilibrium in the process of iteration, and the chaotic map is used to produce a sequence to shuffle pixel position and bits confusion. A network encryption method was proposed in reference [3]. It did diffusion and confusion at the same time, and it improved the computing speed. By integrating more than one chaotic map, the chaotic ability of encryption systems was improved in reference [18] and reference [27]. e result shows that by integrating different chaotic maps, the improved system has better sensitivity and larger key space.
As high-dimension chaotic systems were introduced to the encryption system, people mainly focused on the good encryption effects of high-dimension selection. But the initiated value decision which is also important was ignored. is paper presents the common flows and the sensitive flows of the maps used in image encryption. en, it describes how to choose the sensitive flow of the chaotic system. e sensitive flow is used to generate random sequences in this paper. Experiment results and analysis prove that the image disposed by this algorithm has ability to resist brute-force attack, differential attack, statistic attack, chosen-plaintext attack, noise attack, and so on. e rest of the paper is organized as follows: Section 2 introduces the preliminaries of the high-dimension system; Section 3 describes the encryption scheme with sensitive flows of the high-dimension system; Section 4 presents the experimental performance evaluation and analysis; Section 5 is the summary part.

Preliminaries
High-dimension systems have complex performance in space. To describe the character well, the Lorenz system is used as an example to do the analysis.

ree-Dimension Lorenz System.
e Lorenz system is defined as F x n+1 , y n+1 , z n+1 � δ y n − x n , rx n − y n − x n z n , x n y n − εz n .
e Lyapunov exponents (LE) are shown in Figure 1. Figure 1(a) presents the LE value of δ in the interval [0, 100], while keeping r � 8/3 and ε � 26. Figure 1(b) refers to the LE value of r in the interval [0, 100], while keeping δ � 10 and ε � 26. Figure 1(c) shows the LE value of ε in the interval [0, 100], while keeping δ � 10 and r � 8/3. e Lorenz system has good chaotic behavior when δ � 10, r � 8/3, and ε � 26 because the LE values of the three parameters are all positive at that point. Figure 1 shows that the chaotic behavior happens not just at that point. As long as the parameters' LE satisfy LE1>0, LE2>0, and LE3>0, where LE1 denotesδ's LE, LE2 means r's LE, and LE3 stands for ε's LE.

Manifold
Computing for Sensitive Flows. Chaotic maps have good ability to generate a sequence with little correlation. But there is no reference describing the essence of the sequence used for encryption. In fact, the sequences are discrete points on flows in manifold of the maps. Taking Lorenz equations as an example, we find no analytical solutions of Lorenz equations in three-dimension space. e only way to describe the system is by numerical fitting by equation (1). Figure 2(a) shows the global manifold of the Lorenz system in three-dimension space by numerical fitting. e global manifold consists of lots of flows starting from the local manifold, as show in Figure 2(b). e sequences that we used for encryption are just a segment of one flow after hundred times iteration by equation (1) with an initiate value. Equation (2) defines the manifold of the chaotic system: where W s (S 0 ) is the manifold, ϕ(x) is the equation of the system, W s loc (S 0 ) is the local manifold, S 0 is the singular point of the system, and t ⟶ ∞ stands for infinite iterations.
From Figure 1(c) and Table 1, we can see that the flows starting from the local manifold are not in uniform distribution. Most flows are from a little angle range as shown in reference [5,28,29]. is means that the flows starting from a small angle range have less correlation compared with others. It also shows that the flows from the small angle range are sensitive. It has been proved in reference [29] that the sensitive flows start from the direction of the eigenvector of the manifold. So, the small angle range is just the directions of eigenvectors. e eigenvalues and eigenvectors of the Lorenz system are given in the following equation: e local manifold can be decided as  International Journal of Optics where V → 1 and V → 3 are the eigenvectors of the manifold and λ 1 and λ 3 are the corresponding eigenvalues. e flows starting from the local manifold can be computed as For every encryption chaotic system, scrambling and diffusing by the sequences of the sensitive flows can achieve better effect. e way to get the most sensitive flows follows the steps given below (take the Lorenz system as an example): Step 1. Get the eigenvalues and eigenvectors as equation (3) by the Jacobian matrix of the system.
Step 2. Decide the local manifold by equation (4), as seen as the blue ellipse in Figure 2 Step 3. Choose the initiate value on the ellipse in the eigenvector direction and do iteration along that flow. e initiate value sensitivity is 10 − 16 for the Lorenz system.
Step 4. Choose a segment of the flow computed in step 3 and have the right points which the key sequence need. e sequence of the chaotic map from the sensitive flows of the maps for the encryption system can be obtained.
As can be seen from Figure 2, the global manifold is composed of flows in two small angle regions (1.5708-1.5709 and 4.7124-4.7125). e flows of these two regions are intertwined in space but never intersect. After these streams are discretized, these discrete points are of confusion and complexity, which are convenient to become the initial sequence of the encryption system.

