Image encryption based on Fisher‐Yates shuffling and three dimensional chaotic economic map

IET Image ProcessingVolume 13, Issue 12 p. 2086-2097 Research ArticleFree Access Image encryption based on Fisher-Yates shuffling and three dimensional chaotic economic map Abdelrahman Karawia, Corresponding Author abibka@mans.edu.eg Computer Science Unit, Deanship of Educational Services, Qassim University, Buraidah, 51452 Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516 EgyptSearch for more papers by this author Abdelrahman Karawia, Corresponding Author abibka@mans.edu.eg Computer Science Unit, Deanship of Educational Services, Qassim University, Buraidah, 51452 Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516 EgyptSearch for more papers by this author First published: 17 September 2019 https://doi.org/10.1049/iet-ipr.2018.5142Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Recently, few researchers investigated algorithms of image encryption using different chaotic economic maps (CEMs). However, the authors investigated the effect of these maps on the encryption of the plain image. In the current study, an image encryption algorithm via Fisher–Yates shuffling (FYS) combined with a three-dimensional (3D) CEM is given. FYS is used to generate the random permutation of a finite sequence. First, it is used to shuffle the rows and the columns of the plain image. Second, the 3DCEM is used in the substitution stage to confuse the pixels of the shuffling image. The proposed algorithm is applied to several types of images. Many measurements are performed to check the security and performance of the proposed algorithm. In addition, numerical simulations and experimental results have been implemented to verify that the proposed algorithm can resist different attack types. 1 Introduction In present time, transmitting image using different modern communication methods is necessary for various fields such as military image transmission etc. At the same time, it must be secured. It is, therefore, image encryption algorithms that are displayed to solve this problem with high security. In the beginning, a traditional algorithm called data encryption standard (DES) was used for image encryption [1]. However, its efficiency is reduced when large images encryption is needed [2, 3]. So the authors searched for new algorithms and used the theory of chaos. It is a great theory that has complex features such as unpredictability, bifurcation and sensitivity to the initial conditions. Chaos has introduced some famous maps which have some complex properties that assist in providing an efficient encryption algorithm. Logistic map, Lorenz attractor, Henon map and many chaotic economic systems [4, 5] are examples of these maps. Image encryption algorithms designed using chaotic maps have shown an efficient way to solve the problem of image transmission via modern communications with highly secured and fast methods. Matthews [6] published his first image encryption algorithm in 1989 that was based on some chaotic characteristics. Many authors have developed image encryption algorithms using chaotic maps [4, 5, 7-12]. For example, Askar et al. [4] proposed an algorithm of image encryption using one-dimensional (1D) chaotic map. A 2D chaotic map has been presented in [5]. In [7], a multi-chaotic system has been introduced into the process of image encryption. According to architectural principles, the image encryption algorithms are divided into three different classes: permutation–substitution, permutation-only and substitution-only. In [13], Fridrich was the first who presented the permutation–substitution. It is the famous architecture of image encryption [14-17]. According to this structure, to decrease the strong correlation between pixels, pixel positions are first shuffled in the process of permutation. After that, pixel's values are modified by the substitution process to obtain the breakdown effect. These steps could be repeated many times to reach a good security level. For instance, an algorithm ideal of using permutation–substitution structure and two chaotic maps were presented in [4, 5]. In this algorithm, the chaotic logistic map was applied to the permutation step to modify the pixel's positions while the 2D chaotic economic map (CEM) was used in the substitution step. In [18], Liu and Miao presented a new algorithm that depends on the combination of chaotic shuffling and chaotic substitution. An encryption algorithm using the N-phase logistic chaotic sequence was suggested in [19]. Also, the authors used both shuffling and substitution algorithms. In [20], Wang and Liu suggested the encryption algorithm using chaos and DNA encoding rules. A new image encryption