Color Image Encryption Algorithm Based on Hyper-Chaos and DNA Computing

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I. INTRODUCTION
With the digitalization of the world, cyber security of digital data that flows through the Internet in various forms is one of the main concern for the future of digitalization. Use of digital images is increasing in expressing and sharing sensitive information, whether in military, banks or the personal moments of an individual. Numerous image encryption schemes have been developed based on different chaos theories, maps and systems to hide plain images from unwanted eyes or unauthorized/illegal access [1]- [3]. Hackers are continuously trying to break these schemes. The battle between cryptosystem and cryptoanalysis is never ending. Thus, to improve and develop more secure schemes is essential and inevitable for image security.
Usually, the images have high correlation between adjacent pixels, large data capacities and high redundancy, making it The associate editor coordinating the review of this manuscript and approving it for publication was Orazio Gambino . difficult for conventional security encryption schemes like DES, IDEA, RSA and AES to maintain higher standards of security [4], [5]. Whereas, the encryption schemes based on chaotic maps and systems [6]- [13] are more up to the job due to their properties like high sensitivity to initial values and other system parameters, plaintext sensitivity, large key space, chaotic behaviour and ergodicity. But many encryption schemes have been successfully broken with use of classical attacks [14]- [16].
Man must learn from nature in order to solve his problems. Among natural structures, DNA, contains genetic code, is the book of nature containing instructions of life, determining what we are today. The DNA model is a good way to store and manipulate large amount of data. Therefore, many encryption schemes [17]- [21] based on DNA computing and chaos have been developed. But, there are certain drawbacks or rooms for improvements for the existing encryption algorithms. One aspect of these schemes is the dimension of chaotic maps/systems used in them, either low dimension (one or two) or high dimension (more than 2). Each category has its own merits and demerits. Encryption algorithms with low dimension chaotic maps/systems [17], [18], [22] are easy to implement with simple structure and low computational cost, but unable to maintain high security standards. With growing computing power, their small key space is vulnerable against brute force and other attacks [23]- [25]. Although, the high dimensional chaotic maps/systems generate more complex and random chaotic sequences with a large keys space. Being more sensitive to the initial values, they enable us to develop more secure image encryption schemes [7], [19], but with a higher computational cost. Generally in various steps of encryption process, different chaotic data is used without repetition, adding more complexity to encryption scheme.
Apart from quality of chaotic maps/systems, there are other problems regarding the use of DNA model. There are eight rules to convert a decimal pixel value to a DNA sequence and visa versa. In [18], [26], [27], the same rule is used for all the pixels in encryption and decryption. In [22], [28]- [32], conversion rules are part of the secret key set. But with the static conversion rule and small keys space of conversion rules (1)(2)(3)(4)(5)(6)(7)(8), these encryption schemes are vulnerable against the plaintext and brute force attacks [33]. Although, there are some improvements in recent encryption algorithms. For instance, Wu et al. [34] generated DNA conversion rules based on respective entropies of all the three red, green and blue channels separately, but used the same rule of conversion for all the pixels of a channel. In another example, Chen et al. [35] generated random conversion rules based on rand function, but with the problem that they did not depend on plain images. Different plain images have same conversion rules, making their scheme insecure against known plaintext attack, chosen plaintext attack and statistical attack. Furthermore, DNA operations like DNA addition, DNA subtraction, DNA XOR, etc. (key components of a DNA model), are used widely for the diffusion of plain images [21], [22], [28], [29], [31], [36], [37]. Again, there are eight rules for applying these operations among DNA sequences, but most of the time, same rule is used that makes outcome predictable and less random. Thus, a lot of encryption algorithms have been broken due to less amount of plaintext sensitivity. For instance, encryption scheme [38] was broken by Zhang [39] with the help of a chosen plaintext attack. Liu et al. [40] showed that encryption algorithm [41] is vulnerable against the chosen plaintext attack due to its non-sensitivity to plain image and key set. Recently, an encryption scheme [42] has been cryptanalysed by Akhavan et al. [43] finding that the cipher image did not reasonably depend upon plain image and key set, making it vulnerable against known plaintext attack. They demonstrated that with the use of two or four known images, the key set can be found. Thus, we need a more efficient, robust and secure encryption scheme to overcome the problems, discussed above.
