A novel multi-image cryptosystem based on weighted plain images and using combined chaotic maps

- This paper introduces a new multi-image cryptosystem based on modified Henon map and nonlinear combination of chaotic seed maps. Based on the degree of correlation between the adjacent pixels of the plain image, a unique weight is assigned to the plain image. First, the coordinates of plain images are disrupted by modified Henon map as confusion phase. In the first step of diffusion phase, the pixels content of images are changed separately by XOR operation between confused images and matrices with suitable nonlinear combination of seed maps sequences. These combination of seed maps are selected depending on the weight of plain images as well as bifurcation properties of mentioned chaotic maps. After concatenating the matrices obtained from the first step of diffusion phase, the bitwise XOR operation is applied between newly developed matrix and the other produced matrix from the chaotic sequences of the Logistic-Tent-Sine hybrid system, as second step of diffusion phase. The encrypted image is obtained after applying shift and exchange operations. The results of the implementation using graphs and histograms show that the proposed scheme, compared to some existing methods, can effectively resist common attacks and can be used as a secure method for encrypting digital images.


Introduction
Multimedia data security including digital images, is one of the most basic requirements of computer and telecommunication networks [1].Among the various protection methods, image encryption is one of the most effective and common methods for data protection [2].Although traditional cryptographic methods such as AES, DES, and RSA are also used for image encryption, image data has some features that make these cryptographic methods inefficient, especially if a large number of images are to be sent, such as videos [3].Some of the unique features of the image are: the volume of image data is very large compared to textual data [4], there is a high correlation between the pixel content of the image, the volume of duplicate data in an image is larger than the text [5], and real-time image data transmission should be maintained due to the high volume of data [6].
In recent years, various methods have been proposed for effective image encryption.A new design for image encryption was presented in [7], in the first step, the original image is confused with two keys obtained from a pseudo random Lorenz sequence.In the second step, by performing rotation operations on the confused matrix, the diffusion operation is performed and the OR operation is performed to decode the matrix with a binary key and the encrypted image is obtained.In order to encrypt a complex image, Zhang et al. in [8] proposed a chaotic system with rotation operations simultaneously to DNA sequences.The diffusion of pixels is done through DNA code, and the selection of coding rules is done with the decimal string generated by Chen's chaotic system.To increase the complexity and security of cryptography, the Lorenz system is used to weaken the correlation power between adjacent pixels of the image.The diffusion effect is amplified by rotational operations for DNA sequences, and the XOR operator is performed between the decoding matrix and the binary matrix with the chaotic system.For simultaneous encryption of RGB components of a color image, a new encryption method based on logistic map was introduced in [9].In this method, the randomness of the chaotic sequences produced by Logistic map was not appropriate.Hao li et al. proposed a new scheme based on precision limited chaotic system in [10].They introduced a novel precision limited piecewise linear and Logistic chaotic map, and also they proposed an image encryption system which is robust against statistical attacks, differential attacks and brute force attacks.Jian wang et al. proposed an adaptive image encryption based on lifting wavelet optimization in [11].In their proposed method, based on the adaptive wavelet, the improved wavelet optimization scheme based on improved threshold method and particle swarm optimization algorithm was proposed to improve the evaluating parameters.A novel encryption algorithm for quantum images based on chaotic maps was proposed in [12].In this method, the image is converted into a scrambled state using quantum circuits using the principle of quantum Hilbert image scrambling algorithm.The confused image is encrypted using the quantum XOR using the chaotic maps algorithm.