algorithm is suggested based on double chaotic maps, secure hash algorithm (SHA)-3 and auto-updating system [21]. Wang et al. in [22] suggested a novel scheme of image encryption via a chaotic system and DNA sequence operations. Block scheme of image encryption using chaotic maps and dynamic random growth method was presented in [23]. In [24], Wang and Luan suggested a novel scheme of image encryption using reversible cellular automata and chaos. Stream-cipher algorithm using one-time keys and chaotic maps is designed in [25]. In [26], Liu and Wang suggested the image encryption algorithm based on a bit-level permutation and high-dimension chaotic map. The novel confusion–diffusion method is presented for image encryption [27]. Wang et al. in [28] presented an image encryption system using the high-dimension Lorenz chaotic and perceptron model within a neural network. The goal of the current paper is to increase the efficiency and security of the image encryption by applying the permutation–substitution structure to 3DCEM. In the permutation stage, Fisher–Yates shuffling (FYS) algorithm is applied to reduce the relationship between the plain pixels. The 3DCEM is used in the substitution stage to increase the security of the image encryption algorithm. The parameters of the 3DCEM have a high effect on the speed of convergence to the equilibrium. Using chaotic systems which are multi-dimensional in the process of encryption has some advantages. For instance, such systems contain several parameters than those with one dimension. This means the number of parameters increase and hence the chaotic attractor of those systems becomes more complex. Therefore, breaking the encryption based on those systems becomes more difficult. On the other hand, the key space of the proposed algorithm is larger than the 3D chaotic cat map [2] and most of the other methods [18-28]. During the substitution step, the exclusive OR (XOR) was applied multiple times between the values of the 3DCEM to avoid the classical attacks. The current paper is organised as follows. In Section 2, the FYS algorithm is given. A brief introduction to 3D chaotic is given in Section 3. In Section 4, the proposed image encryption/decryption algorithm is presented in more details. In Section 5, simulation results are given. Security analysis and some comparisons with the algorithms in the literature are presented in Section 6. In Section 7, the conclusion is introduced. 2 FYS algorithm To reduce the relationship between the plain pixels and to enhance the efficiency and security of the image cipher (Fig. 1). Fig. 1Open in figure viewerPowerPoint Algorithm 1: FYS algorithm Modern FYS algorithm will be used to generate a random permutation of numbers for the rows and columns (a finite set). It is illustrated as in Algorithm 1. The procedure used in the shuffle follows the following features: (i) FYS is unbiased (every permutation is equally likely). (ii) FYS needs solely time proportional to the items number being shuffled and no further storage space. The old version of FYS has time complexity. On the other hand, the modern FYS (Algorithm 1) is more efficient since it has time complexity. In this algorithm, the rolled number refers the position of the number extracted from the existing list and placed in the shuffled list and number at the end of the existing list is placed at the same position that produces more shuffling at each iteration. So, the system will be more dynamic. When the list includes one number, it is extracted and placed in the shuffled list. In the current algorithm, the logistic map is used to generate a random number x. 3 3D chaotic economic map In [29], Ding et al. studied the following dynamical system: (1)Such systems own many parameters influencing the behaviours of these systems and are appropriate for the procedures of encryption. The security keys of the encryption procedures consist of those parameters. The important feature of such systems is the chaotic behaviour for which encryption algorithms are considered. On the other hand, Lyapunov exponent is one of the most important dynamical features for the chaotic system. It can to a great extent express the chaotic maps performance. It is utilised as a quantitative measure for the sensitive reliance on initial conditions. If Lyapunov exponent has a negative value, the system will not be chaotic. If it has zero, the system will be neutrally stable. If Lyapunov exponent has a positive value, the system will be chaotic and will be sensitive to initial conditions. In this paper, the 3DCEMs (1) is applied to the proposed algorithm. System (1) displays the chaotic behaviour of an economic map with nine distinct parameters. In economy, those parameters