According to our hypothesis, If the same chaotic data can be used multiple times to generate random key streams without compromising the data security, then the computational cost can be reduced significantly even with higher dimension.
To address the problems of lower dimensional schemes, a four dimensional hyper chaotic dynamical system (CDS) [44] is used in the proposed novel RGB encryption algorithm. Whereas, to reduce the computational cost, Chaotic Dynamical State Variables Selection Procedure (CDSVSP) designed by Bashir et al. [1] is used to select cross states chaotic data of CDS. More precisely to encrypt a color image of size M × N , usually more than 3MN chaotic variables are used in each step of an encryption process to maintain higher standards of security. For example, Wu et al. [21] generated 12MN +2400 chaotic variables for encryption and Xiuli et al. [7] generated 16MN + 3200 chaotic variables for encryption. In our case, with the help of CDSVSP, we only need MN + 300 chaotic variables in the encryption process. Thus, reducing the computational and memory cost significantly and resulting into a much faster encryption process. The cross state data is more random and chaotic in comparison with using a single state to generate key streams or pseudo random numbers. Furthermore, to use full potential of DNA model, pseudo random numbers are generated based on the pixels of plain image that govern the rules of DNA operations, conversion between DNA sequence and decimal numbers. Each pixel is converted with a different rule that is totally random and dependent on the pixel value. Also, the outcomes of DNA operations are entirely different for different rules, this greatly improves the randomness of cypher image. Even a single pixel value change of a plain image will result into entirely different DNA sequences, thus making the proposed scheme more secure against the plaintext and statistical attacks. Following this idea, an image encryption scheme based on hyper chaotic dynamical system [44] and DNA computing is designed with the following novel highlights: • To overcome the shortcomings of lower dimensional chaotic maps/systems, a four dimensional chaotic dynamical system is employed. Further, to reduce the computation cost (a major concern of using higher dimensional chaotic maps/systems), CDSVSP is used to generate multiple key streams from the same chaotic data obtained by the iterations of chaotic dynamical system. As a result, the size of chaotic data used in the proposed encryption scheme is much less, even less than one tenth of what existing encryption schemes are using.
• To use the full potential of DNA computing, based on key streams different sequences of pseudo random numbers are generated to be used as rules for DNA operations and conversion of decimal pixel values to DNA and vice versa. This improves greatly the unpredictability and randomness of resultant DNA sequences or cipher image.
• To diffuse plain image properly, first we diffuse it with mask images (key streams) at decimal level and then at DNA level, the three channels red, green and blue are inter blended with different combinations of DNA operations randomly.
• To confuse plain image properly, the three channels are combined as one dimension array and then permuted, so that the pixel values are randomly distributed in all three channels.
• To ensure the maintenance of high standards of security for the proposed novel image encryption scheme, pixel values of plain image are involved in all the steps of encryption algorithm, i.e., generation of key streams (pseudo random numbers), DNA conversion, DNA operations and decimal conversion. Together with the large key space and hyper chaotic data being highly sensitive to secret keys, a small change in the plain image and key set would result into an entirely different cipher image, thus making the encryption scheme more secure against the known attacks. In the light of above discussed features, proposed image encryption is highly efficient and secure for practical use with much less cost and higher security in comparison with existing encryption schemes. The outcomes of extensive security analysis and tests done in section 4 confirm our claim.
The paper is organised as follows. Section 2 presents preliminary theories about CDM, DNA computing and CDSVSP. Section 3 showcases the proposed novel encryption scheme for RGB images. In section 4, simulation and the ensuing security analyses of the proposed scheme have been performed. Finally, Section 5 concludes this study.

