A Novel and Efficient Multiple RGB Images Cipher Based on Chaotic System and Circular Shift Operations

In real time scenarios, the single image encryption schemes are not that efficient when a bunch of color images is to be encrypted. This problem should be addressed due to computational cost incurred by multiple executions of single color-image encryption scheme. The objective of this research work is to provide a new and efficient multiple color images cipher scheme. This cipher is based on chaotic system, circular shift operations and SHA-384 hash codes. First of all, an arbitrary number of RGB images are input. They are combined in a rectangular fashion to get their hash code. Then this big color-image is decomposed into its three basic components, i.e., red, green and blue. To scramble the pixels of these components, they are shifted circularly both the row-wise and column-wise. Further, XOR operation has been performed to realize the effects of diffusion in these components. Lastly, these components are joined together to get the final RGB cipher image. Apart from the 384-hash codes, a salt key of 384-bit has also been used in the proposed cipher to heighten the security effects. The SHA-384 hash codes render the required key space and plaintext sensitivity. The simulation results and the performance analyses carried out at the end clearly prove the efficiency, effectiveness, robustness and the real-world applicability of the proposed multiple RGB images cipher. Besides, the encryption throughput 7.716 Mb/s of the proposed scheme outperforms many of the existing ones.


I. INTRODUCTION
The present era of digital technologies, internet, internet of Things (IoT), Big Data etc. have revolutionized virtually all the cultures of the nations. In all of this, digital images have assumed a lot of importance. These images are being generated in the diverse areas like military, business, commerce, research, manufacturing, designing, space science, medical science, government and diplomatic missions, weather forecasting, medicine, personal affairs to name a few. Besides, traffic cameras installed on the highways in almost all over the The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. world routinely shoot thousands of images per day. Moreover, a single lens camera has the capability to take many pictures per second. So images are all around of us. We frequently save them and transmit them through some public channel like internet. This storage and transmission of images have grave implications in case they are very sensitive and important and are hacked by some intruders and adversaries. Therefore, it has become an urgent and a serious task to safeguard these images. Traditionally, the objective of the integrity of data has been realized by encrypting it by classical ciphers like RSA, DES, AES etc. But unfortunately, these ciphers cannot be used for images encryption due to their different characteristics like strong bulky volume, correlation among the adjacent pixels, and high redundancy [1], [2]. On the other hand, chaotic maps/systems have proved very excellent in generating the random data which was in turned used for image encryption. The reason of their excellence lies in the natural properties of pseudo-randomness, mixing, aperiodicity, huge key space, unpredictability and high sensitivity to both the systems parameters and initial conditions. Dozens of image ciphers were developed [1], [3]- [9] by using these chaotic maps. But many ciphers contain security flaws. For example, these ciphers [7], [10], [11] were broken by [12]- [14] through the invocation of differential attack, chosen ciphertext attack and chosen-plaintext attack respectively. The reason of the breakage of these ciphers was that no matter which image was input, the same random data was generated. In particular in [12], it was investigated that the secret key was only dependent upon the permutation key instead of all the keys being used. This resulted into the reduction of the key space. So corrective measures are needed to adopt to address these problems by introducing the plaintext sensitivity stuff in the potential ciphers.
