## 4 Simulation Results and Analysis
### Peak Signal-to-Noise Ratio (PSNR) and Mean Square Error Analysis (MSE)  
Evaluating the quality of encrypted images is an important task in ensuring that the original image is accurately represented after being processed through an encryption algorithm. Two commonly used metrics for this purpose are the mean square error (MSE) and the peak signal-to-noise ratio (PSNR) [40]. The MSE is a measure of the similarity between the encrypted image and the original image. A low MSE value indicates that the two images are similar and that the encrypted image is of high quality, while a high MSE value indicates that the encrypted image is of poor quality. The MSE is calculated by subtracting the encrypted image from the original image, as shown in Equation (18). The PSNR is another useful metric for evaluating the quality of encrypted images. The higher the PSNR, the better the image quality; a lower PSNR value is an indicator of poor image quality. The PSNR is calculated mathematically, as shown in Equation (19). \(P(i,j)\) and \(E(i,j)\) represent the original and encrypted image pixels consecutively in this context. The mean square error (MSE) is always zero when the pixel values of the original and encrypted images are exactly the same (when \(P(i,j)\) and \(E(i,j)\) are equal). Table 7 demonstrates the effectiveness of the proposed image encryption method with the high values of the mean square error (MSE). The PSNR is calculated mathematically, as shown in Equation (19).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Image**} & \multicolumn{3}{c}{**Plaintext**} & \multicolumn{3}{c}{**Ciphertext**} & \multirow{2}{*}{**Average**} \\ \cline{2-2} \cline{4-6}  & **Horizontal** & **Vertical** & **Diagonal** & **Horizontal** & **Vertical** & **Diagonal** \\ \hline House(128 \(\times\) 128) & 0.9507 & 0.9396 & 0.9058 & \(-\)0.0001 & \(-\)0.0085 & 0.0009 & \(-\)0.0025 \\ Cameraman(64 \(\times\) 64) & 0.8908 & 0.9315 & 0.8443 & 0.0056 & \(-\)0.0075 & 0.0159 & 0.0047 \\ Lena (128 \(\times\) 128) & 0.8988 & 0.8573 & 0.9481 & 0.0049 & \(-\)0.0022 & \(-\)0.0042 & \(-\)0.00053 \\ Ref [29] & - & - & - & 0.001136 & 0.00080 & 0.00147 & 0.00113 \\ Ref [30] & - & - & - & 0.002225 & 0.003075 & 0.001625 & 0.00230 \\ Ref [39] & - & - & - & 0.0021 & 0.0029 & 0.0023 & 0.00243 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of average correlation of ciphertext.  
Figure 13: Correlation (128 \(\times\) 128): (**a**–**c**) correlation of horizontal, vertical, and diagonal directions of the Lena graph, respectively; (**d**–**f**) correlation of horizontal, vertical, and diagonal directions of Lena encrypted graph, respectively.  
squared error (MSE) and low values of the peak signal-to-noise ratio (PSNR), which was conducted using images that were 128 pixels by 128 pixels in size. These results suggest that the proposed method is capable of effectively protecting the confidentiality of images, and thus, has the potential to make a significant contribution to the field of image encryption.  
\[MSE=\frac{1}{M\times N}\sum_{i=1}^{M}\sum_{j=1}^{N}(P(i,j)-E(i,j))^{2} \tag{18}\]  
\[PSNR=10\times\log_{10}\frac{M\times N}{\sqrt{MSE}} \tag{19}\]