Circulate Matrix and Compression Sensing Based Multi-Level Image Encryption

Circulate Matrix and Compression Sensing Based Multi-Level Image Encryption

Ranjeet Kumar SinghGanesh Gupta Tej Singh Kalka Dubey Anjula Mehto 

Department of Computer Science & Engineering, Madhav Institute of Technology and Science, Gwalior 474005, India

Department of Computer Science & Engineering, Chandigarh University Mohali, Punjab 140413, India

Department of Information Technology, Madhav Institute of Technology & Science, Gwalior 474005, India

Corresponding Author Email: 
ranjeets@mitsgwalior.in
Page: 
853-862
|
DOI: 
https://doi.org/10.18280/ts.390310
Received: 
26 February 2022
|
Revised: 
5 May 2022
|
Accepted: 
16 May 2022
|
Available online: 
30 June 2022
| Citation

© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).

OPEN ACCESS

Abstract: 

Digital data security is a broad research area in the field of science and technology. A lot of research was focused on information security-based mechanism for secure communication. This paper presents a novel image encryption as well as compression based on measurement matrix, pixel exchange and logistic cat map, which includes the permutation, compression, and diffusion processes. Initially the image is divided into four equal sizes of blocks and then each block is transformed into horizontal and vertical low and high frequency band. Then a random matrix multiplication function is applied to achieve an encrypted and scrambling frequency component and apply inverse DWT procedure to get first level of scrambled blocks, and further we apply the second level of security mechanism. Here each adjacent block pixel is exchanged by using the random matrices. For providing the high level of compression we design measurement matrices in compressive sensing by utilizing the circulate matrices and controlling the original column vectors of the circulate matrices with Arnold cat map. With the help of measurement matrix again the blocks are encrypted. Experimental results and performance analyses validate the good compression performance and high security of the given algorithm.

Keywords: 

cryptography, sensing matrix, compressive sensing, random matrix, Arnold cat map

1. Introduction

With the significant developments of internet, digital communication media, and digital data communication exchange over internet network, security is the very important issue. To overcome this issue Symmetric cryptography algorithms (Data Encryption Standard and Advanced Encryption Standard (AES)) are widely used but it is used for only text data not for image. For the purpose of security of image-based data, some new image encryption algorithms are intended, like the chaos [1-4], deoxyribo nucleic acid (DNA) coding [5-7], and S-box [8, 9].

Now a days, Chaotic system is mostly used in image security purpose due to its initial value sensitivity, randomness and unpredictability. Normally, chaotic maps are decomposed into one-dimensional and high-dimensional map. One dimensional chaotic map may be simply and easily predicted [10] because of its simple trajectory and few initial conditions. But in the high dimensional case their computational cost will be increased, so to reduce this problem sensing matrix-based encryption is one of the choices.

Sensing matrix is a new updated research field in the area of computer science. The sensing matrix provides the updated security mechanism to the digital data. The best feature of this approach is recovery of signal is easier. This approach provides compression as well as encryption, so it is comparative good to other conventional technique. Designing of the sensing matrix is easier and less computational therefor it is rapidly used in image processing, signal processing etc.

The newly proposed approach of compressive sampling (CS) or compressed sensing, shown by Candés et al. [11-13], is a new updated image processing approach. Here It permits the signal to be sampled at too much lower a rate than the Nyquist-Shannon rate and makes the signal to be sampled and compressed in a single-step mechanism. Similarly, a chaotic sequence-based approach is shown by Rong Huang and Sakurai [14]. It explained a tool where the original image is projected in a low-dimensional space. This paper used a logistic map for the purpose of generating chaotic sequences. Arnold scrambling is used to measure matrix scrambling. The main drawback of this approach is computational cost, and it takes a more significant number of variables to design the measurement matrix. Now optimization is one of the ways to reduce the computational cost.

Hence Endra et al. presented a research work based on the optimization of the sensing matrix by the MC-ETF method. The optimized matrix is more robust compared to the random sensing matrix. The quality of reconstruction of a signal is comparatively good compared to the random sensing matrix [15]. A more secure approach was proposed by Xu et al. [16] It deeply explained a digital image scrambling procedure based on CS. Here a novel 2D-SLIM hyper chaotic map is designed for the purpose of generating random sequences. The SHA-512 hash values of the digital image and the primary conditions of the proposed hyperchaotic map are used to create the secret key of the algorithm. Then two different directions, CS is used and then re-encrypted using the row and column encryption procedure.