have significant meaning. The parameter is assigned to catch the size of the economic market , where are the marginal costs and w is the weighting coefficient such as while and are constant values for the speed of adjustment. Fig. 2 shows the chaotic behaviour of the map (1) at , , , , , , , and . Moreover, Fig. 3 displays the Lyapunov exponent of the 3D chaotic maps (1) at and . It is clearly evident that when all Lyapunov exponents are ≤0. The Lyapunov exponents are >0 when . So, the system is chaotic in this interval. In [29], Ding et al. discussed the other dynamical features. Fig. 2Open in figure viewerPowerPoint Chaotic behaviour of the map (1)at and Fig. 3Open in figure viewerPowerPoint Lyapunov exponent of the 3D chaotic map (1)at and 4 Encryption algorithm via FYS-3DCEM One of the key requirements of an encryption process is the shuffling of pixels in the plain image to reduce the inherently strong correlations which exist. The pixel shuffling algorithm is applied to the plain image. So, FYS algorithm will apply twice, one for shuffling the rows and the second for shuffling the columns. Now, suppose that the plain image has the dimension and is the array of its pixel positions. The proposed image encryption algorithm is given as follows (Figs. 4 and 5). Fig. 4Open in figure viewerPowerPoint Algorithm 2: image encryption Fig. 5Open in figure viewerPowerPoint Algorithm 3: image decryption In the image decryption, the same chaotic economic sequences are used on encrypted images to recover original images and use the inverse steps of Algorithm 1. Also, the diagram of E-FYS-3DCEM scheme is displayed in Fig. 6. Fig. 6Open in figure viewerPowerPoint Diagram of E-FYS-3DCEM scheme The proposed image decryption algorithm will be referred to as D-FYS-3DCEM algorithm. 5 Simulation results To show efficiency and robustness of the E-FYS-3DCEM algorithm, ten images of different types are used; four images with , five images with and one image with . The histograms of the plain images are shown in Figs. 7b–16b. Fig. 7Open in figure viewerPowerPoint Histograms of the plain images (a) Lena , (b) Histogram of lena , (c) Cipher image of lena , (d) Histogram of cipher image at and The security keys of our encryption are chosen as follows. In the permutation step, logistic map with , for rows shuffling and , for columns shuffling. In the substitution step, 3DCEM (1) with and . Experiments are implemented under MATLAB R2016a running program in a laptop with the following features: Intel (R) Core (TM) i7-4700MQ 2.40 GHz, 12.0 GB memory and 1.0 TB capacity of hard disc. 6 Performance and security analysis An acceptable encryption algorithm has the following characteristics [5]: approximately uniform histogram; large enough key space; approximately no correlations between neighbouring pixels; sensitive to the proposed protected key; acceptable information entropy; and high resistance for statistical and differential attacks. All these characteristics are tested for the E-FYS-3DCEM algorithm. 6.1 Statistical analysis Plain image must have no statistical similarity with cipher image to face most attacks. 6.1.1 Histogram analysis The distribution of pixel's values of an image has been shown via histogram analysis. The cipher image histogram should be uniform or approximately uniform. In this case, obtaining statistical information will be difficult. Figs. 7c and d–16c and d have displayed cipher images and their corresponding histograms. On the basis of the results of Figs. 7-16, the plain image histograms and the cipher image histograms are completely different. Indeed, the cipher image histograms are uniform or approximately uniform. In [30], Kwok and Tang presented the Chi-square test to check the uniformity of the cipher image histograms as follows: (2)where the grey-level number=256; refers to the observed occurrence frequencies of and refers to the expected occurrence frequencies of , . By choosing as a significance level, Table 1 displays the values based on the output cipher images of the E-FYS-3DCEM algorithm. Additionally, and all the calculated values in Table 1 are less than that. So, the cipher image histograms have a uniform distribution. Table 1. values of the plain and the cipher images at and Plain image Cipher image Lena 227.0781 barbara 251.2031 moon_surface 242.5781 resolution_chart 266.7266 Fig. 8Open in figure viewerPowerPoint Histograms of the plain images (a) Aeroplane , (b) Histogram of aeroplane , (c) Cipher image of aeroplane , (d) Histogram of cipher image at and Fig. 9Open in figure viewerPowerPoint Histograms of the plain images (a) Baboon , (b) Histogram of baboon , (c) Cipher image of baboon , (d) Histogram of cipher image at and Fig. 10Open in