In this demanding and challenging modern world, time is a very critical factor. In the past, many ciphers were developed which took too much time for their execution [15]. This time consumption is not in line with the requirements of the modern day. So onus of responsibility lies on the shoulders of cryptographers to develop speedier ciphers to meet the demands of the current era. In other words, encryption throughput (ET) needs to be increased.
Hundreds of Single-Image-Encryption (SIE) schemes exist. No doubt, one can encrypt multiple images one by one by using these schemes. But the performance and efficiency of these schemes will be undesirable given the urgency of the modern day. The strict requirements of the modern world dictate to develop such ciphers which may encrypt an arbitrary number of images efficiently. So Multiple-Images-Encryption (MIE) schemes have gained an increasing attention in these years. For instance, the authors of [16] gave an MIE using 2-D chaotic economic-map and Mixed Images Elements (MIES). As a modus operandi of their scheme, they combined together the given input and split the given input image into its pure image elements (PIES). Then, to permute these PIES, logistic map was used. Confusion effects were realized through the usage of chaotic economic map to get the MIES. Lastly, these MIES were grouped to get the big encrypted image which was later broken into sub images with the original size. A new MIE scheme using chaotic system and DNA has been proposed in [17]. They merged images to a single image followed by scrambling it using chaotic map. The diffusion effects were achieved through DNA conversion and applying DNA XOR operation. SHA-256 hash values were used to calculate the starting parameters of the map. In an other MIE [18], authors developed an index-based permutation-diffusion scheme through the usage of DNA computing. In the first stage, they joined together multiple images. Then this big image was converted into one dimensional array. First half of array indexes were utilized to scramble pixels. Same indexes were used for the DNA computing in the permutation phase, which resulted into the realization of the diffusion effects. Besides, by using the chaotic system and bit-plane decomposition, [17], [19] presented an MIE with the one limitation, that it could encrypt only four images in a single session. So measures must be taken to develop such ciphers which have the capability to encrypt an arbitrary number of ciphers. Recently, a yet another MIE was reported in [20] through the usage of phase mask multiplexing in the fractional Fourier domain. The only problem to this MIE is that, the quality of output decrypted images gets degraded as the number of input images increases. Such defects must be addressed while engineering any new MIE.
Keeping in view the above literature review, we have proposed a novel MIE addressing the accompanying issues. The following highlights characterize the main contributions of the paper: • Through the very simple method of circular shifting of the pixels, the confusion effects have been realized.
• The proposed MIE is very efficient as far as computational time is concerned. Further, a phenomenal rise has been achieved regarding the encryption throughput. So it has ample prospects for some real world application.
• Sixteen color images (equivalent to 48 gray scale images) have been encrypted and decrypted to demonstrate the feasibility of the proposed MIE. To show the capability of the proposed cipher, six different sizes of the combined image have been taken.
• Many steps of the algorithm are of repetitive nature. So this trait can be easily harnessed to parallelize the proposed algorithm in some parallel setting.
• The key space and the plaintext-sensitivity, both are very competitive in the proposed scheme.
• No degradation in decrypted images have been witnessed.
The paper is arranged as follows. Section-2 gives an overview of chaotic map/system and circular-shift operations. The proposed encryption/decryption schemes and generation of system parameters and initial values are discussed in section-3. Then the section 4 highlights the simulation results and section-5 presents security analysis of the proposed scheme. Section-6 is for the discussion of our contribution. In the last, section-7 describes the conclusion of the paper.

