## 4 Simulation Results and Security Evaluation
### Differential Attack  
In accordance with the findings reported by [PERSON] et al. [27], even a minor modification in the pixel values of a plain image can lead to a substantial transformation in the resulting cipher image, thereby highlighting the efficacy of the encryption method. To quantitatively assess this effect, the normalized pixel change rate (NPCR) and the unified average change intensity (UACI) metrics are employed. NPCR, calculated using Equation (22), and UACI, \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Cameraman** & **Lena** & **Pepper** & **Rice** & **House** \\ \hline Original & 3886.6300 & 2785.4833 & 3238.600 & 1805.6977 & 2118.0286 \\ Encrypted & 5671.4695 & 5698.5852 & 5672.7690 & 5600.6353 & 5764.8519 \\ \hline \end{tabular}
\end{table} 
Table 4: Variance of images. Figure 13: Histogram for both plain images and encrypted images. determined through Equation (23), mathematically capture the extent of change in pixel values between the plain and encrypted images:  
\[NPCR=\frac{\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}\beta(i,j)}{M\times N} \times 100\% \tag{22}\]  
\[UACI=\frac{\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}e_{1}(i,j)+e_{2}(i,j)}{25 5\times M\times N}\times 100\% \tag{23}\]  
where  
\[\beta(i,j)=\left\{ \begin{matrix}0 ife_{1}(i,j)=e_{2}(i,j)\\ 1 ife_{1}(i,j)\
eq e_{2}(i,j)\end{matrix} \right.\]  
\(M\times N\) represent the size of image \(\beta\), and the cipher images \(e_{1}\) and \(e_{2}\) have one-pixel difference on their plaintext when encrypted with the same key. It is essential to ensure that any image encryption technique is secure and reliable, regardless of the size of the image being encrypted. To achieve this, the UACI (unified average changing intensity) and NPCR (number of pixel changing rate) values must meet certain critical values. Based on industry standards and best practices as stated by [PERSON] and [PERSON] [30], the ideal value for UACI in an encrypted image is greater than 33%, while the ideal value for NPCR is greater than 99%. These critical values have been determined through rigorous analysis and testing and are considered to be the minimum requirement for ensuring the security and confidentiality of the encrypted image. Therefore, it is of utmost importance that the UACI and NPCR values meet these critical values in order to consider an image encryption technique secure. When these values are achieved, we can be confident that the encryption process has met its critical value and can be deemed secure. Hence, it is crucial to prioritize these values when evaluating and selecting image encryption techniques to ensure the confidentiality and security of sensitive images. Table 5 provides the NPCR and UACI value for images of size \(256\times 256\) which has been used in this study. Table 6 provides a comparison of the UACI and NPCR values obtained from this study with those of other similar studies. The findings demonstrate that the proposed method is suitable for the encryption process, even when there is a minor alteration of the pixel values in the images. Peak Signal-to-Noise Ratio (PSNR), Signal-to-Noise Distortion Ratio (SNR), and Mean Square Error Analysis  
In the study conducted by [PERSON] and [PERSON] [32], the authors mention that the mean square error (MSE) and peak signal-to-noise ratio (PSNR) are commonly utilized to assess and evaluate the quality of images. The MSE is employed to measure the similarity \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Image**} & **Proposed** & **Proposed** & **Proposed** & **[PERSON] et al. [31]** & **[PERSON] et al. [31]** & **[PERSON] et al. [31]** \\  & **Entropy** & **NPCR** & **UACI** & **[31] Info. Entropy** & **NPCR** & **UACI** \\ \hline Lena & 7.9968 & 99.6399 & 33.4308 & 7.9975 & 99.5193 & 33.581 \\ Cameraman & 7.9974 & 99.5773 & 33.3542 & 7.9939 & 99.0039 & 33.102 \\ Pepper & 7.9969 & 99.6094 & 33.5607 & 7.9958 & 98.4972 & 32.940 \\ House & 7.9972 & 99.6017 & 33.3707 & - & - & - \\ \hline \hline \end{tabular}
\end{table}