Integral Imaging Based Optical Image Encryption Using CA-DNA Algorithm

Under the computational integral imaging-based security system, the application of Deoxyribonucleic Acid (DNA) encoding algorithm will cause silhouettes in the cipher image, thereby reducing the security of the system. To solve this problem, we introduced a cellular automata-based DNA (CA-DNA) algorithm, which effectively hides the distribution information of the original scene. It can prevent attackers from obtaining any valid information based on the statistical characteristics of the image, which makes our encryption system more security. At the same time, an improved high-resolution reconstruction algorithm is applied to achieve a high-quality decrypted scene. We conducted relevant experiments to certificate the effectiveness of proposed method. Experimental results verify that the scheme has high security and robustness.


Introduction
In the Internet era, massive multi-dimensional image data are transmitted on the network every moment. To protect the privacy of the authorized users, information protection has become a key task in image transmission. The cryptosystem is an effective means to ensure information security [1]- [5]. Among them, the optical encryption method is widely applied in the field of image encryption due to its significant advantages. It has become a major topic in the research of image security. There are many optical technologies used in image protection, and then a variety of optical image encryption algorithms have been proposed [6]- [11]. As an excellent optical imaging system, computational integral imaging (CII) can display full-color images with a wide field of view and continuous parallax [12]- [18]. Because this algorithm can record scenes into a hologram-like characteristic element image array (EIA) [18]- [22], it has attracted attention in the field of optical data security. Some cryptosystems based on the CII framework have been proposed [23]- [26]. These algorithms have significant advantages of high processing speed and high robustness.
Recently, due to the advantages of the parallelism and high information density of deoxyribonucleic acid (DNA) algorithm, some DNA-based encryption methods have been proposed [27]- [28]. Relevant researches have proved that DNA coding technology can effectively resist chosen plaintext attacks, thereby improving the security of cryptosystems [29]- [31]. Many image encryption methods that combine DNA coding with chaos have been proposed [32]- [33]. DNA sequence operations and Logistic mapping are used in image encryption algorithms. In addition, in order to overcome the security shortcomings of low-dimensional chaotic systems, hyperchaotic systems are used for image encryption [34]- [37]. Most conventional DNA coding-based encryption schemes use fixed coding rules, which has the advantages of being easy to implement and having a low time load. However, for some special images, such as element images, fixed DNA coding rules cannot change the bit distribution of these images. Therefore, it leads to contour problems in the cipher-text. It will reveal the distribution information of the original scene and provide potential clues for attackers to establish a cryptographic system.
To overcome this problem, we introduced the cellular automata (CA) algorithm. CA is a dynamic system based on a finite state set [38]- [40]. The CA-based encryption method is a natural choice for secure transmission and has significant advantages [41]- [43]. In the case of large-scale parallel computing, CA can calculate pseudo-random numbers in parallel, which is of great significance in image encryption. Moreover, benefiting from the mathematical characteristics of CA, compared with the existing algorithms, it can provide a significant advantage algorithm [44]- [46]. The CA encoding algorithm has a large key space because its neighbor size, rules, maximum state, and the initial values can all be used as key parameters for the encryption system. Meanwhile, the complexity of CA makes it difficult for CA-based encryption system to be attacked.
In this paper we propose a high security optical encryption approach based on computational integral imaging and CA-based DNA (CA-DNA) encryption algorithm. The high-quality random sequences generated by the CA algorithm are used to define which encoding or decoding rules are applied to each pixel of the EIA. Therefore, in our algorithm not all pixel coding rules are fixed, which better hides the distribution information of element images. The attacker cannot obtain any valid information based on the statistical distribution of the images, which makes the encryption system more secure. And the CA-DNA complementary operation is applied to further improve the security of the system. In other words, it provides a high-security encryption scheme. Furthermore, a modified computational integral imaging reconstruction algorithm is applied to improve the view quality of the decrypted scene. To prove the feasibility and security of the encoding approach, numerical experiments are conducted, and the results are discussed.

EIA Captured by the Integral Imaging System
CII [13] algorithm is an important part of the imaging system, which plays an important role in 3D image processing. Because of its high security and flexibility, it has received attention in the field of information security. It contains two parts, one of which is the process of picking up the original scene, and the other part is the process of EIA reconstruction. Fig. 1 demonstrate the process of picking up the original scene. Through the lenslet array and image sensor, the original scene to be encrypted is converted to EIs. The (i, j)th elemental image can be obtained by the formula [13]: where x and y denote the lenslet array coordinates, ϕ is the size of each lens, and z is the distance between the original scene and the lenslet array.