II. PRELIMINARIES A. CHAOTIC-MAP
Due to the marvelous characteristics of the chaotic ssystems/maps like ergodicity, aperiodicity, extreme sensitivity to system parameters and initial conditions, mixing, etc., these maps are extensively used in writing many image ciphers. The Intertwining Logistic Map (ILM), whish is a three dimensional chaotic-map was selected for this proposed scheme. As the researchers refined the one dimensional (1D) and two dimensional (2D) logistic maps, ILM came into being with larger key space and is defined below [21]: where n = 0, 1, 2, . . . . ., x 0 , y 0 , z 0 are the map initial values and 0 < µ ≤ 3.999, |k 1 | > 33.50, |k 2 | > 37.97, |k 3 | > 35.7. The Intertwining Logistic Map (ILM) generates best chaotic behavior than its predecessors with no empty values and has significant even distribution as shown in the Figure 1

III. ENCRYPTION SCHEME
The proposed encryption scheme has been developed to encrypt an arbitrary number of RGB images say N in one session. The corresponding flowchart diagram has been shown in the Figure 3.
The proposed multiple RGB images encryption scheme consists of seven stages. In the first stage, an N number of color (RGB) images, each of the same size m × n × 3, are input. These color (RGB) images are arranged in a rectangular way to form a big color image. The SHA-384 hash code has been taken of the so formed big color image to introduce the plaintext sensitivity in the proposed scheme. This action will serve two purposes. Firstly, the potential differential attacks launched by the adversaries would be averted. Secondly, it will help increase the key space. This big key space will help to counter brute force attack from the attacker and intruders. To boost the security effects of the proposed encryption scheme a 384-bits salt key has also been introduced. In the second stage, XOR operation has been conducted between the SHA-384 hash code and the 384-bits salt secret key. The resultant 384-bits or 48-bytes code is  reshaped into 4 × 12 matrix. In the third stage, twelve values of each column of the 4 × 12 matrix are added for all the four rows to get the four values. These four values have been used in tempering the starting values of four system parameters of ILM being used in the scheme. The three initial values of ILM are also modified by using these four system parameters. Now the chaotic system ILM is iterated to obtain three random numbers streams. These three streams of random numbers are further tailored in the way that suits to the peculiar logic of our proposed cipher and finally the three streams of random numbers obtained have been named as selection − rotation1, selection−rotation2 and key−image. In the fourth stage, the big color (RGB) image formed in the first stage is decomposed into its integral color channels of red (R), green (G) and blue (B). In the fifth stage, effects of confusion have been realized. The random numbers produced through selection − rotation1 decide the columns whose pixels are to be circularly shifted. The amount of shifting is decided by the random numbers produced through selection − rotation2.
In the same fashion, the random numbers produced through selection − rotation2 decide the rows whose pixels are to be circularly shifted. The amount of shifting is decided by the random numbers produced through selection−rotation1. This process of shifting is applied for all the three channels one by one to get the scrambled images. In the sixth stage, these scrambled images are further XORed with the key − image key stream one by one to get the encrypted images. In the seventh and the last stage, these encrypted images are merged together to obtain the final big encrypted RGB image.

A. GENERATION OF THE SYSTEM PARAMETERS AND INITIAL VALUES
In the proposed multiple images encryption scheme, the SHA-384 hash function is used to generate a 384-bit external secret key. This act increases the relationship of secret key with the input plain images being used in the scheme. This will augment the security of the proposed image encryption scheme. According to the characteristics of hash function, as an extremely tiny change is made in the given input image, say a single bit, it will generate a totally different hash value. Moreover, a 384-bit salt key has also been used to increase the security effects of this multiple RGB images encryption scheme, as has already been described. The K which is 384-bit secret key for hash code is split into 8-bit blocks which can be stated as follows: subject to k a = k a,0 , k a,1 , . . . , k a,7 , where in k a,b , a indicates the character number and b is the bit number in k a,b . Likewise, a 384-bit salt-key, say SK, is also further broken into 8-bits blocks which can be written mathematically as: Subject to: sk a = {sk a,0 , sk a,1 , . . . , sk a,7 }, where in sk a,b , a represents the character number, and b is the bit number in sk a,b . The initial values of the chosen chaotic map i.e. ILM and key streams have been generated using the following steps. VOLUME 8, 2020 Step 1: The K and SK are reshape into 4 × 12 matrices.
Step 2: Between K and SK, XOR operation is performed, as where ⊕ represents an XOR operation. K is the new key of 384-bits or 48-bytes.
Step 3: The column values for each four rows are added, we get the following: Step 4: The Intertwining Logistic Map system parameters are calculated as follows: Step 5: The systems parameters already calculated in the step 4 are used to calculated the initial values of Intertwining Logistic Map as follows: In the above equations, mod(a, b) is an arithmetic operator which renders the remainder upon dividing a by b.
Step 6: The sequences of three chaotic streams were generated by chaotic system (1), which is repeated for (MN +n 0 ) We discard the first n 0 values of the chaotic-sequence in order to remove the transient effect of the chaotic system. Where n 0 is part of secret keys and its range is: n 0 ≥ 500.
Step 7: Moreover,the streams of random numbers u, v and w are again passed through the following equations.

B. ENCRYPTION PROCEDURE
The N color (RGB) images each of the size of m × n × 3 are encrypted using the proposed scheme in one session. The following steps explain encryption procedure in detail.
Step 1:(Inputting and joining color plain images) Take input of N color images with each of the same size m × n × 3. Join the input images in such a way that one big image img of size M × N × 3 is obtained.
Step 2: (Decomposing big color image) Decompose big image img into its component channels i.e. red(R), green(G) and blue(B) for the (RGB) red, green and blue components each of size M × N .
Step 3.2: Select row number and column number for the red channel and shift the pixels circularly.
Step 3.4: Repeat steps 3.2 to step 3.4 while index ≤ MN ρ Let the scrambled image obtained is red .
Step 4:(Diffusion operation) Reshape red into the size 1 × MN. Take the XOR operation between red and the key − image.
Reshape red into M × N to get the final encrypted red channel. Repeat steps 3 and 4 for the other green and blue channels as well to obtain the encrypted green and blue channels as green and blue . Combine these three encrypted-channels red , green and blue to obtained the encrypted-color (RGB) image say img of size M × N × 3.

