# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 3 The Proposed Scheme
\begin{tabular}{c c c c} \hline \hline
\begin{tabular}{c} **Image** \\ **Index** \\ \end{tabular} & **Correlation (Red)** & **Correlation (Green)** & **Correlation (Blue)** \\ \hline
1 & 0.0018 & 0.0016 & \(-8.2\times 10^{-4}\) \\
3 & \(-0.0026\) & \(-0.0023\) & \(-0.0028\) \\
4 & \(-0.0011\) & \(-0.0023\) & \(-0.0026\) \\
6 & 0.0020 & 0.0028 & 0.0021 \\
7 & \(-0.0026\) & \(-0.0019\) & \(



### Information Entropy Analysis

Information entropy analysis plays a pivotal role in evaluating the security and performance of image encryption algorithms. This measure assesses the level of randomness or unpredictability in a dataset. Specifically, image encryption measures the unpredictability of pixel values in an image. A higher entropy value indicates a more uniform distribution of pixel intensities. This uniformity makes it harder for attackers to detect patterns or predict pixel values, making the encrypted image more challenging for attackers to analyze or decrypt by recognizing patterns. Consequently, a high entropy value means that the encryption technique effectively conceals the information in the original image and offers a robust defense against different types of attacks [44; 45]. Information entropy analysis involves determining and comparing the entropy values of the original and encrypted images to assess an encryption algorithm. An efficient encryption method should result in encryption with an entropy value nearing the theoretical maximum for 8-bit images. Achieving a high entropy value is a paramount indicator that an encryption method is performing effectively by producing a secure and impenetrable encrypted image. Equation (11) provides the formula for calculating information entropy:

\[H(I)=-\sum_{i=0}^{255}p(i)\lg p(i) \tag{11}\]

where \(p(i)\) represents the likelihood that the pixel intensity \(i\) will exist in the image and \(H(I)\) represents the information entropy of the image. In an 8-bit grayscale image, the summing covers all potential pixel intensity values between 0 and 255. The entropy values for both plain and encrypted images are shown in Table 7. Plain and encrypted images obtained mean entropy values of 7.1149 and 7.8879, respectively. The noticeably higher entropy values of the encrypted images show that the encryption procedure was successful in adding a significant amount of randomness, which improves the security of the final encrypted images.

Table 8 compares the \(H\) (information entropy) values for several algorithms, including the proposed method and six benchmark alternatives. The proposed method achieved an \(H\) value of 7.8879. This table provides a detailed breakdown of the results, offering a clear and thorough understanding of the data.

Figure 7: The correlation coefficient analysis across the three color channels.

### Key Sensitivity Analysis

In order to test the robustness of the proposed method against any slight change in the key, a small experiment was conducted where the key used for encryption was slightly modified and then used in the decryption phase. Table 9 illustrates the difference between using the correct, identical key in both the encryption and decryption phases and using a key with slight changes in both phases. Results show that this minor change in the key produces a significant impact on the decrypted image, as shown in Figure 8.

\begin{table}
\begin{tabular}{c c c} \hline \hline
**Image Index** & **Plain Image** & **Cipher Image** \\ \hline
1 & 7.161 & 7.894 \\
3 & 7.092 & 7.862 \\
4 & 7.120 & 7.805 \\
6 & 7.458 & 7.890 \\
7 & 7.024 & 7.885 \\
8 & 7.622 & 7.916 \\
11 & 6.857 & 7.865 \\
13 & 7.433 & 7.878 \\
16 & 7.188 & 7.912 \\
18 & 6.976 & 7.890 \\
20 & 6.34 & 7.938 \\
24 & 7.115 & 7.907 \\
**Average** & **7.115** & **7.888** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Information entropy of plain images vs. encrypted images.

\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Information Entropy** \\ \hline [PERSON] et al. [46] & 7.9893 \\ [PERSON] et al. [15] & 7.9974 \\ [PERSON] et al. [47] & 7.9976 \\ [PERSON] et al. [44] & 7.9976 \\ [PERSON] et al. [45] & 7.9973 \\ [PERSON] et al. [48] & 7.9026 \\ [PERSON] et al. [7] & 7.8793 \\ Proposed Method & 7.8879 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of the \(H\) values between the proposed method and alternative methods.

Figure 8: The impact of a slight change in the private key on the decrypted image.

\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Image** & **MSE** & **PSNR** & **SSIM** \\ \hline Decrypted with the correct key & 0 & \(\infty\) & 1 \\ Decrypted with a slight change in the key & \(1.09\times 10^{4}\) & \(17.8507\) & \(-6.59\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Key sensitivity analysis.