The Encryption and Decryption Algorithms
Image encryption refers to pixel transformation based on different schemes. e transformation can be divided into position scrambling and value diffusion, generally both position and value of pixels will be changed in the encryption process. One scheme of the transformation indicates finding the right ways to choose sequences, which will be used to position scrambling and value diffusion. e less the relationship among the sequence, the harder for the attackers to find rules of the encryption. e chaotic system can generate sequence with little relationship. Another scheme for transformation is to get the right computation algorithm, which can do computation with pixels and sequences. In the paper, the high-dimensional manifold computing will generate a sequence and block dividing algorithm which can do transform computations. e encryption process is as shown in Figure 3. It contains sensitive flow computing to produce key sequences, two scrambling operations, and one divided blocks matching diffusion operation.

e Encryption Algorithm.
e encryption process follows the steps given below: Step 1. Generating sequences: by selecting sensitive flows of the manifold, we can get three sequences S 1 , S 2 , and Step 2. Image scrambling: the scrambling matrices will be produced by the following equation: e image I was transformed to I ′ by expanding the size as (M + m) × (N + n). en, the scrambling can be finished by Step 3. Block matching: we divide image I ″ into 16 blocks (I 1 ″ , I 2 ″ , . . . , I 16 ″ ). e sequence S 2 ′ was changed into matrix S 2 ″ and then divided into 16 blocks S 2 ″ � (S 2 ″ (1), S 2 ″ (2), . . . , S 2 ″ (16)). For each block, it has International Journal of Optics the size of (M + m)/4, (N + n)/4. We also have different diffusion rules for each block as the following: e pixels in different blocks can be encrypted by different rules. After the diffusion section, we can get the new image I diffusion .
Step 4. Image scrambling: the scrambling matrices will be produced by the following equation:

e Decryption Process.
Decryption is the reverse of encryption. It can be finished as given in the following steps: Step 1. Encrypted image scrambling: by equation (11), we can get the diffusion image: Step 2. We divide the diffusion image into 16 blocks I diffusion (1), I diffusion (2), . . . , I diffusion (16) and do block decryption with different rules as following: After the block decryption by equation (12), we can get image I ″ .
Step 3. Image scrambling: by equation (13), we can restore the expanded image I ′ :  International Journal of Optics Step 4. By removing the redundant rows and columns, we will get the plain image I in normal size.

Experiment and Analysis
e encryption algorithm should withstand different kinds of attacks such as brute-force attack, statistical attack, differential attack, chosen-plaintext attack, chosen chipper text attack, known plaintext attack, and known chipper attack. Different kinds of attack refer to different kinds of performance indicators of the encryption system. To be easily compared to the results in references, "lena," "baboon," "peppers," and "airplane" are chosen as the plain text in this paper. ey all have the same size 512 * 512. Several experiments and different kinds of security analysis are presented to evaluate the robustness of the proposed encryption method.

Statistical Analysis.
Histogram is always used for statistical analysis. A histogram is a graphical method for displaying the image's exposure accuracy by using the graphical parameter and describes image's distribution curve. As we all know that the confusion operation does not change the histogram, but the diffusion makes the distribution of the histogram uniform. Taking the image "airplane" as an example, we can see the results as following.  Figure 4(f ), which means the pixels value is changed to be equal in numbers.

Correlation Analysis.
A good encryption algorithm should have the ability to break correlations between adjacent pixels. It may cause information leakage by statistical attacks if this feature is ignored. e correlation coefficient of an image is calculated as where x, y denote the pixel values of two adjacent pixels, E(x) is the function for mean value, D(x) is for the variance and cov(x, y) is the function for correlation, and c denotes the correlation coefficient of the image. Taking image "lena" as an example, Figure 5(a) shows the correlation in the horizontal direction of the plain image. Figure 5(b) shows the correlation in the vertical direction of the plain image. Figure 5(c) shows the correlation in the diagonal direction of the plain image. We can see that in all the three directions, the pixels have strong correlation in the plain image. Figures 5(d)-5(f ) show correlation in the horizontal direction, vertical direction, and diagonal direction of the cipher image. Pixels of the encrypted image in all three directions have little correlation, which means there are no rules for attackers to find. Table 2 shows the correlation value in three directions of different images. It has proved that the novel encryption scheme proposed in this paper has generality for all images.