C. DECRYPTION PROCEDURE
As it is well known in the context of cryptography, the decryption routine is the antonym of the steps of the encryption, because the proposed encryption scheme is private in nature. Figure 4 is for the decryption routine of the potential scheme.
The decryption procedure is described in the following steps. Step 1: The encrypted-color big image img of size M × N × 3 is provided as input to the decryption procedure.
Step 2: selection-rotation1, selection-rotation2 and keyimage, being the three key streams, are generated as defined in the section III.A.
Step 3: Decompose the img into three color RGB components of red(R), green(G) and blue(B).
Step 4: (Nullifying the diffusion effects) Reshape the color channel red into the size 1 × MN and perform the following XOR operation.
i = 1, 2, . . . , MN. red has now only the scrambled effects because the diffusion effects have been taken back from the encrypted image through the equation (26).
Step 5.2: Select row number and column number for the red channel and shift the pixels circularly.
Step 5.3: index = index + 1 Step 5.4: Repeat steps 5.2 to step 5.4 while index ≤ MN ρ Let the unscrambled or plain channel obtained is red . Reshape this image to M × N .
Repeat steps 4 and 5 for the other green and blue channels as well to get the unscrambled or plain green and blue channels as green and blue . Combine these plain channels to get the final plain color image say img . Lastly separate the different smaller plain images from this big color plain image.

IV. SIMULATION
Any good image cipher should have the capability to withstand the entire array of known attacks launched by the potential adversaries. These attacks normally include statistical attacks, histogram attack, differential attacks, brute-force-attack, entropy attack, ciphertext attack, chosen plaintext etc. To practically demonstrate the utility and performance of our potential multiple images encryption scheme, we have chosen sixteen images, i.e., Lena, Baboon, Vegetables, Crane, Elephant, Butterfly, Bird, Wheelbarrow, Radio Telescope, Brain, Camel, Cannon, Duck, Tree, Fire Truck and House each of size 256 × 256 as shown in Figure 5

V. SECURITY ANALYSIS
The performance analyses based on various yardsticks will be conducted in this section.

A. KEY SPACE ANALYSIS
One of the most vital characteristics of any image cipher is key space. Key space consists of the set of keys upon which the cipher depends. To successfully deal with the brute force attack launched by the hackers, it must be sufficiently big. The proposed cryptosystem consists of four salt keys sk 1 , sk 2 , sk 3 and sk 4 each consisting of twelve bytes or twenty four hexa-decimal numbers. It contributes (2 8 ) 48 = 2 384 to the key space of our proposed encryption scheme. Apart from that, the ILM being used in the proposed cipher has three initial state values and the four system parameter values totaling seven variables. So this contributes (10 14 ) 7 = 10 98 to the key space of our proposed encryption scheme, if 10 −14 is VOLUME 8, 2020  taken as computer precision. So 2 384 × 10 98 ≈ 3.94 × 10 213 comes out to be the total key space of the proposed cipher. In order to avert any brute-force-attack, this key-space is more than sufficient. Besides, it fulfills the minimum requirement of 2 100 [17], [22]. Moreover, the comparison between other published works and our proposed cipher key space is shown in Table 1.