Shruthi et al. [17] proposed a chaotic function-based image encryption mechanism. In this research authors design a linear feedback shift register for the way of controlling the randomness of sequences. The main advantage of this approach is the key sequences are stored offline in advance. Gong et al. [18] proposed a compression and encryption based mechanism by applying discrete fractional random transform and hyper-chaotic system. Here DCT is used to convert an image into spectrum and spectrum cutting is applied to compressed the data. Chaotic sequence which is originated from the hyper-chaotic system is used to controlling the random matrix, then discrete fractional random transform is apply to encrypt the compressed spectrum. The computational cost of this algorithm is not moderate and this algorithm is going under plain text attack.

To overcome this weakness a new compressive sensing based simulates compression and encryption mechanism is again proposed by Gong et al. [19], for a linear image. Here authors used Arnold transform to permutate the original image and the bitwise XOR operation is used to measures the change in pixel value.

Ponuma et al. [20] present a research work on hyper-chaos based simultaneous compression-encryption mechanism. Here authors simultaneously compress and encrypt 2D image by using two measurement matrices. Hyper chaos is used for the purpose of improve the security mechanism of digital data. Zhang et al. [21] focus a secrete orthogonal transform-based encryption mechanism. Encrypted image is compressed by using a linear operation. In this article, compressive sensing approach is applied to recover the signal. Recon structed image quality is depending on the rate of compression.

Chai et al. [22] represent an image based data encryption mechanism with the help of compressive sensing, memristive chaotic system and elementary cellular automata. Initially author transform the image into frequency component by discrete wavelet transform, and a zigzag scrambling approach is applied to obtained sparse matrix. Here measurement matrix produced by the memristive chaotic system which is used to compress the data. To improve the recovery of the signal, Chen et al. [23], explain a simultaneous image compression as well as encryption mechanism. This algorithm explains the combined approach of random matrix and compression sensing based permutation-diffusion type image scrambling approach. Here Three-dimensional cat map is used for key stream creation. But this work also not able to reduce the computational cost.

A new image encryption mechanism is proposed by Chen et al. [24], Here authors multi-image encryption mechanism is explained which is based on compressed sensing and optical wavelet transform. In this paper low and high frequency component of four images are merged into a low and high frequency fusion image respectively. After this high frequency fusion image is decomposed into two matrices by CS. Afterward, the two matrices and the low frequency fusion image are scrambled and encrypted to a single ciphertext by phase truncation and phase reservation in the Fresnel domain. A Hybrid concept of cryptography and watermarking concept are also shown in the studies [25-27].

Based upon the above survey, the current image encryption algorithms have the following shortcomings:

  1. The compression and encryption of plain images can be handled efficiently through some CS-based image encryption algorithms. Further, the pixel values can be modified through the linear measurement of CS, and the adjacent pixel coefficients can be eliminated by fusing the scrambling operation. The cipher images thus obtained will be devoid of high randomness, thereby making it susceptible to image crypto system attacks.
  2. Both the security performance and compression are equally crucial for a real-time image transmission. These are crucial particularly in the areas of battlefield medical online transmission due to bandwidth considerations. But at the same time, these cryptographic techniques do not serve well to encrypt compressed images and their ciphertext. This is due to the removal of redundancy in the encryption procedure.
  3. To enhance the encryption security, encryption methods that clubbed fusion with nonlinear operations was proposed. But such techniques inherited issues related to low decrypted image quality and resolution caused by poor high-quality information.

To reduce this problem, here we show the lossless compression and multi-level security mechanism, where computational cost is also moderate’s mathematical model of this algorithm is shown in next section. Here, the main goal of author to reduce or compressed the data and also provide the security. Measurement Matrix, pixel exchange and Arnold cat map are used to achieved the research goal.

2. Proposed Method

In this section of the research work, proposed an image encryption and decryption approach. This section also shows the detail working of measurement matrix-based image encryption and decryption in the section 3.5 and section 3.6.