DNA Sequence Operations
DNA sequence contains four kinds of nucleic acid bases, namely adenine (A), thymine (T), cytosine (C) and guanine (G). If they are defined as numbers "00", "11", "01" and "10", then the digital image can be represented as a corresponding nucleotide string. Among all the 24 types of DNA coding rules, only eight types meet the Watson-Crick complementarity rule. They are recorded in Table 1 Fig. 1. The pickup process of the integral imaging system.   [27]. During the encryption process, we will select these encoding rules through CA sequence to randomly encode each pixel.
In the DNA method, there are addition, subtraction, and exclusive-OR (Ex-OR) operations. The calculation rules of these operations depend on the corresponding DNA coding rules. Table 2 shows the corresponding addition, subtraction, and Ex-OR operations when the DNA coding rule is one [27]. Therefore, for DNA-level images, these three DNA operations can be utilized to scramble adjacent bases, or adjacent DNA sequences.
Moreover, for DNA complementation operation, the complementary rules between four DNA bases must satisfy the following conditions [29]: where F p (Y B ) is the base pair of Y B . Each base of the DNA sequence has six main complementary rules. In this method, one of the six complementary rules are selected to complement the diffusion process through the CA random sequence of each DNA sequence, thereby improving encryption efficiency.

High-Quality Pseudo-Random Sequence Generated by CA Algorithm
The core of the CA algorithm is a discrete dynamic model composed of the arrangement and rules of cells. Benefiting from the mathematical characteristics of the CA algorithm, it can provide significant advantages compared to the existing algorithms. The CA algorithm has a large key space, because its neighbor size, rules, maximum state, and initial values can all be used as the key parameters of the cryptosystem. Moreover, the keys of the CA encoding system can select a generating function with an avalanche effect. For the same image, a small change in the CA key will result in a great change in the pixel distribution of the cryptographic image. And a tiny change in the original image using the same key will also greatly affect the information distribution of the cipher image. The complexity of CA algorithm makes the CA-based encryption system difficult to be attacked. For a one-dimension (1D) CA with two states and three sites neighborhoods, the update of the next state of each unit depends on the state of its neighborhood. The value of each unit is calculated by a prescribed rule. Through the Boolean function can calculate the value of the next state: where F B () denotes the Boolean function with defined rules, c m (k), c m-1 (k), and c m+1 (k) represent the states of m-th cell and its neighbors at time k, respectively. According to Wolfram's theory, a CA with two states and three sites has 28 types of Wolfram rules [38], whose range is defined as 0 to 255. Among these Wolfram rules, only eight types of rules are linear. However, rules 0, 60, 102, 170, 204, and 240 cannot generate random sequences that meet the cryptography requirements. By combining rules 90 and 150, the efficient maximum length random sequence can be generated. The calculated high-quality random sequence will be utilized to encrypt the plaintext. The equations of rules 90 and 150 are written as: and the corresponding characteristic polynomial is:

Modified Computational Integral Imaging Reconstruction Algorithm
The 3D scene can be recovered by a computational reconstruction algorithm. According to the principle of geometrical optics [22], [23], each elemental image is inversely mapped according to the magnification factor. However, with the computational integral imaging reconstruction (CIIR) scheme, the superposition of the pixels will reduce the quality of the recovered scene. Fig. 2 shows that the superposition process of the 3D scene reconstruction. In order to mitigate the effect of fuzzy noise caused by the pixel's superposition, we applied a modified reconstruction algorithm [39]. Each pixel in the recovered scene can be calculated, and the following formula can be utilized to reconstruct the original scene: where T z is the superimposed matrix at the distance z, M×N represents the numbers of element images, m and n are the size of the imaging device, p denotes the size of the pinhole, and u denotes the magnification parameter.