B. KEY SENSITIVITY ANALYSIS
Whatever image cryptosystem is developed, it should be equipped with an extreme sensitivity to a key. This concept refers to a phenomenon that upon making a very little change in any key, the output should be absolutely different from the previous output. In the context of image encryption technology, the key sensitivity can be test with two ways. In the first scenario, the process of encryption is done by making a very minute change in the given key set. The entirely different output should be produced from the previous output. In the second scenario, the process of decryption is not invoked with the correct key set, rather, there should be a very small alteration in the key set. The ciphered image should not be obtained unless the correct values of the key set are not given to the decryption machinery.
In the sensitivity tests for a key in the first case, through the minutely distinct two key sets, one single plain image is encrypted. Suppose K 0 = {x 0 , y 0 , z 0 , µ, k 1 , k 2 , k 3 } be the initial key set. In the Figure 7(a), for encryption of given image the key K 0 has been used and the output ciphered image in Figure 7(b) has been got. After it, a very minute tempering has been made to x 0 , say = 10 −14 i.e., x 0 = x 0 + , and in this process, the remaining keys have not been changed. A different key set say K 1 has been obtained through this action. The big image in Figure 7(a) is now encrypted using the key K 1 which resultantly produced the cipher image as shown in Figure 7(c). The differential image of the cipher images (Figures 7(b) and 7(c)) is obtained and is shown in Figure 7(d). This image is obtained by taking  the difference between the absolute value of corresponding pixel intensity values. Apart from that, there exists very slight difference of 10 −14 in one of the parameters of the keys, the difference of pixels in between the two encrypted images of the Figures 7(b) and 7(c) is 99.6144%. To probe the matter of key sensitivity in a more detail, difference rates were calculated between the two ciphered-images got by K 0 and K a (a = 1, 2, . . . , 14). The results so obtained are presented in the Table 2. This table further reports that 99.5748% is the minimum difference rate which is better than [?], [23], [24]. Moreover, the value of 99.61% has been calculated to be the average key sensitivity which is equal to the [25] and better than [23], [24]. Therefore, we can assert that the reported encryption algorithm is superior than the others.
As far as the evaluation of the key sensitivity of the second case is concerned, K 0 and K 1 have been used to retrieve the original images from the encrypted images in Figures 7(b) and 7(c), respectively. The retrieved images so obtained have been depicted in the Figures 7(e) to 7(h). The figures give the clear picture that the encrypted images can only be retrieved by employing the correct key sets. Even a very little tempering in the keys will end up with categorically different results VOLUME 8, 2020  and relevant plain images will not be got. Concludingly, we can assert with full confidence that the reported image cryptosystem have the benefits of high key sensitivity for both of its encryption and decryption machineries.

C. STATISTICAL ANALYSIS
Another very significant metric is the statistical analysis. The histogram analysis and correlation coefficient analysis are the two statistical analysis tests that are usually conducted by researchers.

1) HISTOGRAM ANALYSIS
The spread of the pixel intensity values of the given image is shown via histogram. A plain image is normally a meaningful arrangement of the pixels. Resultantly, its histogram is full of information. A potential antagonist can easily tap to recover the original image from this information depicted in the form of the slanting bar of the histogram. The cipher image should have the uniform bar. Figures 8 and 9 show the histograms of the original plain combined image and cipher image. Both these figures have entirely different histograms which makes the statistical attack very difficult to succeed.
Besides, we also calculated the variance of the histograms which is a measure about the uniformity of the histogram of the encrypted images. Relatively higher the uniformity, smaller the variance values will be, which in turn depicts the good security effects of the encryption algorithm [34], [35]. The variances of the histograms of the selected sizes encrypted images are shown in the Table 3. As can be seen from the Table 3, the variances calculated through the key K 0 are shown in the first row, whereas defined in the section V.B, the variances of the other rows have been obtained after the modification in the keys which are K a (a = 1, 2, . . . , 14). The average variance value is 5459 which is less than [36]. Apart from that, 621,874 is the calculated average value for the variance regarding the histograms of the plain images.
Moreover, the effect created after altering the key upon the smoothness/uniformity of the encrypted images by calculating the percentages of variance differences between the two cipher images has also been investigated. The results so obtained have been shown in the Table 4. An average variance fluctuation for the entire table comes out to be 0.19%. Further, the maximum variance fluctuation is 0.74% which is less than the [23]- [25], [37]. Besides, the minimum variance fluctuation is 0.00% which is again less than [23]- [25], [36], [37]. These comparions show the potentially good security effects of the proposed MIE.
The uniformity in the cipher images is further verified by using Chi-Square test [24], [25], [38], [39]. The lower chi-square values are indicative of higher uniformity of a cipher image and vice versa [38]. The mathematical formula  characterizing this concept is: where obf j is observed frequency of j, and exf j is expected frequency of j which is expressed as: here M × N represents the resolution of an input image. The Chi-Square test results of the potential cipher and a comparison carried out with other ciphers [24], [25], [38], [39] are shown in Table 5. The significance level are kept 1% and 5% in the test. At 5% and 1% significance level and 255 degrees of freedom, the chi-square value is 293.2478 and 310.457 respectively [40]. For both 5% and 1% level of significance the hypothesis is accepted as can be deduced from , which means that the encrypted image histograms through our scheme the gray scale values are uniformly distributed.