Now, here we also shown the detail working structure of pixel exchange, designing procedure of measurement matrix, Logistic Map and frequency component scrambling are in section 3.1, section 3.2, section 3.3 and section 3.4:

3. Pixel Exchange Procedure

Initially, two random matrixes A1 and A2 are created whose elements are varied to 0 to 1. Random matrix A1 is used for pixel change between block B1 and B2, and similarly, A2 is used for pixel exchanged between block B3 and block B4. Here the size of the random matrix is the same as the size of the image blocks. Assume the size of the block is m×n therefore; the size of the random matrix is also m×n. But in this experiment, the size of the random matrix is represented by M×N. The output of the pixel exchange procedure is represented by B1p, B2p, B3p, and B4p. Here, B1p represent the block B1 after getting the result of pixel exchange with the help of random matrixes A1, similarly B2p, B3p, and B4p. represents the result of pixel exchange of block B2, block B3 and block B4.

For the purpose of successful exchange, the pixel, the most important thing is calculation of modified pixel position. New position (m’, n’) is created with round function, the detail mathematical expression is given below.

$\begin{aligned}

&\left.m '=f_{1}(m, n)=1+\operatorname{round}\{(M-1) R(m, n)]\right\} \\

&n ' f_{2}(m, n)=1+\operatorname{round}[(N-1) R(m, n)], \\

&1 \leq m \leq M, 1 \leq n \leq N

\end{aligned}$

In Eqns. (1) and (2), the f1(m,n) function is used to calculate the modified value of m and similarly, f2(m,n) is used to calculate the modified value on n, which is represented by m’ and n’. After deciding the new location of pixel we developed a algorithm for exchange procedure based on mean value of random matrixes A1 and A2. Now, calculate the mean value of random matrix to change the pixel values between appropriate positions of blocks of host image. The mathematical function for calculating the mean value is given below:

In Eqns. (1) and (2), the f1(m,n) function is used to calculate the modified value of m and similarly, f2(m,n) is used to calculate the modified value on n, which is represented by m’ and n’. After deciding the new location of pixel we developed a algorithm for exchange procedure based on mean value of random matrixes A1 and A2. Now, calculate the mean value of random matrix to change the pixel values between appropriate positions of blocks of host image. The mathematical function for calculating the mean value is given below:

$A_{1}{ }^{c}=\frac{1}{M \times N} \sum \forall _{m, n} A_{1}(m, n)$

$A_{2}{ }^{c}=\frac{1}{M \times N} \sum \forall _{m, n} A_{2}(m, n)$

After getting the A1c and A2c values of random matrix A1 and A2, we exchange the pixel of blocks. The detail procedure of pixel exchange is given in below algorithm and the detail working structure is also shown in Figure 1. In the Figure 1, B1 and B2 represents the Block B1 and Block B2 and A1 is the first random matrix.

Figure 1. Pixel exchange procedure

Algorithm for Pixel Exchange:

Step 1: Select the pair of Block B1 and Block B2 for pixel exchange.

Step 2: Generate a random matrix A1.

Step 3: Find the new location or position of pixel i.e. (m’, n’).

Step 4: Find the mean value of random matrix A1.

Step 5: {

$\text { if } A 1(m, n)>A 1^{C}$

{

$\begin{aligned}

&\operatorname{Block~} B_{1}\left(m^{\prime}, n^{\prime}\right) \rightleftarrows B \operatorname{lock} B_{2}(m, n) \\

&\operatorname{Block~} B_{2}\left(m^{\prime}, n^{\prime}\right) \rightleftarrows B \operatorname{lock} B_{1}(m, n)

\end{aligned}$

}

else

  { 

$\begin{aligned}

&\text { Block } B_{1}\left(m^{\prime}, n^{\prime}\right) \rightleftarrows B \operatorname{lock} B_{1}(m, n) \\

&\operatorname{Block} B_{2}\left(m^{\prime}, n^{\prime}\right) \rightleftarrows B \operatorname{Block} B_{2}(m, n)

\end{aligned}$

}

Step 6: We get the B1P, B2P, B3P, and B4P.

For the recovery of the original frequency sub-bands inverse pixel exchange procedure is applied. If $A_{1}(m, n)>A_{1}{ }^{c}$, the pixel at the position $(m, n)$ and $\left(m^{\prime}, n^{\prime}\right)$ are exchanged to each other for two modified blocks B1P and B2P and, if $A_{1}(m, n)<A_{1}^{c}$, the pixel exchanged is made in the inner pixel of modified blocks B1P and B2P. Similarly, if A1(m, n) >A1c, exchange the pixel present at the position, (m, n) and (m’, n’) to each other for two modified $\text { blocks } B_{3 \mathrm{P}} \text { and } B_{4 \mathrm{P}}$. If $A_{1}(m, n)<A_{1}{ }^{c}$exchange the inner pixels of B3P and B4P blocks.