II. PRELIMINARIES
Baker map, Lorenz map and logistic map are a few of the many maps which generate random data used in different chaos based applications. Our choice is the chaotic tent map [45] known as: where a ∈ (0, ρ) is an integer. It will be used in the proposed image encryption scheme.

A. CHAOTIC FOUR DIMENSIONAL DYNAMICAL SYSTEM
The chaotic dynamical systems are well known source to produce the chaotic data with desired properties. Lorenz and Rossler did the preliminary work by developing their respective three dimensional chaotic dynamical systems. Later, Yong and Yun-Qing [44] developed a much better four dimensional chaotic dynamical system (CDS) as follows.ẋ This new system is symmetrical, dissipative and possesses very rich dynamical structures. The Figure 1 shows the

2) A new variable say slt(L) is defined as the selected variable in
Through the usage of this variable, a key stream element for P(L) will be generated. An indicator index(L) will make the decision, defined as: In the above equation f (a, ρ, x) is a tent map. a and ρ contribute to the secret keys. For the first pixel value, P(−1) has been set as a seed value. The CDSVSP works as follows: Choose i 0 , j 0 , k 0 , l 0 sufficiently large, different from each other and act as secret keys. Now, we create the initial state S = {X i 0 , Y j 0 , Z k 0 , W l 0 }, for the first pixel P(0). Then, we calculate the selected variable slt(0) for P(0) by computing index(0). System state is updated as follows: Again, calculate the selected variable slt(1) from updated state S for P(1) by computing index(1) and update the system state. Inductively, get the updated state variable set S for P(n) and select the state variable slt(n) from S by computing index(n). Suppose, we have the state {X i , Y j , Z k , W l } for the P(n) and without loss of generality, we assume that index(n) = 0. In this case, the state value X i is chosen for ciphering P(n), and the combination of state variables is {X i+1 , Y j , Z k , W l }. Similarly, calculate index(n + 1), and let index(n + 1) = 1. The state value Y j will be selected to cipher P(n + 1), and the combination state variables trans- Similarly, we supposed that index(n + 2) = 2. The state value Z k will be selected for ciphering P(n+2). Then, state variables combination changes without loss of generality. The state value W l will be chosen for ciphering P(n + 3).

C. DNA MODEL
DNA encoding is done using four basic nucleic acids. These are called Adenine (A), Cytosine (C), Guanine (G) and Thymine (T). These nucleotides complement themselves. For example, upon attributing '01' to C, '10' will be attributed to G. Analogously, upon attributing '00' to A, '11' will be attributed to T. There are total 4! = 24 combinations out of which just 8 combinations comply with the Watson-Crick rule of complementary nucleotides. Eight rules of encoding have been shown in the Table 1. To use in encryption process, DNA−Conv and DEC −Conv are two conversion functions from 8-bit pixel value to DNA sequence of length four and vice versa based on rules Table 1 There exist other operations in DNA cryptography like addition, subtraction and XOR. These operations resemble the operations in the binary number system. Since we have eight kinds of encoding rules, in the same way, we have eight kinds of addition, subtraction and XOR operations. Table 2 presents these three operations according to rule 1. For further use, we denote +, − and ⊕ for DNA addition, DNA subtraction and DNA XOR, respectively. We can these operations between two DNA sequences of length four based upon rules (1)(2)(3)(4)(5)(6)(7)(8).

III. PROPOSED IMAGE ENCRYPTION SCHEME
The given color plain image of size M × N × 3 comprising of three channels (R, G, B) is first converted into three one dimensional arrays I = (RO, GO, BO), by inserting one row after the other, respectively. Each of the one dimensional MN sized arrays RO, GO and BO consists of eight bit pixel values of red, green and blue channels, respectively. The CDS II.2 is iterated (MN )/4 + n 0 times with n 0 ≥ 100 for the initial values x 0 , y 0 , z 0 and w 0 that will be part of secret keys.