2) CORRELATION COEFFICIENT ANALYSIS
For any plain image, normally high correlation among the adjacent pixels exists. Any two consecutive pixels are called adjacent pixels. These pixels may be diagonally, horizontally, or vertically adjacent. To dismantle the strong nexus among the pixels is one of the basic tasks of any image cipher. In order to calculate the correlation coefficient, In these three directions, we have randomly chosen 8,000 pairs of the consecutive pixels. These directions include diagonal, horizontal VOLUME 8, 2020 and vertical. The formulation in the language of mathematics for calculating this coefficient is In the equation (35), x and y represent the intensity values of the pixels and the total number of pixels are N in the image. Figure 10 depicts the distribution of the correlation coefficient diagonally, horizontally and vertically adjacent pixels of the input original RGB image and encrypted image.
The correlation coefficients between two adjacent pixels for the encrypted images of sizes 1280 × 1280, 1024 × 1024, 768 × 768, 512 × 512, 256 × 256 and 128 × 128 are shown in Table 6. This table depicts that the correlation coefficients between the adjacent pixels of the cipher images is very close to 0. Further, a comparison with other existing MIEs has also been performed. Both the Table 6 and Figure 10 depict that after the application of our proposed encryption procedure at the input images, the relation between input plain images and output cipher images have been minimized abundantly. Put in other words, it refers to dismantling the correlation in between the adjacent pixels in their entirety.

D. INFORMATION ENTROPY(IE) ANALYSIS
IE or information entropy characterizes the degree of uncertainty and arbitrariness in the given information source. The scientist named as Shannon [41] translated this concept into a mathematical formula, which is In the above equation the H (m) refers to the IE of the information source m. Moreover, the probability of the symbol m i is represented by p(m i ). For an ideally randomized image pixel-wise with 256 grey values, 8 (Eight) is the maximum value of information entropy. Naturally, the encrypted image information entropy should be very near to 8. Table 7 shows the values of IE of the images with different sizes. One can see that the average value for the entropies is very close to 8 for all the images, which depicts the good security effects as far as the potential information entropy attack is concerned. Further, different existing MIEs have also been compared with ours. Our results are better.

E. LOCAL SHANNON ENTROPY
The randomness of encrypted images can be measured using the Local-Shannon-Entropy. According to Ref [43]- [45], mathematically, local Shannon entropy can be represented as: where H 1 , H 2 , H 3 , . . . , H l are non-overlapping blocks with U E randomly selected pixels of the encrypted image. The S H j (l = 1, 2, 3, . . . , l) are obtained by equation 37. In the experiment, we assign l = 30, U E = 1936, which mean that 1936 random pixels are selected for each 30 blocks of cipher image. According to Ref [45], with respect to αlevel confidence of 0.05, the obtained value of local-Shannonentropy will be in between 7.901901305 to 7.903037329. The results obtained on test images for local Shannon entropy can be seen in Table 8, which represents ideal degree of randomness for the encrypted images.