### Robustness Analysis

#### 4.7.1 Special Plain-Image Attacks

The proposed encryption technique was used to encode simple black-and-white images in a recent experiment. Certain images were chosen as inputs, and the test images were similar in size to those in the cited dataset, measuring \(512\times 768\times 3\). The findings indicate that the information entropy of the encrypted images increased significantly compared to that of the plaintext images, reaching values nearly equal to the theoretical maximum. Figures 9 and 10 present the encryption results for selected plaintext images using the proposed method. In Figure 9, an entirely black image is encrypted, showcasing how the technique transforms a uniform black image into a randomized pattern, effectively obscuring any trace of the original content. Similarly, Figure 10 displays the encryption of an all-white image, where the uniform white pixels are converted into a chaotic, unrecognizable encrypted form. These examples emphasize the algorithm's robustness against attacks on plaintext images with uniform values, ensuring secure encryption without exposing identifiable patterns or hints of the original image.

Table 10 presents a comprehensive analysis of the encryption and decryption operations for all-black and all-white images, using various metrics to evaluate the performance of the proposed encryption method. These metrics include the MSE, PSNR, UACI, NPCR, SSIM, and information entropy. The results indicate that the information entropy of the cipher images increased significantly compared to that of the plaintext images, reaching a value of 7.99, which is very close to the theoretical maximum of 8. This high entropy value suggests that the encryption method effectively transformed the plaintext images into cipher images with highly unpredictable and random distributions, enhancing security. The other metrics, such as the MSE, PSNR, UACI, NPCR, and SSIM, further demonstrate the robustness and effectiveness of the encryption process in altering the original image characteristics while maintaining the integrity of the decryption process.

Figure 10: All-white image encryption using the proposed method.

Figure 9: All-black image encryption using the proposed method.

#### 4.7.2 Salt-and-Pepper Noise

Salt-and-pepper noise, or randomly distributed black-and-white pixels, can significantly reduce image quality and disrupt image encryption methods. Researchers frequently add different amounts of salt-and-pepper noise to images to test how resilient these systems are to this kind of noise. They then evaluate how well the system can reduce or correct the distortions that arise. Noise reduction techniques, such as median, mean, and adaptive filtering, are often applied and evaluated to determine their efficacy in handling this noise. Quantitative metrics like the Mean Square Error (MSE) and peak signal-to-noise ratio (PSNR) are used to measure the system's robustness under different noise levels. Carrying out a comprehensive robustness study can find possible flaws in image encryption systems and improve their dependability by creating algorithms that are resistant to noise, guaranteeing reliable and accurate performance in real-world situations [7].

The robustness of the proposed encryption method against salt-and-pepper noise is evaluated in Table 11 and Figure 11. Table 11 presents the calculated PSNR and MSE values, comparing noisy encrypted images to the original versions. As the noise intensity increases, the PSNR values drop, reflecting reduced image reconstruction quality, while the MSE values rise, indicating more significant errors and distortions. A higher PSNR corresponds to better reconstruction accuracy, whereas a lower MSE indicates fewer discrepancies between the original and noisy images. The data in Table 11 confirm that the proposed encryption system effectively withstands salt-and-pepper noise, preserving the quality of decrypted images, even under substantial noise levels.

Figure 11 demonstrates the effect of incorporating salt-and-pepper noise at varying levels into an encrypted image. Even with noise, the decrypted images maintain a high level of similarity to the original ones, highlighting the method's ability to retain image quality and fidelity. The efficiency of the proposed method in reducing the effects of noise is confirmed by the strong alignment between the original and decrypted images, acting as a reliable anti-noise algorithm. These findings emphasize the robustness and dependability of the encryption system in managing noise interference, ensuring accurate image reconstruction.

\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Encrypted Image**} & \multicolumn{3}{c}{**Decrypted Image**} & \multicolumn{3}{c}{**Info.**} \\  & **MSE** & **PSNR** & **UACI** & **NPCR** & **SSIM** & **Entropy** & **MSE** & **PSNR** & **UACI** & **NPCR** & **SSIM** & **Info.** \\ \hline All & \multirow{2}{*}{\(2.17\times 10^{4}\)} & \multirow{2}{*}{10.9} & \multirow{2}{*}{0.49} & \multirow{2}{*}{0.99} & \multirow{2}{*}{\(1.16\times 10^{-15}\)} & \multirow{2}{*}{7.99} & \multirow{2}{*}{0} & \multirow{2}{*}{NaN} & \multirow{2}{*}{0} & \multirow{2}{*}{1} & \multirow{2}{*}{0} \\ All & & & & & & & & & & & \\ white & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 10: Special plaintext image results.

#### 4.7.3 Gaussian Noise Attack

Gaussian noise is a type of statistical noise that follows a Gaussian distribution, also known as a normal distribution. This noise is characterized by its probability density function, defined by the mean and standard deviation of the noise values. In digital images, Gaussian noise typically arises during acquisition due to sensor noise, which can be caused by poor lighting conditions or high temperatures, and during transmission due to electronic circuit noise. Gaussian noise in images results in random variations in pixel values, which can affect the quality and clarity of the image. Gaussian noise can be mitigated in image processing using various filtering techniques, such as Gaussian smoothing. However, these methods may inadvertently blur fine details and edges in the image [7].