Key Sensitivity Analysis.
Key sensitivity is an important feature for any good encryption system. " e good sensitivity means that the encrypted image cannot get the right image even though there is a tiny change in the secret key." Taking the image "couple" as an example, tiny changes Δ � 10 − 15 , Δ � 10 − 16 , and Δ � 10 − 18 are chosen as the key disturbance, and the experiment results are shown as the following. We can see that the slight change Δ � 10 − 15 leads to a complete wrong result, but Δ � 10 − 16 and Δ � 10 − 18 make the right result. It can be proved that the sensitivity of the encryption system is Δ � 10 − 15 . Table 3 shows the difference rate of the slightly changing key with different images. For all the images chosen as plain images, the sensitivity always keeps as Δ � 10 − 15 .

Key Space Analysis.
e key space reflects the ability of the encryption algorithm to resist brute-force attack. If the key space is larger than 2 128 , it means the system can stand for brute-force attack. It has been proved that the sensitivity is Δx � 10 − 15 . e sensitive flow computing make the space to be 10 88 . e encrypted system is three dimensional with two confusion steps. e image is divided into 16 blocks in diffusion operation. So the key space can be calculated as keyspace � 10 88 * 2 3 * 2 4 * 2 3 * 2 3 ≈ 2 216 .

Differential Attack Analysis.
e following equations are used to analyze the effectiveness of the differential attack resistant: 6 International Journal of Optics where NPCR refers to the number of pixels changed, UACI stands for the average changing intensity, D(i, j) is the Boolean function, M is equal to the row number of each color field matrix, N is equal to the line number of each color field matrix, and I is the encrypted plain text, while I ′ is one pixel changed encrypted plain text. Table 4 shows the comparison of NPCR and UACI of the image "lena" with the method in this paper and the methods in the references. It demonstrates that our encryption method has strong ability of withstanding the differential attack. We got a larger NPCR value than that in Reference [30] and Reference [5] and a slightly smaller UACI value than that of Reference [30] and Reference [5].

Noise
Attack. An efficient image encryption algorithm should be robust against different kinds of the noise. In this experiment, we first add salt and pepper noise, speckle noise, and Gaussian noise with density 0.000001 to the cipher image to see if the algorithm can decrypt the right plain text.   International Journal of Optics 4.7. Information Entropy Analysis. Information entropy is employed to measure the uncertainty in a random variable, which can be described as where p r (I � i) represents the probability of the symbol i.
For an 8 bit truly random image, the ideal entropy is 8. e closer H(I) gets to 8, the better the randomness of I. Table 5 shows the Shannon entropy analysis of different images by this algorithm.

Quality Evaluation Metrics of Decryped Image.
A general requirement for all image encryption schemes is that the encrypted image should be greatly different from the plain image, while the decrypted image should be as same as the plain image. We use peak signal-to-noise ratio (PSNR) to measure the difference between the plain image and the decrypted image and the normalized cross correlation (NCC) and structural similarity index metric (SSIM) to see if there is information leakage in the encryption process: PSNR � 10 log 10 D 2 max MSE , e PSNR is used to see if the encrypted image contains useful information.
e higher the peak signal-to-noise ratio, the more confusing the encrypted image. If the peak signal-to-noise ratio approaches infinity, the encrypted image approximates to noise. NCC and SSIM are used to measure whether the decrypted encrypted image was the same as the original plain text image. eir intervals are [0, 1], and if their values are 1, then the information is 100% reserved in the both encryption and decryption process. is indicator measures the efficiency and accuracy of the decryption process [31][32][33].
I(x, y) is the plain image. D(x, y) refers to the decrypted image. M and N stand for the row numbers and the column numbers. μ 1 and δ 1 are the mean value and variance value of I(x, y), while μ 2 and δ 2 are the mean value and variance value of D(x, y). δ 12 is the covariance of I(x, y) and D(x, y). C 1 and C 2 are constants to avoid zero value of the denominator. We made C 1 � 2.25 and C 2 � 2.25 * 3 in this experiment.
From Table 6, we can see that PSNR is close to ∞, which means the plain image and the decrypted image are completely the same. NCC and SSIM are both equal to 1, which means there is no information loss through the encryption and decryption process.

Conclusion
In this paper, the sensitive flows of the manifold in the chaotic system which is always ignored were presented first. e way to get the sensitive flows was shown then. By the sensitive flows, the sequences were generated as the keys of the encryption system. e encryption process contains two scrambling steps and one diffusion step. In the diffusion section, the image was divided into 16 blocks, and different encrypted rules were used for different blocks. In this way, the key space was improved and the correlation was reduced. In the encryption process, the high-dimensional manifold is taken to do calculation, and only the local manifold region with the best sensitivity is used as the initial secret key to generate the pseudorandom sequence. e scrambled image is processed in blocks, and different random sequences are used to encrypt different regions, which makes the encryption system more complicated. e decryption process is the reverse of the encryption process. e final original image can only be obtained by partitioning the ciphertext image and then restoring the position. Simulation results and analysis show that the novel algorithm has ability to encrypt the image into random-like cipher images. It can also resist common kinds of attack.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e author declares that there are no conflicts of interest.