F. PLAINTEXT SENSITIVITY ANALYSIS (DIFFERENTIAL ATTACK)
There are numerous ways through which cryptanalyst can access to secret key. Differential attack is one of them. The mechanism of this particular attack works as follows. Hacker manages to have two copies of some image. He tempers one image by doing a very little change in it. Afterwards, these two plain images are passed through the encrypted procedure to obtain the cipher images. In this way, the relationship between these two images can be found if they are handled intelligently which can further lead to the discovery of the secret key. This explains the term ''differential attack''. To check the robustness of the algorithm against this attack, two metrics are traditionally used, i.e., unified average changing intensity (UACI) and number of pixels change rate (NPCR). These measures find the effect of changing a single pixel in the plain image over the cipher image. The mathematical formulae are In the above equation, (M , N ) is the size of the image. F(i, j) is defined mathematically as: where G represents the ciphered image before one pixel change and G represents the cipher image after one pixel change of the plain image. Table 9 shows the NPCR and UACI values of the chosen images with different sizes. The averages of NPCR and UACI VOLUME 8, 2020  for the three color components of the different sizes are 99.6077% and 33.4158% respectively, which is symptomatic to the fact that the potential image cipher is too much potent to endure the differential attacks. Apart from that, a comparison with other schemes has also been made.
The Tables 10 and 11 shows the critical values [46] about the NPCR and UACI respectively. The critical values N * 0.05 , N * 0.01 , N * 0.001 to reject the null hypothesis with the α = 0.05, α = 0.01, α = 0.001 (three significance levels) are shown in Table 10. This means that if the value of NPCR for these two cipher images is less than N * α , two cryptic images will be not sufficiently arbitrary and random with an α-level of significance. It is clear in the Table 12 that for all the sizes of images for the proposed encryption scheme the values of NPCR at all the levels of confidence fulfil the critical (theoretical) threshold of randomness test of NPCR. For UACI the critical value U * α , is composed of two parts, i.e., U * + α , the right value and U * − α , the left value as shown these values in Table 11. For the proposed encryption scheme, the null hypothesis gets rejected, if any value for the UACI is outside the interval (U * − α , U * + α ). For all the sizes of images over the proposed cipher, the values of UACI shown in Table 13 fulfil the critical values of UACI randomness test.

G. PEAK SIGNAL-TO-NOISE RATIO ANALYSIS
The recurrent idea of any image encryption scheme is to achieve maximum discrepancy between the input plain image and its output cipher image. To deal this situation, a metric called Peak Signal-to-Noise Ratio in short ''PSNR'' is usually employed. This measure has been designed to gauge the difference between the two images. Its mathematical formula is: where the width is represented by M and height is represented by N for the test image. The pixel values are represented by: P 0 (i, j) for original image and P 1 (i, j) for encrypted image. The Mean Squared Error or simply MSE is the departure or error between the given input original image and its output encrypted image. The smaller the PSNR value, the larger the MSE value, the better the encryption security.
The PSNR values as shown in Table 14 are obtained by different techniques. One can see from the afore-mentioned table that its values are infinite (∞) in between the original image and the decrypted image. This infinite value was caused due to MSE = 0. It naturally implies that the generated output decrypted image is exactly identical to the original image. This also proves that our proposed encryption scheme is lossless. Further, in terms of the similarity between the input original plain image and the cipher image, the PSNR values of our images obtained by our proposed algorithm is better than [47], [48]. This means that our proposed encryption algorithm has better encryption effect.

H. MEAN ABSOLUTE ERROR (MAE) ANALYSIS
The main objectives of any image cipher scheme is to maximize the difference between the plain image and cipher image. Mean-Absolute-Error or simply MAE is basically used for this purpose. The MAE can be mathematically represented as follow: For each plain image P, and cipher image C, the M is used for width and N being used for height. The larger value of MAE the better will be the proposed scheme. The MAE results VOLUME 8, 2020  have been shown in Table 15 generated by our proposed encryption scheme and a comparison has also been drawn with some other published works.  On the other hand for data loss attack, the encrypted images have been shown in Figures 12(a) to 12(d). The amount of the loss of data in these figures is 0%, 25%, 50% and 75% respectively. Using our proposed scheme after deploying the decryption algorithm to these cropped cipher images, the corresponding decrypted images are shown in Figures 12(e) to 12(h). Again, one can easily judge that the retrieved images are so much intact that they can be easily discerned. These results cogently say that the proposed image cipher has the ability to avert the attacks of noise and data loss.

J. TIME COMPLEXITY ANALYSIS
Along with the security considerations, a proposed cipher scheme must be efficient as well vis-à-vis its running time. In real time environment such efficient algorithm has  more applications and demand. Using the machine with the given specifications: Intel(R) Core(TM) i7-3740QM CPU @ 2.70 GHz, 8 GB Memory (RAM) and 500GB Hard-disk with Windows 10 Education, MATLAB R2018a, the proposed encryption scheme has been developed and simulated. The Table 16 compares the time taken between our algorithm and the other schemes.
Apart from the encryption and decryption speeds, there is an other related concept called encryption throughput which measures the image size in the give time. Mathematically, it is described as follows The encryption throughout of our proposed scheme and a comparison conducted with the existing schemes is shown in Table 17. One can see that our scheme vividly outperforms these schemes vis-à-vis ET. We couldn't find any MIE analyzing its ET in the literature. Thus we compared our MIE with the single image cipher schemes regarding ET.