3.1 Measurement matrix

In this section we are going to design a measurement matrix with the help of logistic map. Here, author create two measurement matrix which is represented by MM1 and MM2, these matrices are useful to provide second level of data scrambling and data encryption. Initially we generate N  number of sequences by using logistic map. Let us consider $y=\left[y_{1}, y_{2}, y_{3}, y_{4} \ldots y_{n}\right]$. Sequences are generated by using logistic map. These sequences are used to fill the column vector of the measurement matrix. The measurement matrix MM1 and MM2 is calculated with the help of original column vector $y=\left[y_{1}, y_{2}, y_{3}, y_{4} \ldots y_{n}\right]$. The first element of the measurement matrix MM1(1, j) is calculated by multiplying $M M_{1}(N, j-1)$ by λ, where $2<j<N$ and $\lambda<1$ The mathematical function for designing a measurement matrix MM1 and MM2 is given below:

For measurement matrix MM1: 

$M_{1}(1, j)=\lambda \cdot M M_{1}(N, j-1)$

$M M_{1}(2: N, j)=M M_{1}(1: N-1, j-1)$

For measurement matrix MM2:

$M M_{2}(1, j)=\lambda \cdot M M_{2}(N, j-1)$

$M M_{2}(2: N, j)=M M_{2}(1: N-1, j-1)$

3.2 Logistic map

Logistic map is a non-linear mathematical quadratic expression defines as:

Xn+1=r.Xn (1-Xn)

where, Xn∈(0,1) and 0≤r≤4.

Here Xn and r represents the system variables and n represents the number of iterations. Basically, it is a recursive function which is used to generate a number of sequences. The value of Xn+1 is dependent on value of Xn and (1 - Xn) where Xn contain only 0 and 1 and r lies between 0 to 4. The mathematical function which is used to create the Column vector of the measurement matrix by using the logistic cat map is given below:

$M M_{1}(i)=r * M M_{1}(i-1) *\left(1-M M_{1}(i-1)\right);$

$M M_{1}(1, i)=M M_{1}(i) ;$

$M M_{2}(i)=r * M M_{2}(i-1) *\left(1-M M_{2}(i-1)\right) ;$

$M M_{2}(1, i)=M M_{2}(i);$

The above mathematical expression used the initial condition MM1(0) = 0.11, MM2(0) = 0.23 and r =3.99.

3.3 Frequency component scambling

In this section, frequency scrambling is explained in detail. Initially, host image is decomposed into four frequency sub-bands by using discrete wavelet transformation. Here, four random matrices R1,R2,R3 and R4 are generated whose size is equal to the size of all frequency sub bands. In this experiment random matrix R1 is selected for scrambling the frequency sub-band LL, similarly random matrix R2, R3 and R4 is selected for scrambling the frequency sub-band LH, HH and HL. The mathematical function is given below.

{

f unction f = encrypt (matrix,sub - band)

find row and column of matrix and sub - band.

create a matrix X = zerows(no of rows = no of rows of matrix,

no of coloum = no of coloum of sub - band matrix)

for i = 1∶ no of rows of matrix

for j = 1∶ no of coloum of sub - band matrix 

for k = 1: no of coloum of sub - band matrix

Y (i,j) = Y (i,j) + sub - band (i,k) * matrix (k,j);

}

The mathematical function of inverse procedure of frequency scrambling is given below:

{

          Function f= encrypt (matrix,encrypted sub - band)

    find row and coloum of matrix and encrypted sub - band.

create a matrix M = zeroes (no of rows = no of rows of matrix,no of column = no of column of encrypted sub - band matrix)

                                      for i = 1∶ no of rows of matrix

                                      for j = 1: no of coloum of encrypted sub - band matrix f or k = 1: no of Coolum of encrypted sub - band matrix

Y (i,j) + encrypted sub - band (i,k) * matrix (k,j) = Y (i,j)

Now, inverse discrete wavelet transformation is applied to each unscrambled frequency component of blocks and get the blocks of original image. Finally, reassembled the all blocks of original image and get the decrypted original image. 