A nonlinear optical multi-image encryption scheme based on chaotic system and two-dimensional linear canonical transform was proposed in [13].In this method, low-frequency sub-bands of the four grayscale images are extracted with the contourlet transform, and the phase truncation and the bitwise XOR operation, as nonlinear processes, improve the robustness of the presented multi-image encryption scheme against chosen-plaintext attack.A new image encryption scheme based on hybrid chaotic maps was proposed in [14].In the proposed method, the confusion phase is based on Arnold's two-dimensional map and the diffusion operation is performed using hybrid chaotic maps.The composition of the chaotic maps is selected based on the correlation coefficient of the original image as well as the characteristics of the bifurcation diagram of the combined chaotic maps.Bhaskar Mondal et al. proposed an image encryption scheme based on a novel two-dimensional sine-cosine cross-chaotic map in [15].The proposed chaotic map generates two pseudo-random sequence which are used in permutation, and diffusion phase.The confusion layer is implemented by shuffling the image pixels, and the diffusion layer is designed by bitwise XOR operation.In [16], an image encryption algorithm based on parallel permutation-and-diffusion scheme was proposed.In the parallel row step, firstly, permuting each row, then performing row forward diffusion and row reverse diffusion on each row, are implemented.In the parallel column step, firstly, permuting each column, and then conducting column forward diffusion and column reverse diffusion to obtain the cipher image, are implemented.Yang et al. introduced a new algorithm in [17], based on fractional order hyper chaotic system and DNA approach for double image encryption.The proposed algorithm provides a variety of DNA coding and operation modes.Chaotic sequences are used to control the coding and operation mode in order to improve the complexity of the encryption process.Fractional order and initial values of fractional order hyper-chaotic system are used as the key of the proposed algorithm.A pseudo-random number generator based on multiple chaotic maps (i.e.Logistic map, Sine map and Chebyshev map) was proposed in [18].Subsequently, a pseudo-random sequence, whose size is the same length as plain image, was produced by the generator.The index sequence obtained by sorting the pseudo-random sequence is used to shuffle the image vector, and the original pseudo-random sequence is used to diffuse the shuffled vector by the addition modulus operations.Then, the diffused vector is circularly left-shifted to get the ciphered image vector.
Chaos has received a lot of attention in recent years for its unique features for image encryption [19,20].Some reasons for researchers to use chaos for image encryption are the randomness of the generated sequences, high sensitivity to the initial conditions, ergodicity, and simplicity of implementation [21,22].Chaotic maps are divided in to one-dimensional and high dimensional categories [23,24,25].Implementation of one-dimensional chaotic maps are simple but not very secure and can be easily hacked [26,27].High dimensional chaotic maps have higher security, but are difficult to implement and have high processing overhead [28,29].
To solve the mentioned problems, in this paper, one-dimensional Logistic, Tent and Sinusoidal chaotic seed maps, are employed with their non-linear combinations.The proposed method consists of two main phases, confusion phase to disrupt pixel coordinates and diffusion phase to change pixels content.The confusion phase is implemented using reversible and two-dimensional modified Henon map to change the coordinates of the pixels, randomly.The diffusion phase is implemented using the XOR operation between the pixels of the plain image and the chaotic sequences produced from the appropriate combination of seed maps with shift and exchange operations.In the proposed method, two plain images are encrypted and sent simultaneously.For each image, a weight is assigned according to the correlation rate, and a proper combination of chaotic maps is selected according to the weight of each images.Low-correlation images are encrypted with simpler maps, and vice versa.
This paper is organized as follows: Section 2 presents a brief survey on Logistic, Tent, Sine, and Henon chaotic maps.Section 3 gives complete description and discussion of the proposed cryptographic method which is followed by experimental verification results in Section 4. Finally, section 5 and Section 6 discusses the conclusion and future scope of the work respectively.

Preliminaries
In this section, modified Henon map, three simple chaotic maps (Logistic, Tent, and Sine maps), and nonlinear combination of these simple maps are described briefly.We call these simple maps as seed maps.
Where  and  are the Henon map parameters.The behavior of the Henon map depends on the value of  and b.For a ϵ (0.54, 2), and b ϵ (0, 1), it shows chaotic behavior [30,31].Due to the fact that this paper uses Henon map for confusion phase and changing the coordinates of the image pixels, so for the N×N image:   ,   ,  +1 ,  +1 , … ϵ{0,1,2,…,N-1}.Therefore, in order to keep the coordinates of the pixels in this range, we use the modified Henon map according to equation 2.
Where f is the number of floating points, and N is the size of N×N image.