K. COMPUTATIONAL COMPLLEXITY
In the literature, theoretical analysis for the computational complexity have also resorted by the researchers. The mathematical theory of Asymptotic [52] is normally used for this purpose. Both the generation of the chaotic data and encryption scheme contribute to the computational complexity. Intertwining logistic map (1) has been iterated MN times to generate the three streams u, v and w of random numbers, where (M , N ) is the resolution of the input image. Since there are MN iterations and, in each iteration, three instruction are being executed, thus obtained complexity is (3MN). Further, the three streams of selection-rotation1, selection-rotation2 and key-image have been generated out of the ''raw'' streams u, v and w and this again contributes  (2MN/ρ + MN) to the complexity, ρ being the geometric mean between M and N . In the confusion phase, both the column and row shifting operations cost MN/ρ in each iteration and there are six instructions so the total costs is (6MN/ρ). The diffusion phase costs (MN). By adding all these costs, the complexity comes out to be (5MN + 8MN/ρ). This complexity is only for one channel. For all the three color channels, the total complexity comes out to be (15MN + 24MN/ρ) or (15MN + 24 √ MN ). After putting the value of ρ, this complexity beats these studies [4], [37] since the computational complexity of both of these studies is (24MN). The proposed cipher beats [4], [37]   at M = 256 and N = 256. As the dimension of the input image increases, this factor will also increase.

VI. DISCUSSION
In the context of images encryption and decryption, two contradictory things exist, i.e., security and efficiency. They are reciprocally interrelated. This is the job of the cryptographer while engineering any new cryptographic product, to strike a proper balance between these two factors. The primitives of the enterprise of cryptography have been knitted and cohered in such a way that both the requirements are satisfied. This is what we have achieved in the study under consideration. The pressing demands of this modern world is to encrypt maximum images in the minimum possible time. This concept is formalized through the encryption throughput, and we got this as 7.716 Mb/s. This result outshines many of the existing image encryption schemes present in the literature. One can see from the results of the validation metrics, i.e., information entropy, noise and data loss attack, differential attack metrics (NPCR, UACI), correlation coefficient, that they all are very competitive. So we are justified in saying that through a clever engineering, we have struck a proper balance between the two competing requirements of security and efficiency.

VII. CONCLUSION
The schemes for encrypting single images for the security purpose abound in the literature. There exist few schemes which address the encryption purpose for multiple images. The salient feature of the proposed MIE is the enhancement of the encryption throughput. In this study, a novel encryption/decryption scheme for multiple images has been proposed based on the chaotic system and simple circular row and column shifting operations. Two streams generated by the chaotic map have been used for the selection of row or column and the amount by which the selected row or column is to be circularly shifted. This shifting addressed the requirement of confusion in the image encryption technology.
The third stream generated by the chaotic map has been used to embed the diffusion effects in the scheme. To achieve the plaintext sensitivity in the proposed scheme, SHA-384 hash codes have been used. Apart from that, a 384-bit salt key has been used to enhance the security effects. Sixteen color images have been used to demonstrate the practical utility and effectiveness of the proposed framework. Besides, six different sizes have been taken to show the capability of the proposed cipher to handle the different sizes for the big image. The comprehensive security analyses vividly portray the effectiveness, efficiency and practicability of the proposed scheme. EHAB MAHMOUD MOHAMED (Member, IEEE) received the B.E. and M.E. degrees in electrical engineering from South Valley University, Egypt, in 2001 and 2006, respectively, and the Ph.D. degree in information science and electrical engineering from Kyushu University, Japan, in 2012. From 2013 to 2016, he was a Specially Appointed Researcher with Osaka University, Japan. Since 2017, he has been an Associate Professor with Aswan University, Egypt. Since 2019, he has been an Associate Professor with Prince Sattam Bin Abdulaziz University, Saudi Arabia. His current research interests include 5G, B5G, and 6G networks, cognitive radio networks, millimeter-wave transmissions, Li-Fi technology, MIMO systems, and machine learning applications in wireless communications. He is a technical committee member in many international conferences and a Reviewer in many international conferences, journals, and transactions. He is the General Chair of the IEEE ITEMS'16 and the IEEE ISWC' 18. VOLUME 8, 2020