## 5 Security Analysis, Performance Evaluation, and Discussion
### Performance Parameters  
Performance parameters are essential to find out the quality of the proposed security model. The assessment is carried out based on the parameters: correlation coefficient, NPCR, and UACI values, in which an original image is compared with a cryptcoded image. Peak signal-to-noise ratio (PSNR), mean square error (MSE), bit error rate (BER), and structural similarity (SSIM) [23] indicate the efficacy of the channel coding technique. This is calculated between an original and received decrypted image. PSNR estimates the noiselessness of the cryptcoded image. The greater value of PSNR suggests a better quality of the received decrypted image. The PSNR is calculated as [15]:  
\[PSNR=10\log\biggl{[}\frac{l^{2}}{MSE}\biggr{]} \tag{17}\]  
where I is a pixel value in the image. The maximum value for a grayscale image is 255. Mean square error (MSE) [15] is the parameter that estimates the error between the original and received decrypted image. The \(MSE\) is computed as:  
\[MSE=\frac{1}{\left[M*N\right]^{2}}\sum_{i=0}^{n}\sum_{j=1}^{m}\bigl{(}\;X_{ij} -Y_{ij}\bigr{)}^{2} \tag{18}\]  
\(M\) and \(N\) are two dimensions of an image. \(X_{ij}\) and \(Y_{ij}\) are pixel intensity of an original and recovered image. The main advantage of MSE and PSNR is to find the noise level in a reconstructed image. These parameters also help us analyze the efficiency of channel coding techniques used. Table 5 shows that we are achieving good results for PSNR and MSE, as the PSNR value is high and MSE is low. Figure 6: Bitwise uncorrelation test (BUCT). Bit errors occurring during transmission are indicated by BER [23]. This is the number of bits received in error, divided by the total number of bits transmitted. \[\text{BER}=\text{Errors}/\text{Total Number of Bits}\]  
The structural similarity index (\(SSIM\)) measures the structural similarity between two images. The value of 1 suggests nearly identical images. \(SSIM\) is calculated as follows [16]:  
\[SSIM=\frac{[2*mu_{1}(p)mu_{2}(p)+c_{1}]}{\left[\,mu_{1}(p)^{2}mu_{2}(p)^{2}+c _{1}\right]}*\frac{[2*cov(p)+c_{2}]}{\left[s_{1}(p)^{2}+s_{2}(p)^{2}+c_{2} \right]} \tag{19}\]  
where, \(mu_{1}(P)\) and \(mu_{2}(p)\) are the mean value of seq\({}_{1}\) and seq\({}_{2}\) computed over a small \(XY\) window located around \(P\); \(s_{1}(p)\) and \(s_{2}(p)\) are the standard deviations of \(seq_{1}\), and \(seq_{2}\) computed over the same window; and \(cov(p)\) is the covariance between seq\({}_{1}\) and \(seq_{2}\). One of the purposes of encryption is to reduce the association between two pixels [17]. The reduced correlation value suggests an improved encryption effect and better security. The correlation coefficient is calculated using the equations given below:  
\[E(x)=\frac{1}{N}\sum_{i=1}^{N}x_{i} \tag{20}\]  
\[cov(x,y)=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-E(x))(y_{i}-E(y)) \tag{21}\]  
\[D(x)=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-E(x))\;^{2} \tag{22}\]  
\[D(y)=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-E(y))\;^{2} \tag{23}\]  
\[D(y)=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-E(y))\;^{2} \tag{24}\]  
where,  
\(x\) and \(y\): values of the two adjacent pixels  
\(N\): Number of selected adjacent pixels  
A study on the algorithm's sensitivity is accomplished by finding the number of pixels changing rate (NPCR) and the unified average changing intensity (UACI) [17]. E\({}_{1}\) and E\({}_{2}\) are the two encrypted images derived by changing a single bit of the original image. \[X(i,j)= \begin{cases}1,&if\;\;E_{1}(i,j)\
eq\;E_{2}(i,\;j)\\ 0,&if\;\;E_{1}(i,j)=E_{2}(i,\;j)\end{cases}  \tag{25}\]