Logistic Map
Logistic map is a one-dimensional and discrete chaotic map.This map is one of the simplest chaotic maps, and is mathematically defined as follows [14]: Where  is the control parameter also known as bifurcation parameter.According to the bifurcation diagram of Logistic map, when  is in the range of [0.9, 1], the map will behave chaotically [30]. 0 ϵ (0, 1) is initial value for   .

Tent Map
Tent map is another one-dimensional chaotic map that is used in many applications [30].This map is also known as triangular map.The definition of Tent map [30] is presented as equation 4.
Where  is the control parameter and according to the bifurcation diagram of Tent map, if  ϵ [0.5, 1], then the map will have chaotic behavior [30]. 0 ϵ (0, 1) is the initial value for   .

Sine Map
Sine map is one of the most widely used one-dimensional chaotic maps [14], that can produce complex chaotic sequences with a range of [0, 1].The definition of Sine map is: Where  is the control parameter and according to the bifurcation diagram [14],  ϵ [0.87, 1] and  0 is the initial value for   .

Combined Chaotic Maps
To increase the unpredictability of chaotic sequences, we use seven nonlinear combinations of simple Logistic, Tent, and Sine maps, shown in Table 1.These nonlinear systems are used to generate complex chaotic sequences.
The cascade operator is applied to seed maps, which improves complexity level of the chaotic structure, and the mod operation ensures the output is restricted to [0, 1].
In this paper, in order to quantify all the produced chaotic sequences between 0 and 255, except for Henon map, equation 13 is employed.
Where Vi is the generated chaotic sequence, f is the number of floating point, and Ui is the chaotic sequence between 0 and 255.

Proposed Scheme
Figure 1 shows the block diagram of the proposed scheme.For simplicity, the proposed method is implemented for two N×N gray images, but it can be generalized to more than two color images of any size.The introduced method has been structured in such a way that it can be used in parallel in multiprocessor systems.This method uses two basic parts for cryptography: confusion and diffusion.The confusion operation is used to disrupt the pixel coordinates, and the diffusion operation is used to change the content of the pixels.One of the main goals in image encryption is to keep the correlation between the pixels in the encrypted image as small as possible and close to zero.In our scheme, to enhance efficiency, we proposed that a certain weight be assigned to the input images according to the degree of correlation of the adjacent pixels, and these weights be effective in the cryptographic process.Plain images with smaller correlation coefficients close to zero, should be lighter encoded and have lower weights.So different images are encrypted in different ways.Low-correlation images are encrypted with simpler maps, and vice versa.
The proposed algorithm is summarized in 7 steps as follows.
Step 1: The absolute value of correlation coefficient for 2000 random horizontally, vertically and diagonally adjacent pixels for two images I1, I2 with a size of N×N are calculated separately, and are called r1 and r2, respectively.
Step 2: The coordinates of the pixels of the two plain images I1, I2 are changed separately, using the modified Henon map using equation 1, so the H1 and H2 matrices of size N×N are generated.
Step 3: Using the correlation coefficients calculated in step 1 (r1, r2), the appropriate weight (w) for each image is assigned according to Table 2, and also the proper combination of chaotic maps are selected to produce the appropriate chaotic sequences, and formation of C1, C2 combined chaotic matrices of size N×N.These weights are selected based on bifurcation diagram of seed and hybrid chaotic maps, and also the correlation rate of plain images.
After quantizing the values of C1 and C2 matrices at [0, 255] by equation 13, bitwise XOR operation are performed between H1 and C1, and also between H1 and C2 separately.The matrices of I'1 and I'2 are produced as equations 14, 15.
Step 4: In this step, I'1 and I'2 matrices with size N×N are concatenated, and a new matrix (R) with size 2N×N is produced.
Step 5: Suppose the weight of the first image is w1 and the weight of the second image is w2.The value of t is calculated by equation 16.
For all 2N×N components of 8-bit grayscale pixels in the R, the left circular shift operation is performed for t time, and matrix S is produced.