figure viewerPowerPoint Histograms of the plain images (a) Boat , (b) Histogram of boat , (c) Cipher image of boat , (d) Histogram of cipher image at and Fig. 11Open in figure viewerPowerPoint Histograms of the plain images (a) Lena , (b) Histogram of lena , (c) Cipher image of lena , (d) Histogram of cipher image at and Fig. 12Open in figure viewerPowerPoint Histograms of the plain images (a) Barbara , (b) Histogram of barbara , (c) Cipher image of barbara , (d) Histogram of cipher image at and Fig. 13Open in figure viewerPowerPoint Histograms of the plain images (a) Moon surface , (b) Histogram of moon surface , (c) Cipher image of moon surface , (d) Histogram of cipher image at and Fig. 14Open in figure viewerPowerPoint Histograms of the plain images (a) Resolution chart , (b) Histogram of resolution chart , (c) Cipher image of a resolution chart , (d) Histogram of cipher image at and Fig. 15Open in figure viewerPowerPoint Histograms of the plain images (a) House , (b) Histogram of house , (c) Cipher image of house , (d) Histogram of cipher image at and 6.1.2 Information entropy analysis To show grey values distribution, the information entropy is used. The entropy H has the following formula [31]: (3)where the source symbol is X; is the probability of symbol ; and H appeared in bits. The theoretical value of entropy H is 8. Table 2 displayed the information entropy of the cipher images based on E-FYS-3DCEM algorithm. The calculated values of Table 2 are close to the exact value 8. On the other hand, the statistical test for the block entropy is presented in [32] and both qualitative and quantitative results are introduced. In reality, 100 non-interlocked blocks of size are chosen haphazardly from the cipher image. The information entropy for each block is evaluated based on (3) and the average entropy is recorded. Table 2. Information entropy for cipher images at and Image Global entropy Actual block entropy Theoretical block entropy 7.9882 7.1690 pass pass 7.9972 7.1779 pass pass 7.9992 7.1840 pass pass 7.9969 7.1826 pass pass 7.9993 7.1838 pass pass 7.9993 7.1773 pass pass 7.9979 7.1726 pass pass 7.9994 7.1745 pass pass 7.9975 7.1786 pass pass 7.9974 7.1727 pass pass On the basis of Table 2, one can conclude that the cipher images are random-like after applying the E-FYS-3DCEM algorithm. 6.1.3 Correlation for the neighbouring pixels In the plain image, each pixel has a high correlation with its neighbouring pixels in all directions: horizontal, vertical and diagonal. It can be considered a measurement to compare the high correlations in the plain and the cipher images. This measurement can be executed by selecting 5000 pairs randomly of two neighbouring pixels from the plain and the cipher images. The correlation is computed using the following formula: (4)where and where n and m are the grey values of the two neighbouring pixels in the image. Fig. 17 demonstrates the relationship distribution of two horizontally neighbouring pixels in the original and the cipher images. The relationship of two neighbouring pixels in the original and the cipher images is given in Table 3. Fig. 17 presents those correlations for the plain and the cipher images. The correlation coefficients of the cipher image for the E-FYS-3DCEM algorithm and other algorithms are shown in Table 4. Table 3. Correlation coefficients of two neighbouring pixels of plain image and cipher image at and Image Plain image Cipher image H 0.8906 0.0001 V 0.9518 0.0033 D 0.8512 0.0099 H 0.9351 0.0009 V 0.9692 0.0004 D 0.9181 0.0010 H 0.9659 0.0025 V 0.9837 −0.0016 D 0.9552 −0.0037 H 0.8105 0.0073 V 0.8797 0.0006 D 0.8335 −0.0019 H 0.9566 −0.0009 V 0.9600 0.0003 D 0.9237 0.0022 H 0.9385 −0.0013 V 0.9669 −0.0020 D 0.9225 −0.0009 H 0.9812 −0.0023 V 0.9660 −0.0037 D 0.9402 0.0016 H 0.9121 0.0005 V 0.8634 −0.0004 D 0.8282 0.0013 H 0.8859 0.0005 V 0.9316 −0.0023 D 0.8988 0.0033 H 0.8828 −0.0023 V 0.8793 −0.0026 D 0.7524 0.0027 Table 4. Correlation coefficients of two neighbouring pixels of cipher image for E-FYS-3DCEM algorithm at and and other algorithms Image Cipher image E-FYS-3DCEM algorithm [4] [33] [34] Case I Case II Case III H 0.0009 0.0122 0.0075 −0.0038 0.0205 0.0024 V 0.0004 −0.0456 −0.0079 0.0093 0.0175 −0.0006 D 0.0010 −0.0188 −0.0093 −0.0189 0.0231 0.0012 Fig. 16Open in figure viewerPowerPoint Histograms of the plain images (a) Lena , (b) Histogram of lena , (c) Cipher image of lena , (d) Histogram of cipher image at and Fig. 17Open in figure viewerPowerPoint Correlation analysis of plain images and correlation analysis of cipher images at and (a), (b), (c), (d), (e), (f) 6.2 Sensitivity analysis Acceptable encryption algorithm should be sensitive to the protected key and the plain image. It is, therefore, with little change in the protected key or in the plain image, a totally various cipher image will be produced [35]. 