# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 3 The Proposed Scheme
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & **NPCR** & **NPCR** & **UACI** & **UACI** & **SSIM** & **SSIM** \\
**Index** & **Encrypted** & **Decrypted** & **Encrypted** & **Decrypted** & **Encrypted** & **Decrypted** \\ \hline
1 & 0.9960 & 0 & 0.2853 & 0 & \(9.24\times 10^{-3}\) & 1 \\
3 & 0.9961 & 0 & 0.2986 & 0 & \(6.78\times 10^{-4}\) & 1 \\
4 & 0.9961 & 0 & 0.2995 & 0 & \(7.37\times 10^{-4}\) & 1 \\
6 & 0.9960 & 0 & 0.3056 & 0 & \(0.0011\) & 1 \\
7 & 0.9961 & 0 & 0.2857 & 0 & \(-3.72\times 10^{-4}\) & 1 \\
8 & 0.9961 & 0 & 0.3111 & 0 & \(-0.0013\) & 1 \\
11 & 0.9960 & 0 & 0.3010 & 0 & \(-2.27\times 10^{-4}\) & 1 \\
13 & 0.9962 & 0 & 0.3078 & 0 & \(4.48\times 10^{-4}\) & 1 \\
16 & 0.9961 & 0 & 0.2889 & 0 & \(4.21\times 10^{-4}\) & 1 \\
18 & 0.9962 & 0 & 0.3329 & 0 & \(5.06\times 10^{-4}\) & 1 \\
20 & 0.9960 & 0 & 0.4008 & 0 & \(5.90\times 10^{-4}\) & 1 \\
24 & 0.9961 & 0 & 0.3033 & 0 & \(4.59\times 10^{-4}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated NPCR, UACI, and SSIM values for encryption and decryption. Figure 6: Histogram analysis. processing capabilities. This impressive performance can assure the reader of the method's effectiveness in securely encrypting image data without causing significant delays. ### Correlation Analysis

The linear relationship between two images is measured by the correlation coefficient (CC), which assesses the similarity between the pixel values of an encrypted image and its original counterpart. A higher CC value indicates a stronger correlation between the two images. The formula for calculating the CC is given by Equation (10) [6]:

\[CC(X,Y)=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})(Y_{i}-\bar{Y})}{\sqrt{\sum_{i=1}^{ n}(X_{i}-\bar{X})^{2}}\sqrt{\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}}} \tag{10}\]

where \(X_{i}\) and \(Y_{i}\) are the pixel values of the first and second images, \(\bar{X}\) and \(\bar{Y}\) are their respective means, and \(n\) is the number of pixels in the images. Table 5 shows the calculated CC values between the encrypted and original images using the proposed method for each color channel. The extremely low values of the CC confirm the absence of any notable relationship between the two images. An example result where the CC value between two image files indicates the relationship between their gray levels is shown in Figure 7. The CCs for consecutive pixels that are horizontal, vertical, and diagonal in a variety of plain images and their encrypted equivalents are shown in Table 5. These coefficients range from \(-1\) to \(1\), with \(1\) indicating a perfect positive correlation between neighboring pixel pairs, \(-1\) indicating a perfect negative correlation, and \(0\) indicating no correlation. Higher correlation coefficients signify stronger relationships between adjacent pixels, leading to lower randomness in pixel distribution. The table demonstrates that the original images exhibit high correlation coefficients, whereas the encrypted images show significantly reduced coefficients, reflecting increased randomness and decreased similarity between pixel pairs. Table 6 demonstrates the correlation coefficients, indicating how the encryption process significantly reduces the correlation between adjacent pixels, thus ensuring increased randomness and security in the encrypted images. This reduction in correlation is essential for making the encrypted images resistant to statistical attacks. These coefficients, which range between \(-1\) and \(1\), reflect the degree of correlation between neighboring pixel pairs: \(1\) represents a perfect correlation, while \(0\) signifies no correlation. Higher values indicate a stronger connection between adjacent pixels, increasing pixel arrangement randomness. The table reveals that the original images have high correlation coefficients, whereas