Fig. 1 Block diagram of the proposed method
Step 6: For all 2N×N components of the S matrix, the contents of the bits in the ⌈ Since all operations performed for encryption are reversible, so the decryption operation is similar to the encryption operation that must be performed in reverse.

Theoretical analysis
To prove the effectiveness of the proposed method, and to prove why we used Table 2 for different correlation coefficients, Lyapunov's exponent (LE) analysis is employed.Maps with a larger positive Lyapunov exponent have a larger chaotic range [14,31,32].Here we prove the efficiency of the proposed method for ST (combined of Sine and Tent maps), and there is a similar proof for other combined maps.We represnt the Lyapunov exponent value of Sine map with  () , and Tent map with  () , and the Lyapunov exponent value of the combined ST map with  () .Let  0 ,  0 are two initial values with small difference. 1 ,  1 are the next iteration of  0 , and  0 .(), () are Sine and Tent maps respectively.We have: After n iteration we have: = () +  ()  We know the larger value of a positive LE has better chaotic performance, so  () and  () must be positive numbers, and it is clear that  () ≥  () and  () ≥  () .

Motivation
One of the goals of image encryption is to make the correlation between adjacent pixels in the encrypted image as small as possible and close to zero.Therefore, as an motivation, if the plain image has a small correlation coefficient, it should be encrypted with simpler methods.As an innovation, we proposed to assign a weight to the plain images.We assigned less weight to low-correlation images and more weight to high-correlation images.In addition, these weights are used in the cryptographic process and are effective in the cryptographic process.Therefore, images with different correlations are encrypted in different ways.This innovation makes the encryption method unpredictable and images with low correlation have less processing time and vice versa.Furthermore, due to the simultaneous and parallel encryption of two images, the performance increases.

Performance analysis and discussion
Evaluation measures are necessary to confirm the effectiveness of image encryption technique.In this section, the various characteristics of an image encryption algorithm are explored using these parameters.

Differential attack
Two evaluation parameters are used to show the effectiveness of the cryptographic method against differential attacks [32]: Number of Pixel Change Rate (NPCR) and Unified Average Changing Intensity (UACI).Differential attack shows the sensitivity of the cryptographic method to small changes in the original image.High sensitivity indicates that if a small change, such as a change of only one least significant bit in the original image occurs, different encrypted images are obtained [33].Table 3 shows the values obtained for these two evaluation parameters for standard images.In Table 4, the NPCR and UACI values of the proposed method are compared with some of the previous methods.This evaluation parameter indicates the percentage difference between the original encrypted image and the original encrypted image with only one bit difference [34,35].The maximum value of NPCR is 100%, and the values closer the maximum value indicate that the cryptographic method is more resistant to differential attacks [36,37].The value of NPCR is calculated using equation 17.
P1 is the original encrypted image and P2 is the original encrypted image with only one bit difference.M is the width of the image and N is the height of the image.

Unified Average Changing Intensity (UACI)
After creating a bit difference in the original image, two images are encrypted.UACI shows the average intensity of difference between two encrypted images [38,39].The value of the parameter is calculated by equation 19.
High values is suitable for this evaluation parameter, because it indicates that the average difference between two encrypted images is large and the cryptographic method is resistant to differential attacks [40].

Statistical analysis
One of the main goals of image encryption is to reduce the correlation between adjacent pixels as much as possible and also to make the histogram of the encrypted image uniform [41,42].In statistical analysis of a cryptographic method, the uniformity of the histogram as well as the value of the correlation coefficient between adjacent pixels in the encrypted image are examined [43,44].

Histogram analysis
The image histogram shows the number of pixels that have the same gray scale level [45].For 8-bit gray scale images, the horizontal axis of the histogram represents the gray levels from 0 to 255, and the vertical axis represents the number of pixels that have specific gray levels.The plain images usually does not have a uniform histogram, and one of the important purposes of cryptography is to make the encrypted image histogram uniform, so that there are approximately equal numbers of pixels on each gray scale level [46,47].The uniformity of encrypted histograms, makes difficult for the attacker to guess the true gray levels of the plain image.Fig. 2, Fig. 3, Fig. 4, and Fig. 5 shows the non-uniformity of the plain images histogram, as well as the uniformity of the encrypted images.