A. KEY STREAM GENERATION PROCEDURE
The key stream generation procedure (KSGP) given below will be used in various steps of the encryption process to generate entirely different and random key streams from the same chaotic data obtained by the iterations of CDS II.2. Let I = (RO, GO, BO) be the three given one dimensional MN sized arrays of eight bit pixel values. In order to make sure that a change in a single pixel value in any channel results into an entirely different cipher image, a linear combination of RO, GO and BO is obtained as follows.
where α, β, γ are integers and part of the secret keys. Thus, S is a one dimensional array of MN eight bit pixels values. Then, we apply CDSVSP to S and get the selected state variable slt that is used to generate the key stream denoted by K as follows: where floor(x) means the nearest integer to x toward minus infinity, abs(x) is the absolute value of x, mod(x, y) is the remainder when x is divided by y, round(x) rounds the value of x and i varies from 0 to MN − 1.

Remark 1:
The brief description of KSGP can be seen in the Figure 2. The CDSVSP [1] is quite efficient in generating entirely different sequences for selected variables even for the change of one pixel. Therefore, KSGP results into random key streams that will allow us to achieve high standards of security with minimal resources.

B. ENCRYPTION PROCESS
The proposed encryption procedure is presented as follows: Step: 1 (Decimal diffusion) KSGP is applied to the plain image I = (RO, GO, BO) three times with different key sets to generate three key streams K 1 , K 2 and K 3 . Then, I will be diffused with these key streams to get where the seed values RO 1 (−1), GO 1 (−1) and BO 1 (−1) are part of secret keys.
Step: 2 (Permutation) The three one dimensional arrays I 1 = (RO 1 , GO 1 , BO 1 ) are combined as one dimensional 3MN sized array In order to scramble the pixel positions, the chaotic tent map f (w, 3MN − 1, x) has been used h times. Here w = 3MN −1 i=0 T (i) mod (3MN − 1) and x = {0, 1, 2, . . . , 3MN − 1}, being the input variable, indexes each pixel of T . Since tent map works on the one-to-one principle, so it will render a permutation say σ (x). As the permutation σ (x) is applied on T , T is obtained. In other words, positions of pixels or index values are changed in σ (x), i.e., T (σ (x)) = T (x). Then, T is split into three one dimensional MN sized arrays Step: 3 (DNA conversion) Once again, KSGP is applied to I 2 = (RO 2 , GO 2 , BO 2 ) four times with different key sets to generate four key streams K 4 , K 5 , K 6 and K 7 . The three one dimensional arrays RO 2 , GO 2 and BO 2 are transformed into DNA nucleotides resulting into the arrays RO 3 , GO 3 and BO 3 by applying DNA conversion rules based on the key streams K 4 , K 5 and K 6 , respectively, as follows: if  Step: 6 (Cipher image making) The three one-dimensional MN sized arrays RO 5 , GO 5 and BO 5 of eight bit pixel values are converted into M × N sized red channel R , green channel G and blue channel B , respectively. Finally, from red, blue and green channels, a cipher image (R , G , B ) is generated as an output of the proposed image encryption scheme.
The decryption is done in the reverse order of encryption. The flow chart of encryption process is expressed in Figure 3.

C. DISCUSSION
KSGP is used seven times to generate key streams K i (i =  1, 2, . . . , 7) based on CDSVSP and linear combinations of all three channels red, green and blue. Therefore, one pixel change in any of the channels will trigger a different sequence of selected variables in CDSVSP resulting into an entirely different key stream. Also, the pixel values are permuted across channels, so pixel change in any position of the channels is equally affected in generating unpredictable key streams. Overall, key streams are directly linked with pixel values of the plain image.
The mask images (Key streams K i (i = 1, 2, 3)) are diffused with I = (RO, GO, BO) to change pixel values and it is apparent from Equations III.2 that they are also intermixed. The sequences of pseudo random numbers rule i (i = 1, 2, 3) are used to convert 8-bit decimal pixel values to DNA sequences of length 4. The use of pseudo random numbers being used as a conversion rule results into completely random and unpredictable DNA sequences as the same pixel value has different corresponding DNA sequences for different rule values (1)(2)(3)(4)(5)(6)(7)(8).
Moreover, a pseudo random number r given in Equation III.9 determines different set of DNA operations III.6, III.7 and III.8 that are used to intermix red, green and blue channels. Further, the key stream K 7 is also used to decide on the DNA conversion rules (1-8) using rule 4   In light of the above discussion, it is clear that the proposed encryption algorithm has the potential to resist any attacks with much less resources as compared to the other existing schemes. The experimental and security analyses done in the next section also support our thesis.