6.2.1 Differential attack The sensitivity to the plain image can be checked by changing 1 px of the plain image. Number of pixels change rate (NPCR) and Unified average changing intensity (UACI) are computed to measure the sensitivity to the plain image by the given equations (5) (6)where (7)The width and height of the two cipher images and are denoted by w and h. In Table 5, 8 bits are used to code each grey level. In addition, NPCR is close to 99.6% and UACI is close to 31.5% for examined images. So, the E-FYS-3DCEM algorithm can resist against differential attacks. Table 5. NPCR and UACI comparisons of Algorithm NPCR, % UACI, % expected value 99.61 33.46 E-FYS-3DCEM algorithm 99.61 31.54 [14] 99.55 28.81 [15] 99.66 28.71 [36] 99.70 29.30 6.2.2 Test of key sensitivity In the current section, the sensitivity to the protected key of the E-FYS-3DCEM algorithm is checked. On the basis of the true protected key, Fig. 18 displays the recovered images of the cipher images for lena, barbara, house and moon surface images. The decrypted images of a little change in a parameter 'a' of the protected key are displayed in Fig. 19. Fig. 20 shows the decrypted images with only change in parameter 'w' of the protected key. Also, a little change in parameter '' leads to the decrypted images in Fig. 21. Therefore, any modicum change in one parameter of the introduced protected key will be given different image and cannot recover the original plain image. So, the E-FYS-3DCEM algorithm owns acceptable security keys. Fig. 18Open in figure viewerPowerPoint Decrypted images of (a) Lena, (b) Barbara, (c) House and, (d) Moon surface at the correctprotected key Fig. 19Open in figure viewerPowerPoint Decrypted images of (a) Lena, (b) Barbara, (c) House and, (d) Moon surface at and Fig. 20Open in figure viewerPowerPoint Decrypted images of (a) Lena, (b) Barbara, (c) House and, (d) Moon surface at and Fig. 21Open in figure viewerPowerPoint Decrypted images of (a) Lena, (b) Barbara, (c) House and, (d) Moon surface at and 6.3 Analysis of the introduced key space The key space must be sufficiently large to accommodate an acceptable encryption algorithm. In [37, 38], Song et al. and Zhu et al. presented the condition to achieve this target, the space of acceptable encryption must be more than . The E-FYS-3DCEM algorithm has a space key that consists of 13 values. Therefore, the assigned key space has a size that equals to if the author uses precision . So, the condition is satisfied since the assigned key space is >2100. Table 6 shows that the key spaces in [35, 39, 40] are less than the introduced key space. So, the E-FYS-3DCEM algorithm is resisting the attacks of brute force. Table 6. Comparison of the introduced key space with other key spaces in the literature Algorithm E-FYS-3DCEM algorithm [39] [35] [40] key space 6.4 Noise attacks In this current section, the E-FYS-3DCEM algorithm has been examined against noise attacks. The cipher image is corrupted with Gaussian and salt and pepper noises. The Gaussian noise is added with variances 0.01 and 0.1. In addition, the salt and pepper noise is added with densities 0.05 and 0.1. Mean-squared error (MSE) and peak signal-to-noise ratio (PSNR) are used to measure the effect of noise attacks on the E-FYS-3DCEM algorithm. Suppose P is referred to as plain image and D is referred to decrypted images. MSE and PSNR are computed by (8)and (9)where the pixel value of the image is denoted by I. The higher PSNR indicates the higher similarity between P and D. Figs. 22-25 and Table 7 display the noise attacks results. The decryption after adding salt and pepper noise gives result better than decryption after adding Gaussian noise since PSNR exceeds 18.5 while the decrypted image histogram is approximately the plain image histogram. Table 7. Results of noise attack Noise MSE MSE PSNR PSNR (E-FYS-3DCEM algorithm) ([39]) (E-FYS-3DCEM algorithm) ([39]) Gaussian noise with variance = 0.01 and mean = 0 2348.4 4410.1 14.4 11.7 Gaussian noise with variance = 0.1 and mean = 0 5272.1 5631.4 10.9 10.6 salt and pepper noise with density 0.05 444.1 869.9 21.7 18.7 salt and pepper noise with density 0.1 909.3 1829.6 18.5 15.5 Fig. 22Open in figure viewerPowerPoint Ciphered images of lena after adding (a) Gaussian noise with variance = 0.01 and mean = 0, (b) Gaussian noise with variance = 0.1and mean = 0, (c) Salt and pepper noise with density0.05 and, (d) Salt and pepper noise with density0.1 Fig. 23Open in figure viewerPowerPoint Histograms of ciphered images of lena after adding (a) Gaussian noise with variance = 0.01 and mean = 0, (b) Gaussian