3.4 Encryption algorithm

For the purpose of compression-based encryption, initially two measurement matrixes MM1 and MM2 is designed. Measurement matrix is treated as a circulate matrix. The column vector is filled by logistic chaos map and row vector is fixed. Algorithm for design measurement matrix is also explained in the previous section 3.2.

At first original image is selected then divided into four equal parts based on row and column vector. After finding four equal sizes of blocks, again each block is divided into horizontal and vertical low and high frequency band by using discrete wavelet transformations. After finding the four-frequency band i.e., LL,LH,HH and HL, we design a four random matrix. Here; the random matrix is used for scrambling the all frequency sub-bands. The mathematical function of scrambling the frequency sub-band is explained in section 3.4.

In this experiment the original image is divided into four equal sizes of blocks and again each block is decomposed into their four frequency sub-bands. All frequency sub-bands are scrambled with random matrix. After the completions of first phase of scrambling, inverse discrete transformation is applied to reassemble the all appropriate frequency sub-bands.

Now, for the purpose of the enhanced the security mechanism random pixel exchange procedure is applied. The detail explanation of random pixel exchange procedure is shown in section 3.1. Initially A1 and A2 two random matrix is generated and their size is equal to the size of image blocks. Random matrix A1 is used to exchange the pixel between block B1 and B2, similarly Random matrix A2 is used to exchange the pixel between block B3 and B4. After the completion of the random pixel exchange procedure to get the scramble blocks $B_{1 \mathrm{P}}, B_{2 \mathrm{P}}, B_{3 \mathrm{P}} \text { and } B_{4 \mathrm{P}}$.

Finally, measurement matrix-based encryption is applied to all scrambled blocks of image. The main advantage of measurement matrix is, it provides compression-based encryption. 

Figure 2. Working structure of encryption mechanism

In this experiment two measurement matrix MM1 and MM2 are designed to encrypt the blocks. The measurement matrix MM1 is used to compressed and encrypt for blocks B1 and B2. Similarly, the measurement matrix MM2 is used to compressed and encrypt the blocks B3 and B4. Now, finally combined all blocks to get the encrypted image.

The detail of the working mechanism of the proposed image encryption algorithm is given in Algorithm 1. The detail of the encryption procedure is also given in Figure 2.

Algorithm 1: The basic algorithm step for Image encryption based on measurement matrix:

1: Select an image (original image)

2: At first, divided original image into four equal size blocks i.e. B1, B2, B3 and Band design a measurement matrix. Size of measurement matrix is depending upon size of block size of image. Let us consider B1 ∈ RM×N, where RM×N is original image signal and Bis a one of the blocks of input image. The measurement matrix MM1, MM2 ∈ RM×N, M × N is the length of measurement matrix.

3: Now, logistic chaos map is used to create a sequence with initial condition initial condition MM1 (0) = 0.11,MM2 (0) = 0.23 and r = 3.99. These sequences are used to fill the column vector of the circulant matrix.

4: Find the frequency based component of each block by discrete wavelet trans-formation. Basically, DWT convert horizontally and vertically low and high frequency component of the blocks, i. e.[LL,LH,H H,H L] = DW T (Block1).

5: Create R1,R2,R3, and R4 four matrices which size is equal to the size of LL,LH,HH and HL sub-band of the blocks of image. Now scrambled all the sub-bands of the blocks by a mathematical function. Here, R1 matrix is used for LL sub-band, R2 is LH, R3 is HH and R4 is HL sub-band. The mathematical function is given below:

{

Function f = encrypt(matrix,sub - band)

find row and coloum of matrix and sub - band.

Create a matrixX = zerows(no of rows =no of rows of matrix, no of coloum = no of coloum of sub − band matrix)

for i = 1∶ no of rows of matrix

for j = 1∶ no of coloum of sub - band matrix f or k = 1∶ no of coloum of sub - band matrix

Y (i, j) = Y (i, j) + sub − band(i, k) ∗ matrix(k, j);


}

6: After finding the scrambled LL,LH,HH and HL sub-band, Now apply inverse discrete wavelet transformation we get scrambled blocks B1,B2, B3, and B4.