IV. SIMULATION AND SECURITY ANALYSIS
In the realm of image cryptography, a plethora of attacks exist. Normally, all these attacks exploit the inherent vulnerabilities lying in the design of ciphers. The list of known attacks normally include chosen plaintext/ciphertext attack, statistical attack, differential attack, brute force attack, noise and data crop attack, entropy attack etc. One of the principal foci of any cipher is to make it defiant to these attacks.
In order to do the security and performance analysis of the proposed scheme, eight different color images have been taken, all of size 256 × 256. The images 'Lena', 'Baboon', 'Peppers', 'Tree', 'House', 'Beans', 'F16' and 'Girl' can be accessed in the USC-SIPI Image Database, available online at: http://sipi.usc.edu/database/). All the simulations are done in Matlab 2016 version with 64-bit double-precision according to the IEEE 754 Standard [46].
To simulate the proposed algorithm, we have taken these initial values and system parameters for the four-dimensional chaotic dynamical system: Furthermore, the step size is taken sufficiently small in solving CDS to avoid any unwanted behaviour [47] and degradation effects [48].
The Figure 4 shows the input plain-images, encrypted images and the corresponding decrypted images. After encryption, the cipher images have been completely changed and have left no clue to reach to the original images.

A. KEY SPACE ANALYSIS
Key space is a very important factor for developing an encryption scheme. No matter how much robust algorithm, one has developed, if the key space is low, then it can be broken through brute force attack. The proposed encryption scheme comprises of twelve different system states and parameters that are x 0 , y 0 , z 0 , w 0 , a, b, c, d, b 1 , b 2 , b 3 , b 4 . Given the computational precision of 10 −15 , the key space of the proposed encryption scheme comes out to be 10 215 ≈ 2 714 as shown in the Table 3.  A comparison of the key space of the algorithm with the other recent algorithms has been made in the Table 4. The key space is sufficiently large for the proposed image encryption algorithm as compared to other recent algorithms.

B. SENSITIVITY TO SECRET KEY
Extreme key sensitivity is a necessary and sufficient condition for a good cryptosystem. Key sensitivity means that a minimal change in any of the keys should results into a radically different output. It can be gauged in both encryption and decryption processes. During the encryption process, a tiny change in only one of the secret keys should result in a VOLUME 8, 2020  completely different cipher image as compared to the one obtained using a key without any change. For decryption, the original plain image should not be recovered from the cipher image unless 100% correct secret key set is used.
In order to measure the key sensitivity of the proposed encryption scheme, two key sets with very tiny difference have been used to encrypt the same input image. Suppose, y 0 , z 0 , w 0 , a, b, c, d, b 1 , b 2 , b 3 , b 4 } is used to encrypt Lena plain image of the Figure 5a and as a result, we got the cipher image that is shown in Figure 5b. In order to demonstrate the key sensitivity, a very tiny change of 10 −14 has been done in just a single variable x 0 , i.e., x 0 = x 0 +10 −14 and we got a resulting new key set, Key 1 . On encrypting the same Lena image (Figure 5a) using the new key set Key 1 , the resultant cipher image is in the Figure 5c. Figure 5d shows an image obtained by taking the absolute value of the difference between the corresponding pixel intensity values in Figure 5b and 5c. An analysis of Figure 5b and 5c shows that a very small change of (10 −14 ) in only one variable has resulted in an approximately 99.62% differences among the encrypted images in terms of the intensities of pixels.
To demonstrate the key sensitivity in a more elaborate way, the rates of differences of pixel intensities between two encrypted images generated by Key 0 and Key t (t = 1, 2, . . . 8) for all 8 test images (USC-SIPI Image Database) have been calculated by introducing a very slight difference in only one key between Key 0 and Key t . The results have been listed in Table 5. It can be observed from the Table 5 that the average rate of difference between two encrypted images is more than 99.60%. Therefore, it can be safely concluded that a very minute change in the secret keys has resulted into the radically different encrypted images.
To demonstrate the key sensitivity for the decryption process, the keys Key 0 and Key 1 have been used to retrieve the original images from the encrypted images in Figures 5b  and 5c respectively. The retrieved original images are shown in the Figures 5e-5h. The figures indicate that we can only recover the encrypted images successfully, if we employ the correct keys. Even a very tiny difference in the secret keys has an enormous impact on the decryption result and cannot provide the valid input image as demonstrated in Figure 5. Therefore, we can conclude that the proposed algorithm is high sensitive to a very minute key change during the encryption and decryption processes.