Correlation coefficient
In the original images, usually there is a high correlation between adjacent pixels in three directions: horizontal, vertical and diagonal [48,49].High correlation makes it easy for the attackers to guess the pixel values of the original image [50].Therefore, another goal of an effective encryption method is to reduce the correlation coefficient between adjacent pixels in the encrypted image as close as possible to zero [51,52].Using equation 23, the correlation coefficient can be calculated., and the value of an encrypted image should be near to 0 [53,54].To test the correlation, we randomly select 3,000 pixels adjacent horizontally, vertically, and diagonally from the images, and use equation 23 to calculate the value of the correlation coefficient.When the histogram of the encrypted image is uniform, the correlation between adjacent pixels also decreases dramatically (Fig. 6).Table 5 shows the results of the correlation coefficients of the original images and the encrypted images using the proposed scheme.Table 6 also compares the results of the proposed method with some of the recently introduced methods.[14] 0.0054 0.0049 0.0042 Ref. [55] 0.0026 0.0051 0.0264 Ref. [56] 0.0016 0.0020 0.0014 Ref. [57] 0.0001 0.0015 0.0013 Ref. [58] 0.0012 0.0003 0.0010

Information entropy
Information entropy is another criterion for evaluating the performance of a cryptographic method [55,56,57].For 8-bit gray images, the information entropy value is in the range of 0 and 8, and the optimal value for this parameter is 8.If all 256 gray levels of the image have the same probability (P(si)= ) in all image pixels, we will reach the maximum value.The information entropy value of an image is calculated using equation 24.
() = − ∑ ((  ) 2 ((  255 =0 )), Where the P(si) indicates the existence probability of a si gray level in the image.It is clear that if the histogram of the encrypted image is more uniform, then the entropy value of the encrypted image is closer to the ideal value of 8. Table 7 shows the results for some of the standard images, and Table 8 compares the proposed method with some of the previous methods.

Security Key Space and key sensitivity
One of the most important parameters for evaluating the performance of a cryptographic method is the secret key space and the sensitivity of the keys to minor changes [58,59,60].The larger secret key space and the higher sensitivity, results more resistance of the cryptographic method to kinds of attacks.An effective encryption scheme should have a key space more than 2 100 to resist the brute force attacks [61].For example, if the key space of a cryptographic method is 10 3 , in the worst case an attacker must test 10 3 different keys to get the correct key, while if the key space is 10 10 then the attacker must test more keys and spend more time to find correct key, so it will be actually impossible to find the right key in a reasonable time for enough larger key space.In this paper, the secret keys are: parameters a, b in Henon map, parameters d, d0 in Logistic map, parameters z, z0 in Tent map, parameters w, w0 in Sine map, correlation coefficient of plain images r1, r2, and finally iter is the number of iterations of the cryptographic process.Since the number of secret keys are 11, if the accuracy of the decimal numbers is 14 floating points, then the key space will be 10 154 , which is sufficiently resistant to attacks.
In key sensitivity analysis, for a high performance cryptographic method, two items are considered.First, with a very small change of 10 -14 in the secret key, a completely different encrypted image should be obtained.Second, if the encrypted image is decrypted with an incorrect key that has a very small difference of 10 -14 with the correct key, the original image will not be obtained [62,63].Fig. 7 shows the results of key sensitivity analysis for two secret keys.Parameter a in Henon map is considered as the first secret key Ke1, and parameter b in Henon map is considered as the second secret key Ke2, and the other secret keys are the same.The implementation results show that the original images can only be retrieved and decrypted with only the correct keys.