7: Random matrix A1 and A2 is used to pixel exchange between the blocks.

8: Now, multiply measurement matrix MM1 to scrambled block1 and block2 to get the compressed and encrypted data CEB1 and CEB2, similarly multiply measurement matrix MM2 to scrambled block3 and block3 to get CEB3 and CEB4. The mathematical function is given below:

Eblock1 × MM1 = CEB1,

Eblock2×MM1=CEB2

Eblock3 × MM2 = CEB3 and

Eblock4 × MM2 = CEB4

9: Finally, combined the all compressed and encrypted block to get the encrypted original image.

Algorithm 2: The basic algorithm step to Decryption of image.

1: Select the encrypted image, and divide into four equal size blocks. For creating equal size of blocks at first find row, column of the image and then divide row and Column into two parts.

2: Multiplying inverse of measurement matrix to scrambled block to get B1P, B2P, B3P and B4P.

$\begin{gathered}

\mathrm{B}_{1 \mathrm{P}}=\mathrm{MM}_{1}^{-1} \cdot \mathrm{CEB}_{1} \\

\mathrm{~B}_{2 \mathrm{P}}=\mathrm{MM}_{1}^{-1} \cdot \mathrm{CE}_{\mathrm{B} 2} \\

\mathrm{~B}_{3 \mathrm{P}}=\mathrm{MM}_{2}^{-1} \cdot \mathrm{CEB}_{3} \text { and } \\

\mathrm{B}_{4 \mathrm{P}}=\mathrm{MM}_{2}^{-1} \cdot \mathrm{CEB}_{4}

\end{gathered}$

3: Now, Inverse random pixel exchange procedure is applied and get un-scrambled blocks.

4: DWT is applied to all un-scrambled blocks to get the frequency sub-bands of un-scrambled blocks.

5: Inverse function of the frequency scrambling is applied to get decrypted LL,LH,HH and HL sub-band of the blocks of image. The procedure of inverse pixel exchange is given below.

{

Function f = encrypt (matrix, encrypted sub − band)

find row and column of matrix and encrypted sub − band.

create a matrix M = zerows(no of rows = no of rows of matrix, no of coloum = no of coloum of encrypted sub − band matrix)

f or i = 1: no of rows of matrix

f or j = 1: no of coloum of encrypted sub − band matrix f or k = 1: no of coloum of encrypted sub − band matrix

Y (i, j) + encrypted sub − band(i, k) ∗ matrix(k, j) = Y (i, j)


}

6: After finding the un-scrambled LL, LH, HH and HL sub-band of the blocks of image. Inverse discrete wavelet transformation is applied to get all decrypted blocks i.e. Block1, Block 2, Block 3, and Block 4 of the original image.

7: Finally, combined all blocks to get the decrypted original image.

4. Result Analysis

In this experiment test the result on different images i.e. Lena, Pepper, Mandrill and Cameraman image. Histogram of Lena image, encrypted Lena image and decrypted Lena image is show in Figure 4. Similarly histogram of Pepper, Mandrill and Cameraman and their encrypted and decrypted image is also shown in Figure 4. Here all the simulations are done by Matlab on a 64-bit computer and Microsoft Windows 10 operating system. This experiment used 256 × 256 pixel of Lena image, pepper image, mandrill and cameraman image. The tested image is shown in Figure 3.

Figure 3. Lena, pepper, mandrill and cameraman image

The initial condition of Logistic map is $M M_{1}(0)=0.11, M M_{2}(0)=0.23 \text { and } r=3.99$. The simulation results are illustrated in Figure 4 to Figure 6.

This work proposed the enhanced N. Zhou model based on frequency-based compression- encryption procedure. Here, discrete wavelet transformation is used to decomposed the blocks of image into their low and high frequency sub-bands.

In this experiment all the frequency sub-bands are scrambled by random matrix and again blocks are scrambled by random matrix. This frame work provides the dual scrambling procedure to enhanced the data security level. After scrambling measurement matrix is used to compress and encrypt the appropriate blocks. This experiment provides the better result compare to results of N. Zhou approach.