C. DIFFERENTIAL ATTACK (PLAINTEXT SENSITIVITY)
A potential hacker tries in every possible way to find the original image. One of the ways is to do a minor change in the input plain image, then encrypt both the plain images and find some meaningful relationship between both the plain as well as encrypted images. To tackle this situation, two measures Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI ) have been developed to test the impact on the encrypted image after changing a single pixel in the plain-image. NPCR refers to the percentage of different intensities of pixels between the two images, i.e., plain and cipher images. Whereas, UACI is the average intensity of the differences between the plain image and cipher image. They are formulated as follows: where M and N are the dimensions of the image. In the above formula, D(a, b) is expressed as: C (a, b).
C and C are respectively the cipher images before and after one pixel of the plain image is changed. The values of the differential attack metrics of NPCR and UACI have been shown in the Table 6.
The average values of these metrics are 99.6300%(NPCR) and 33.5296%(UACI ) clearly demonstrating the defiance of potential differential attacks. Further according to the Table 7, the proposed algorithm performs better than the algorithms described in [4], [21], [35]- [37], [50]- [52] as far as NPCR and UACI for the Lena image are concerned.

D. RANDOMNESS OF GENERATED KEYSTREAMS
The proposed encryption scheme is efficient in a way that it uses less chaotic data. To see that generated keystreams (mask images) are totally random and have absolutely no relation with each other. NPCR and UACI are calculated between them. Table 8 shows that NPCR > 99. 6 and Table 9 shows that UACI > 33.4. Thus, the minimal resources are used without compromising the security.

E. STATISTICAL ANALYSIS
Enduring the statistical attacks should be one of the built-in features of a good image encryption scheme. Histogram and correlation attacks are normally included in the statistical attacks.

1) HISTOGRAM
An image's histogram depicts the pixel intensities distribution. A potential intruder can exploit the histogram of an  image to get some meaningful information for his malicious intentions. So, after the encryption process, the histogram of a ciphered image should be as much uniform as possible. Figures 6 and 7 depict the histograms of Lena image for its plain and encrypted versions respectively. In the plain image, the bars are not uniform, rather they are fluctuating. Hence full of information for some potential adversary. Whereas for the ciphered image, the bars are very uniform giving no meaningful information to an adversary. These uniform bars give a tough time to a potential hackers in launching any statistical attack.
Variance is a measure of average spread of data/information around the mean value. A low the variance means a lower differences between data point in a distribution. Thus, a small variance value is desirable for a uniform histogram. Table 10 gives the variance values for the histograms of the encrypted test images (Lena, Baboon, Peppers, Tree, House, Beans, F16 and Girl).
The variance values in the first column of Table 10 have been calculated through the initial key set Key 0 , while the ones in the remaining columns have been obtained by changing only one secret key of Key 0 . These secret keys are Key t (t = 1, 2, . . . 8) as discussed in the section IV-B. It is clear from the table that the variance values of the ciphered-images are about 800, which are less than those in [21], [36], [49], [53]. In contrast, variance values for the input plain images are about 86,488. Thus, the proposed scheme has a good encryption effect by reducing the variance values.