noise with variance = 0.1and mean = 0, (c) Salt and pepper noise with density0.05 and, (d) Salt and pepper noise with density0.1 Fig. 24Open in figure viewerPowerPoint Decrypted images of lena after adding (a) Gaussian noise with variance = 0.01 and mean = 0, (b) Gaussian noise with variance = 0.1and mean = 0, (c) Salt and pepper noise with density0.05 and, (d) Salt and pepper noise with density0.1 at and Fig. 25Open in figure viewerPowerPoint Histograms of decrypted images of lena after adding (a) Gaussian noise with variance = 0.01 and mean = 0, (b) Gaussian noise with variance = 0.1and mean = 0, (c) Salt and pepper noise with density0.05 and, (d) Salt and pepper noise with density0.1 6.5 Classical attacks analysis The famous four classical attack types are ciphertext only, known as plaintext, chosen plaintext and chosen ciphertext (for more details about these types see [41]). In fact, the most powerful attack is the third one (chosen plaintext attack). If the E-FYS-3DCEM algorithm can resist it, the E-FYS-3DCEM algorithm can resist other attacks. The E-FYS-3DCEM algorithm is sensitive to the initial parameters and initial values . Only a small change in one of them leads to a huge difference in and . In this type of attack, the attacker takes the ciphered image and short time for running the encryption machine. So, suppose that the plain image is P and the corresponding ciphered image is C. Let and be the designed image to discover the decimal codes. So, the chosen plaintext attack using Xu et al. algorithm may be illustrated as follows [42]: Step 1: Encrypt the designed image D using the E-FYS-3DCEM algorithm; the ciphered image is denoted by . Step 2: Perform bit-wise-XOR operation between C and to get the recovered plain image. Step 3: Make a comparison between the plain image and recovered plain image in step 2. Fig. 26 indicates that the recovered plain image is completely different from the original plain image and so the E-FYS-3DCEM algorithm can resist the attack of chosen plaintext. Fig. 26Open in figure viewerPowerPoint Chosen plaintext attack (a) Designed image D, (b) Encrypted image of design imageD, (c) Encrypted image C, (d) Recovered imageP 6.6 Contrast analysis Contrast is a quantity used to recognise objects of an image. The acceptable encryption algorithm gives cipher images of a high level of contrast. Since the randomness with a high level is given by the map (1) is high, the cipher image has high-level contrast. The cipher image contrast is calculated to see the efficiency of the map (1) in the applications of image encryption. The contrast is calculated by (10)where the number of grey-level co-occurrence matrices is denoted by . The contrast analysis for various types of images is given in Table 8. Table 9 shows that the contrast analysis of the E-FYS-3DCEM algorithm is better than the algorithms in [4]. Table 8. Contrast of plain and cipher images at and Image Contrast Plain image Cipher image 0.3563 10.5363 0.1741 10.4460 0.9463 10.6790 0.3095 10.4795 0.2071 10.5026 0.6063 10.5053 Table 9. Contrast of the plain [lena ] and the cipher image for the E-FYS-3DCEM algorithm at and and [4] Image Contrast Plain image 0.3563 — cipher image E-FYS-3DCEM algorithm 10.5363 — — [4] case I 10.1655 — — — case II 10.3909 — — — case III 10.3971 For analyses of quantity for each key, the histogram variances are utilised to estimate uniformity of cipher images. The smaller value of variances denotes the higher uniformity of cipher images. The variances of ciphered images which are encrypted by different protected keys on the same plain image are evaluated. If the values of variances are converged when the protected keys are modifying then, the ciphered images have higher uniformity. The variance of histograms is calculated as follows: (11)where Y are the histogram values vector and and is the number of pixels, in which grey value is equal to i. Two variances of histograms are evaluated for two cipher images by (11) for the same plain image with various protected keys. In such various protected keys, one parameter of protected keys is modified only. The plain images of lena , aeroplane and boat are examined for one round encryption. Table 10 records the histogram variances for cipher images. In the first column of Table 10, the variances are evaluated based on the protected key KEY, the variances in other columns are evaluated by only modifying one parameters of and in KEY. The variance values are about 1100, which shows that the average variation of a number of pixels in every grey value is about 33 px. However, the variance value is 446,750 for the histogram of the plain image lena. Moreover, the variance values for the histogram of the cipher image of len