Figure 4. The Lena, pepper, mandrill and cameraman image encryption decryption and their histogram

Figure 5. (a) Lena Image, (b) Encrypted Lena Image, (c) Correlation distribution between original Lena and encrypted Lena (e)Decrypted Lena (f) Correlation distribution between original Lena and decrypted Lena

Figure 6. (a) Cameraman image, (b) Encrypted cameraman image, (c) Correlation distribution between original cameraman and encrypted cameraman (e) Decrypted mandrill (f) Correlation distribution between original cameraman and decrypted cameraman

4.1 Histogram

Histogram of the image depicts the distribution of intensities in a digital image. Histogram is the one of the measurements of the quality of images. Here, histogram is used only for performance measurement of encryption algorithm. The histogram of the original image and decrypted image is similar to each other that means the decryption algorithm is robust and efficient. Figure 4 shows the histogram of original image, and their encrypted and decrypted image.

4.2 Correlation of two adjacent pixels

Correlation is a one of the other quality measurement approach of the image. In a meaningful image correlation should be 1 or we can say if correlation between two images is 1, that means both images is same. Here, the correlations between two image pixels are measured in vertical, horizontal and diagonal direction. Figures 5 and 6 show the correlation distribution of original image and encrypted image, original image and decrypted image. The mathematical expression of correlation coefficient is given below:

$C C=\frac{\sum_{i=1}^{m} \sum_{j=1}^{n} w(i, j) * w(i, j)}{\sum_{i=1}^{m} \sum_{j=1}^{n} w^{2}(i, j)}$

4.3 Entropy

Entropy of image represent the amount of disorder or randomness of the image. Entropy is used to verify the randomness of the decrypted image. The mathematical function of the entropy is given below.

$H(Y)=-\sum_{i=1}^{n} \operatorname{Pr}\left(y_{i}\right) \log _{2} \operatorname{Pr}\left(y_{i}\right)$

where, Pr (yi) represent the probability of yiand n represents no of bit in each pixel. The entropy of the plain image, decrypted image and encrypted image for different image is given in Table 1.

In Table 1, we see the entropy value of different image i.e. Lena, encrypted Lena, decrypted Lena image are 7.7364, 4.3909 and 7. 6533.Similarly, the entropy value of Peppers, encrypted Peppers and Decrypted Peppers are 5.4816, 5.7539 and 3. 837.The entropy value of cameraman, encrypted and decrypted cameraman are 7.2678, 4.3319 and 7.2547.

In Table 2, represents the correlation coefficient of adjacent pixels along horizontal, vertical and diagonal axis of Lean, encrypted Lena, Cameraman, encrypted Cameraman and Pepper and encrypted pepper image are shown. This table also provides the comparative results to N. Zhou approach.

Table 1. Entropy table

Image

Entropy

Image

Entropy

Lena

7.7364

Cameraman

7.2678

Lena encrypted

4.3903

Cameraman encrypted

4.3319

Lena decrypted

7.6533

Cameraman decrypted

7.2547

peppers

5.4816

Mandril

7.7748

Peppers encrypted

5.7539

Mandril encrypted

4.3684

Peppers decrypted

3.837

Mandril decrypted

7.4112

4.4 Peak signal to noise ratio (PSNR)

It is a one of the well-known quality measurement tests between the two images i.e. host and encrypted or host and decrypted image. In Table 3, its show the PSNR between host and decrypted host image. The mathematical expression for calculation PSNR between two images is given below [28].

$P S N R=10 \log \frac{255 \times 255}{\left(1 / \mathrm{M}^{*} \mathrm{~N}\right) \sum_{i=1}^{M} \sum_{j=1}^{N}(x(i, j)-y(i, j)) 2}$

Table 2. Comparison of robustness

Image (256×256)

Algorithm

Horizontal

Vertical

Diagonal

Lena

Plain Image

0.9724

0.9449

0.9206

our

0.0072

0.0004

0.0031

[29]

[30]

[31]

[32]

0.0064

0.0104

0.0042

0.0069

0.0003

0.0299

−0.0043

−0.0028

0.0026

0.0062

0.0163

−0.0047

Peppers

Plain Image

0.9714

0.9644

0.9388

Our

-0.1121

0.0041

-0.0029

[29]

[30]

[31]

[32]

−0.0117

0.0385

−0.0005

0.0074

0.0039

0.0296

−0.0062

0.0035

−0.0012

0.0069

0.0036

0.0041

Cameraman

Plain Image

0.9592

0.9337

0.9079

Our

0.0048

0.0011

-0.0074

[29]

[30]