2) CORRELATION ANALYSIS
Generally, an image consists of a large number of pixels. Usually, the adjacent pixels are very similar, so they have very high correlation with each other. An ideal image encryption algorithm should break this inherent correlation between the adjacent pixels such that an adversary could not use this information to predict to the original image. Normally, the VOLUME 8, 2020    correlation between horizontal, vertical and diagonal pixels is calculated. The mathematical formula for calculating this correlation is given below [54]: where x and y are the pixel intensity values of two adjacent pixels and the number of pixels has been represented by N . Figure 8 depicts the correlation distribution of adjacent pixels in horizontal, vertical and diagonal orientations for the plain and ciphered Lena image.
The Correlation Coefficient (CC) between two adjacent pixels for plain and encrypted image of Lena have been shown in the Table 11. It can be seen from the Table 11 that the CC between adjacent pixels of the plain input image is nearly equal to 1 showing a high positive correlation between adjacent pixels. It is also clear from Figire 8a, 8b and 8c. Whereas, they are almost equal to 0 in case of ciphered image, showing no or very low correlation between adjacent pixels. Both the Table 11 and Figure 8 clearly demonstrate that the proposed encryption algorithm has broken the correlation among the adjacent pixels between the plain image and ciphered image and does not give even an iota of resemblance between the two. In other words, this signals to a successful breakage of the correlation of adjacent pixels in the plain image. Table 12 has compared the correlation of the original Lena image with its different encrypted versions generated by different encryption algorithms. The proposed algorithm has given the better results as compared to the encryption algorithms reported in [21], [37], [50], [51], [55]- [58].

F. INFORMATION ENTROPY
Entropy is used to measure the degree of randomness and unpredictability of an information source. In 1949, Shannon [59] developed a mathematical formula to measure information entropy: In the above formula, H (m) represents the information entropy of m that is an information source. p(m i ) is the probability of the symbol m i . The maximum value of the information entropy comes out to be 8, if we have an absolutely random image consisting of 256 gray values. The more close to 8 the value of entropy is, the better it always is. Table 13 shows the values of the entropy for our chosen images. The average of the values for all the images is nearly equal to the ideal value 8. Hence, the proposed scheme is very much resistant to any kind of entropy attack. Table 13 further gives a comparison with some other schemes. The proposed scheme performs better than those in [7], [21], [60], [61] as far as information entropy is concerned.

G. PEAK SIGNAL-TO-NOISE RATIO ANALYSIS
To create a maximum divergence between an input plain image and its encrypted version is the chronic idea of any image encryption scheme. To yardstick this measure, researchers have given the idea of Peak-Signal-to-Noise Ratio (PSNR). This metric measures the difference between the input image and its encrypted version. Mathematically it is defined as: where M and N refer to the width and height of the test image respectively. P 0 (i, j) and P 1 (i, j) are the intensity values of the pixels of the original and ciphered images respectively. Further, MSE is the mean squared error between the two images, i.e., the original and the encrypted one. The larger value of MSE will generate a smaller value of PSNR, which in turn renders a better encryption security. Table 14 gives the PSNR values of the proposed and the other encryption techniques. It can be observed from the table that the PSNR values between the plain and the decrypted images are (∞). It signals that there is no difference between the plain input image and its decrypted version owing to the fact that MSE = 0. This refers to the fact that the encryption and decryption algorithms do not lose any pixel intensity value. Results in Table 14 show that the PSNR value given by the proposed scheme is the smallest when it is compared with some other schemes [62]- [64]. Thus, the proposed scheme has a better security.