Noise and cropping attack
An attacker may make noise to destroy the original image, or cut part of the image information that causes the data to be lost [64].An effective cryptographic method must be able to withstand noise and cropping attacks.Usually, to evaluate the cryptographic method against cropping attacks, sizes of the encrypted image are cropped, then the cropped image is decrypted, and finally the quality of the decrypted image is evaluated compared to the plain image.To evaluate the cryptographic method against noise attacks, we add 5% salt and pepper noises to the encrypted image and then the noisy image is decrypted, and its quality is evaluated compared to the original image.Figures 8,  9, 10 shows the results of different cropping attacks.To evaluate the quality of noisy and cropped images compared to the original image, four parameters: Peak Signal to Noise Ratio (PSNR), Signal to Distortion Ratio (SDR), Signal to Noise Ratio (SNR) and Structural Similarity index (SSIM), have been introduced [64], which are explained in the following sections.9, demonstrate the result of evaluating parameters for different iterations and    10 show the results of evaluating parameters for different iterations and 25% cropping attack.11 illustrate the results of evaluating parameters for different iterations and 50% cropping attack.Implementation results show that, higher value of cropping attacks, causes lower evaluating parameters, and in some iterations, some parameters have better values.In Table 11 it is clear that the value of PSNR in 1 th iteration is better.Table 12 shows the comparision of PSNR for different cropping attacks in some references and maximum values of proposed method.This parameter is used to evaluate the effect of noise and cropping attack on the encrypted image.Higher value of PSNR is good [64], because it shows that the noise and cropping attack has low effect on the decrypted image [14].The maximum value for PSNR is infinite.PSNR is calculated by equation 25. ( I is plain image and I ' is decrypted image after cropping attack, and MSE is Mean Square Error.If I and I' were exactly the same, then the value of PSNR value will be infinite.

Signal to Distortion Ratio (SDR)
Another parameter for evaluating the quality of a decoded image after a cropping attack is SDR.Higher value of SDR is suitable [14], because it indicates that the cropping attack has low effect on the decrypted image [64].The maximum value for SDR is infinite.The SDR in db (decibel) is calculated as equation 27.I is plain image, and I ' is decrypted image after cropping attack.If I=I', then SDR will be infinite.

Signal to Noise Ratio (SNR)
This parameter is also used to evaluate the effect of noise and cropping attacks [64].The amount of SNR is in the range of 1 and ∞.High values are suitable for SNR because it indicates that noise and cut have little effect on the encrypted image.If the two original and decoded images after cutting and noise attacks are the same, the value of SNR is infinite.The SNR value is calculated using equation 28.Where O is the original image, and L is decrypted image after noise and cropping attacks.

The Structural Similarity (SSIM) index
This parameter is used to measure the similarity between two images [64].In this research, we use SSIM index between plain image and decrypted image after cropping attack.The resultant SSIM index is a decimal value between -1 and 1.If two images are exactly the same, the value of this parameter is equal to 1. SSIM is defined as equation 29. ( µx is the average of x, µy is the average of y, σ  2 is the variance of x, σ y 2 is the variance of y, σ  is the covariance of x and y, C1=(k1L) 2 , C2=(k2L) 2 , L the dynamic range of the pixel values, k1=0.01 and k2=0.03.

Execution time
Another important factor in evaluating the performance of an encryption method is the processing time for encrypting an image in one round.In order to show the speed performance, implementation results are given in this section.The test was performed via the MATLAB R2016b on a computer running 64-bit Windows 10, the Intel (R) Core (TM) i7-6700 @ 3.60 GHz CPU, and 8 GB of RAM.All 26 images in the USC SIPI image Database (http://sipi.usc.edu/database) are selected as experimental images.Table 13 shows the average processing time for each image sizes and various correlation coefficients.It is clear that plain images with less correlation, have less processing time and vice versa.Table 14 compares the average execution time for the proposed method with some references.