[31]

[32]

0.0040

---

---

−0.0044

−0.0027

---

---

−0.0054

−0.0084

---

---

0.0025

Lake

Plain Image

0.9572

0.9586

0.9289

Our

-0.0161

-0.0071

-0.0061

[29]

[30]

[31]

[32]

−0.0159

---

0.0231

−0.0084

−0.0074

---

0.0140

−0.0028

−0.0005

---

0.0097

0.0033

Man

Plain Image

0.9538

0.9403

0.9097

Our

0.0032

0.0076

-0.0069

[29]

[30]

[31]

[32]

0.0022

0.0272

---

---

0.0089

0.0301

---

---

−0.0066

0.0089

---

---

Table 3. PSNRs (db) of different image

Plain Image

PSNR (db)

Lena

Cameraman

Pepper

Mandrill

Man

33.94

32.01

33.53

33.21

33.72

Table 4. PSNRs (db) of different methods

Algorithm

PSNR (db)

Ref. [33]

Ref. [31]

Ref. [34]

Ref. [21]

Ref. [28]

Our algorithm Mandrill,

26.52

17.42

22.62

26.06

33.92

33.94

Here, x(i, j) represents the host image pixel value, similarly y (i, j) represents the decrypted host image pixel value. Size of the image is here represented by M, N. Normally we know that higher PSNR value show lower distortion. Table 3, show the PSNR value of different host image and decrypted host image. Similarly, in Table 4, we show the PSNR value of different algorithms and images.

4.5 Structural similarity index measurement (SSIM)

Mainly SSIM check the quality between two images in the aspects of brightness, structure and contrast. The measurement value of SSIM lies between 0 to 1. Here, 1 represents the both images are approximate similar and 0 represent the both images are totally different. The mathematical expression of SSIM calculation is given below [28].

$\operatorname{SSIM}=\frac{\left(2 \mu_{x} \mu_{y}+C_{1}\right)\left(2 \sigma_{x y}+C_{2}\right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\right)\left(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}\right)}$

Here,

$C_{1}=\left(\mathrm{k}_{1} \times L\right)^{2}, \mathrm{C}_{2}=\left(\mathrm{k}_{2} \times L\right)^{2}, \mathrm{k}_{1}=0.01, \mathrm{k}_{2}=0.02, L=255$

and $\mu_{x}, \mu_{y}, \sigma_{x}, \sigma_{y}, \sigma_{x y}$ show the mean value, variance and covariances value of the host and decrypted host image. Table 5 and Table 6 show the SSIM of different images and SSIM values of different algorithm. In Table 5, we observed the values of SSIM of all images are near by 1, that means the decrypted image is very similar to host image. Hence, the proposed mechanism has good performance in SSIM and good quality of recover and reconstructed host image.

Table 5. SSIM values of different image

Plain Image

SSIM

Lena

Pepper

Mandrill,

Man

Cameraman

0.9437

0.9186

0.9289

0.9187

0.9184

Table 6. SSIM Values of different algorithm image

Image

Ref. [33]

Ref. [28]

our

Lena

0.6211

0.9373

0.9437

Man

0.5553

0.9101

0.9187

4.6 Time of encryption mechanism

Time analysis is the one of the most difficult and interesting work for development of algorithm in different field. Table 7 and Table 8 shows the time taken to encrypt the different host image, similarly time taken to encrypt the image based on different algorithms.

Table 7. Encryption time with different images

Plain Image

Time

Lena

Cameraman

Pepper

Mandrill

Man

0.019621

0.019787

0.206430

0.036545

0.046548

Table 8. Encryption time of different algorithm and image

Image

Ref. [33]

Ref. [28]

our

Lena

0.03178

0.0198

0.0196

Man

0.10380

0.0545

0.0465

5. Conclusions

This work mainly focuses on the security mechanism of host image information. To secure the information, this paper produced a new compression based image encryption mechanism based on pixel scrambling, random pixel exchange and measurement matrix. Here dual security mechanism is already achieved for multi-level encryption are applied. Initially, host image is decomposed into their frequency component and then scrambled. Each block of host image is scrambled based on pixel exchange procedure. Finally, second level of security mechanism is applied i.e. measurement matrix is used to dual security mechanism to encrypt the host image. The results shown is various table and graphs, it also compares to the existing approach.

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