Description of the Encryption and Decryption Procedure
In the previous DNA-based encryption system, the generated element image can be scrambled using the DNA encoding method. However, the pixel values of the element image have not been changed, which leads to silhouette problems in the encrypted image. It makes the encryption system vulnerable to statistical attacks. A CA-DNA encryption approach can change pixel values of element image and provide a cipher-text with uniform information distribution. Fig. 3 describes the flow chart of our encryption algorithm. The encryption procedure of our scheme is introduced as follows: Step 1: Convert the original 3D scene into the form of 2D elemental image f (i, j) with size M×N using the CII algorithm.
Step 2: Generate two high-quality pseudo-random sequences M 1 (i, j) and M 2 (i, j) with size M×N by two different groups CA rules.
Step 3: Decompose the EIA f (i, j) to three binary matrices R 1 (i, j), G 1 (i, j), and B 1 (i, j) with the size of M×N. Then transform the three binary matrices into three DNA sequence matrices R 2 (i, j), G 2 (i, j), and B 2 (i, j) with size M × 4N based on the DNA coding rules defined in Table 1  random encoding sequence En_M(i, j) generated from pseudo-random sequences M 1 (i, j). The random coding sequence En_M (i, j) can be obtained by: Step 4: Perform the diffusion operations by DNA addition and Ex-OR to get three DNA diffused matrices R 3 (i, j), G 3 (i, j), and B 3 (i, j) with size M×4N.
Step 5: Select the rule from six complementary rules according pseudo-random sequence M 3 (i, j).
Based on M 3 (i, j) and selected complementary rule, perform DNA complementary operation on DNA diffused matrices and obtain three DNA complementary matrices R 3 j).The pseudo-random sequence M 3 (i, j) is described as: Step 6: Decode three DNA matrices R 3 ' (i, j), G 3 ' (i, j) and B 3 ' (i, j) using DNA random decoding sequence De_M(i, j) generated from pseudo-random sequences M 2 (i, j) and the DNA encoding rules, then convert them into the decimal matrices R 4 (i, j), G 4 (i, j), and B 4 (i, j). The random decoding sequence is described as: Step 7: Perform scrambling operations on the decimal three matrices R 4 (i, j), G 4 (i, j), and B 4 (i, j) with three pseudo-random sequences M 1 (i, j), M 2 (i, j), and M 3 (i, j) respectively, and combine them into a cipher image.
The proposed scheme is a symmetric encryption system, so the image can be decrypted by the reverse operation of the decryption process. The decryption process is shown in Fig. 4.

Experimental Results
In this section, we confirm the capability of the proposed scheme through a series of simulation experiments. The EIA is generated from the original scene "Cars" by the CII algorithm. For encryption system, we use two sets of linear CA rules (rules (150; 150; 90; 150; 90; 150; 90; 150) and rules (150; 90; 150; 90; 90; 90; 150; 90) to generate two high-quality random sequences. Fig. 5(a) shows the recorded original elemental image of"Cars" with the size of 355×355, and Fig. 5(b) is the encrypted image obtained by the proposed optical encryption algorithm. Figs 5(c)-5(e) show the recovered scenes obtained using a modified computational integral imaging reconstruction algorithm at different depths, and Fig. 5(f) illustrates the recovered scene obtained at the correct depth (100mm) using the CIIR algorithm. Since the encryption process includes seven steps, the time complexity of the proposed algorithm is determined by all the steps of the encryption algorithm, so the total time complexity of the proposed scheme is O(24×M×N). And To confirm the visual quality of the recovered scene, we utilize the peak signal-to-noise ratio (PSNR) to objectively measure the quality of the decrypted scene. The PSNR can be calculated by the following formula: where M and N represent the size of image, E and E' are the original and recovered scenes, respectively. The PSNR value of the recovered scene generated by the proposed method is 38.56 dB, and that of the recovered scene obtained by the traditional CIIR method is 34.68 dB, respectively. It confirms that the proposed method can obtain a decrypted scene with high visual quality.

Key Security Analysis
According to cryptanalysis theory, a qualified encryption system must require a large key space to resist brute force attacks. In addition to the depth of reconstruction, CA algorithm further increases the key space of the cryptosystem. For a 1D CA with n-cells, two states, and m-site neighborhood, its key space is approximately 2 2M ×2 N ×2 2N (M = 355, N = 355). Hence, the total key space possessed by the scheme is far greater than the security requirement of 2 100 (≈10 30 ), it means that the method can effectively resist brute force attacks.  At the same time, the cryptosystem must be highly sensitive to key change. When the key changes slightly, the corresponding decrypted image becomes completely different. As shown in the following example, the sensitivity of the proposed scheme to key conversion can be seen. Fig. 5 shows the restored plane scene with partial wrong key. Figs 6(a)-6(c) show plane scenes reconstructed with different random sequences generated by wrong CA rules. Fig. 6(d) illustrates the recovered plane images with the wrong distance. From the result, it can be seen that the decrypted scene is very different from the ordinary scene, and the original information is not visually identifiable. The results prove that proposed scheme is sensitive to the key change.
Information entropy is an important characteristic for evaluating the randomness of an encryption system [40]. The information entropy of the ciphertext can be calculated by the following formula [31]: where P(E) is the probability of E. From the theory of information entropy, we can know that the closer the value of information entropy is to 8, the better the randomness of the image. And the entropy of the encrypted image is 7.9995, which proves that the cryptographic system can effectively resist entropy attacks. In order to further verify the security of the proposed algorithm, we compare our method with the DNA-based encryption algorithm and CA-based encryption algorithm. In DNA-based algorithm, DNA encoding and DNA decoding operations are performed using fixed rules (key1 = 2, key2 = 8), and a one-dimensional chaotic sequence with parameters (x 0 = 0.9058, u 0 = 3.6246) is used to complete DNA complementary operation and bit pixel scrambling operations. In CA-based algorithm, a one-dimensional CA mask with rules (150; 90; 150; 90; 90; 90; 150; 90) is used to obtain encrypted image. Table 3 is the comparison of the PSNR, MSE and Information entropy results of the ciphertext image using our proposed algorithm and some other methods. It can be seen from the results that the PSNR of the ciphertext generated by our proposed method is smaller, and the information entropy is closer to 8, so the randomness of the ciphertext is better. It proves that our proposed method has good security.