Conclusions
This paper introduces a new method for encrypting digital gray scale multi-images.The confusion process is implemented using the modified Henon map, and the diffusion process is implemented using two steps XOR operation, also shift and exchange operations.Initially, the coordinates of two images are disrupted by the modified Henon map, and the confusion process is implemented.
As the first step of diffusion process, bitwise XOR operations are applied between the content of the confused images in confusion stage, and proper combined chaotic sequences.These chaotic sequences are selected according to the correlation values of the original images as well as the characteristics of the bifurcation diagram of the Logistic, Tent, and Sine chaotic maps.Depending on the correlation value of the original images, one weight is given to each image.Images with less correlation have less weights and vice versa, and lighter images are encrypted with simpler chaotic maps and have less processing time.After the first stage of the diffusion process, two achieved matrix are concatenating together to form an expanded matrix.Due to the weight of the images, shift and exchange operations are applied as the second and third steps of the diffusion process respectively.Finally, as the fourth step of the diffusion process, the second bitwise XOR operations are applied between the matrix obtained from the previous stages and combined chaotic sequences obtained from the LTS (Logistic-Tent-Sine) system with proper initial values, and the encrypted matrix is obtained.
According to the implementation results, processing time is proper, the information entropy value is close to the maximum value 8, and the correlation between adjacent pixels in encrypted images is close to zero.The histogram of the encrypted images are also uniform, and the proposed method is resistant to ordinary attacks, so it can be used as a secure method for encrypting digital images.

Step 7 :
swapped together and matrix P is obtained.In this step, a new 2N×N matrix Z containing chaotic sequences generated by the nonlinear combination of three Logistic, Tent and Sine maps (LTS system) is achieved.After quantizing the values of matrix Z in the range [0, 255] by equation 13, the bitwise XOR operation between P and the new 2N×N matrix Z is performed and final encrypted matrix E is produced.

Fig. 3
Fig. 3 First column (a), (d) are two 256×256 plain images, second column (b), (e) are decrypted images respectively, (c) is the 512×256 two plain image single encrypted image, (f) is the uniform histogram of double encrypted images

Fig. 7
Fig. 7 (a) and (b) are 256×256 plain images, (c) is the 512×256 encrypted image of two plain images with secret key Ke1=0.78,(d) is the 512×256 encrypted image of two plain images with secret key Ke2=0.83,(e) is the 512×256 decrypted image of two plain images with secret key Ke3=0.78+10 -1 , (f) is the 512×256 decrypted image of two plain images with secret key Ke4=0.83+10 -1 , (g) is the 512×256 decrypted image of two plain images with secret key Ke5=0.93,(h) is the 512×256 decrypted image of two plain images with secret key Ke4=0.63,(i) is the decrypted image of plain image (a) with correct secret key Ke1=0.78,(j) is the decrypted image of plain image (b) with correct secret key Ke2=0.83

Fig. 8
Fig. 8 (a) and (d) are 256×256 plain images, (b) is the decrypted image for (a) after 1 16 cropping attack in one round, (e) is the

Fig. 11 (
Fig. 11 (a) and (c) are 256×256 plain images, (b) is the decrypted image for (a) after 5% salt and pepper attack, (d) is the decrypted image for (c) after 5% salt and pepper attack Fig. 11 shows the results after 5% salt and pepper attack.

Table 2
Assigned weights to images with different correlations, and corresponding chaotic maps

Table 3
NPCR and UACI tests results for cipher standard 256×256 grayscale images

Table 4
Comparison of the NPCR and UACI values between our proposed approach and the other methods

Table 5
Correlation test results for some 256×256 standard images

Table 6
Comparison of encrypted image spatial correlation, and global entropy with some recent image encryption schemes

Table 7
Information entropy test results for standard images

Table 8
Comparison of the entropy value between proposed scheme and other methods

Table 9
Encryption evaluating parameters for

Table 10
Encryption evaluating parameters for 25% cropping attack and different iterations

Table 11
Evaluating parameters for different iterations and 50% cropping attack

Table 12
Comparision of PSNR in 1 round for different cropping attacks in some references and proposed method

Table 13
Average processing time for each image size and various correlation coefficients

Table 14
Comparison between the average execution time for propsed method and some other methods (ms)