H. NOISE AND DATA LOSS ATTACKS
Practically, the transmission of images may get polluted with some kind of noise during its transfer over the network or multi-media network. Furthermore, during the transmission, some portion of the image may be lost. A good encryption scheme should endure the noise and data loss attacks. Figures. 9a to 9c Figures. 9d to 9f show the decrypted images through the usage of the proposed algorithm. It is clear that the original images are easily discerned from the noisy ciphered images. Figure 10a plots the encrypted Baboon image.   In Figure 10b, a block of data from the Baboon ciphered image with size 80 × 170 has been cropped out to depict the data loss. The ciphered image with partially lost data is decrypted using the proposed algorithm. Figure 10c shows the decrypted image. It is clear that the decrypted image still bears so much information that it may be easily recognized. Thus, we can conclude that the proposed scheme is impervious and resistant to any noise and data loss threat.  for judging the computational time of some algorithm, i.e., empirical and theoretical. In empirical method, the computational time of some algorithm is physically measured through some gadget like stopwatch etc. We have calculated the encryption throughput (ET) for the proposed algorithm. ET is calculated by dividing the size of the given image in Megabits to the total time taken for its encryption. The average ET for the eight images is 2.68 MBit/s. The Table 15 shows that the proposed encryption procedure is better than [7], [52], [53], [65].
In theoretical analysis for calculating time-complexity, the mathematical theory of Asymptotics [66] is employed. For this purpose, two aspects will be considered, i.e., generation of the chaotic data and the encryption algorithm. Chaotic data K is being generated by applying CDSVSP to the array S described in the section III. Its complexity comes out to be (4MN ) where (M , N ) is the size of the given input image. The complexity of both the decimal diffusion and permutation operations is (6MN ). The complexity of DNA conversion and decimal conversion is (6MN ). Moreover, the complexity of DNA level diffusion is (3MN ). So, the total complexity comes out to be (19MN ), which is better than (24MN ) [21] and O(64 N 2 ) [67]. If the image taken is a square of size N , then it becomes (19N 2 ).

V. CONCLUSION
The simulations and experimental test results show that the proposed encryption scheme is successful in maintaining high standards of security even though using much less resources in comparison with the existing encryption algorithms. The chaotic data obtained from CDS is of high quality and CDSVSP is quite efficient in generating entirely random selected sequences from it. Also, the pixels of plain image are directly involved in all the encryption steps like DNA conversion, diffusion in decimal, thus DNA sequences have high plaintext sensitivity, making it impossible to break by plaintext attacks and statistical attacks. The full potential of DNA model is demonstrated by using different conversion rules and DNA operations for each pixel that further randomise the ciphered image, improving the existing encryption algorithms based on DNA model. Thus, due to lower time complexity, higher efficiency and security, we are justified in claiming that the proposed algorithm is more suitable for real-time and real-life applications in the filed of image security, multi-media networks and on-line systems.
Funding This study is partially funded by Universal College of Learning, New Zealand (PD20/1323).
M. G. ABBAS MALIK received the Ph.D. degree in computer science with specialization in artificial intelligence and statistical learning from Université Grenoble Aples (UGA), France, the master's degree in computational linguistics from Univesité de Paris 7, Denis Didrot, France, and the Master of Science degree in computer science from the University of Punjab, Pakistan. Since 2009, he has been serving in various universities in France, Pakistan, Saudi Arabia, and New Zealand. He is currently working with the Universal College of Learning, New Zealand. His primary research interests include artificial intelligence, machine learning, data analytics, natural language processing, computer vision, augmented reality, and virtual reality.
ZIA BASHIR is currently an Assistant Professor of mathematics with the Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. His research interests include function spaces, fuzzy Analysis, decision making models, and cryptography.
NADEEM IQBAL received the M.Phil. degree in computational science and engineering from NUST, Islamabad, Pakistan. He is currently working as an Assistant Professor with the School of Computing and Information Sciences, Imperial College of Business Studies (ICBS), Lahore, Pakistan. Prior to joining the ICBS, he worked in various academic institutions and has guided numerous bachelor's and master's students. His research interests include images cryptography, computer graphics, and philosophy of mathematics.
MD. ATHAR IMTIAZ received the master's and Engineering degrees in computing, and the Ph.D. degree in computer science from the University of Auckland. He is currently employed as a Lecturer with Massey University, Palmerston North. His research interests include HCI, machine learning, education technology, and software engineering. His area of specialization is human-computer interaction (HCI). He has industry work experience in some of the largest software multinational companies. VOLUME 8, 2020