Statistical Analysis
In order to confirm the statistical characteristics of our CA-based DNA encryption scheme, we compare our method with the DNA-based encryption algorithm and CA-based encryption algorithm. Fig. 7(b) illustrates the cipher image produced by a DNA encoding algorithm. From the red frame in this figure, we can see the patterns of the object, which can provide hints for attacking encryption method. To address this problem, we introduce a CA algorithm to evenly distribute the energy of obvious patterns. As shown in Fig. 7(d), the cipher image generated by the proposed encryption approach, which is a uniformly distributed noise-like image. Compared with the image generated by the CA algorithm. as shown in Fig. 7(c), the ciphertext generated by our proposed method is more uniform. It effectively solves the silhouette problem.
To prevent data from being illegally obtained by attackers, it is of great significance to ensure that the cipher-text and the plain-text are not statistical similarity. As an important tool, image histogram is applied to analyze the statistical properties of encryption method. Fig. 7(e) illustrates the histogram of the element image. Fig. 7(f) illustrates the histogram of the cipher image produced by the DNA-based encryption algorithm. Fig. 7(g) illustrates the histogram of the cipher image produced by the CA-based encryption algorithm. And Fig. 7(h) shows the histogram of the cipher image generated by proposed approach. The results verify that the histogram of cipher image produced by proposed method becomes flatter. Therefore, the proposed algorithm brings effective performance against statistical attacks.
Meanwhile, the auto-correlation between the pixels of the cipher image should be weak. Fig. 8(a) shows that there is strong auto-correlation between adjacent pixels in the original scene, and Fig. 8(b) illustrates the auto-correlation of the cipher image generated by the DNA-based algorithm. Fig. 8(c) illustrates the auto-correlation of the cipher image generated by the CA-based algorithm and Fig. 8(d) shows the auto-correlation of the encrypted image generated using the proposed method, which certificate s that the auto-correlation of this image is weaker than that of other images. The results undoubtedly verify that the proposed encryption method has qualified decorrelation performance to resist attacks.

Robustness Analysis
A qualified encryption approach also should withstand a given mass attacks. We analyze the robustness to some attacks for encrypted image by comparing our proposed CA-based DNA encryption scheme with the conventional DNA-based scheme. Fig. 7 illustrates the results of decrypted scenes with different density Gaussian noise attacks. Figs 9(a) and 9(d) denote the recovered scenes by proposed approach. Figs 9(b) and 9(e) show the recovered scenes with DNA-based scheme. And Figs. 9(c) and 9(f) show the recovered scenes with CA-based scheme. From the results shown in Fig. 9, although, the encrypted images are severely affected by the noise attack, the proposed method can clearly identify the original scene information.
Next, we utilize PSNRs to quantitatively measure the quality of the recovered scene against noise attacks. Table 4 records the PSNRs calculated using different encryption schemes. These results indicate that the proposed scheme brings better robustness to attacks.

Conclusion
In conclusion, we presented an optical encryption approach using integral imaging and CA-DNA algorithm, which resolve the silhouette problem of cipher image in the traditional DNA encoding algorithms. The DNA random complementary operation and pixel scrambling can further improve the security of the encryption scheme. Meanwhile, the modified reconstruction method is applied to enhance the visual quality of the recovered scenes. We also analyze the robustness of the proposed scheme against different attacks. Experimental results verify that the encryption scheme has better capability than the DNA-based algorithm.