Table 12 analyzes the effectiveness of the proposed method using various Gaussian noise levels by examining the PSNR and MSE metrics. These metrics assess how effectively the encryption technique maintains the integrity of encrypted images when noise is present. With higher noise levels, the PSNR values generally decline, signifying a decrease in image

Figure 11: Robustness analysis of salt-and-pepper noise.

quality, while the MSE values rise, indicating increased error and distortion. Figure 12 illustrates the impact of different Gaussian noise levels on an encrypted image. Even with added noise, the decrypted images closely resemble the original ones, demonstrating the method's ability to maintain image integrity. The positive results highlight a strong correlation between the original and decrypted images, verifying the encryption method's resilience to Gaussian noise. This robustness ensures the accurate recovery of the original image, even under significant noise conditions.

Figure 12: Robustness analysis of Gaussian noise.

#### 4.7.4. Occlusion Attacks

Occlusion attacks, which involve partially obstructing or covering parts of an image, can pose significant challenges to image encryption systems. Our study shows that the proposed algorithm can effectively withstand up to 50% occlusion attacks. Figures 13 and 14 illustrate the effects of occlusion on decrypted images, showing results from encrypted images with varying levels of occlusion: 5%, 25%, and 50%. In these cases, although the decrypted images exhibit some distortion, they remain recognizable to the human eye, indicating the resilience of the proposed encryption method. Beyond 50% occlusion, the decrypted images become unrecognizable, suggesting that the method's robustness is compromised. These findings underscore the importance of the proposed method's balance between security and image recognizability, making it highly relevant for applications where partial image loss is a concern.

Table 13 analyzes how different levels of occlusion impact various image quality metrics, including the SSIM, PSNR, and MSE. The data clearly show that as there is a rise in the occlusion, these metrics decline. At 0% occlusion, the SSIM is high at 1, indicating excellent similarity to the original image. At the same time, the MSE is low at 0, and the PSNR is high at \(\infty\), reflecting minimal distortion and high image quality. As the occlusion increases to 5%, 10%, and 25%, the SSIM decreases to 0.765, 0.6511, and 0.3977, respectively, showing a progressive decline in structural similarity. Concurrently, the

\begin{table}
\begin{tabular}{c c c} \hline
**Noise Level** & **MSE** & **PSNR** \\ \hline
0.5 & \(8.90\times 10^{3}\) & 19.88 \\
0.15 & \(4.01\times 10^{3}\) & 27.87 \\
0.05 & \(2.52\times 10^{3}\) & 32.52 \\
0.005 & \(2.28\times 10^{3}\) & 33.50 \\
0.0005 & \(2.28\times 10^{3}\) & 33.52 \\ \hline \end{tabular}
\end{table}
Table 12: Robustness analysis of Gaussian noise.

Figure 13: Low occlusion values.

MSE rises to 527.6988, 927.653, and 2.28 \(\times\) 10\({}^{3}\), indicating increasing error, and the PSNR drops to 488.14, 42.4987, and 33.4866, reflecting reduced image quality. At 75% occlusion, the SSIM plummets to 0.0782, the MSE surges to \(6.8897\times 10^{3}\), and the PSNR falls to 22.4474, highlighting significant degradation. These results illustrate the method's ability to maintain some level of image quality up to 50% occlusion, beyond which the quality metrics indicate substantial deterioration.

#### 4.7.5 Clipping Attacks

Clipping attacks involve altering the dynamic range of an image by truncating pixel values, which can lead to a loss of detail and contrast, potentially compromising the effectiveness of image encryption systems [7]. Table 14 and Figure 15 clearly show that the algorithm is robust against clipping attacks, particularly when the changes in dynamic range values are minor. As the clipping range increases, various quality metrics show degradation in their values, indicating some loss of image quality. Despite this degradation, the recovered images remain understandable to the human eye, suggesting that the images' essential features and overall structure are preserved. This robustness is crucial for applications where maintaining the recognizability of an image is important, even in the presence of attacks that alter its dynamic range. The data in Table 14 and the visual representations in Figure 15 confirm that the proposed method can effectively withstand

\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Occlusion \%** & **SSIM** & **MSE** & **PSNR** \\ \hline
0\% & 1 & 0 & \(\infty\) \\
5\% & 0.77 & 527.70 & 48.14 \\
10\% & 0.65 & 927.65 & 42.50 \\
25\% & 0.40 & \(2.28\times 10^{3}\) & 33.49 \\
50\% & 0.20 & \(4.57\times 10^{3}\) & 26.54 \\
75\% & 0.08 & \(6.89\times 10^{3}\) & 22.44 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Occlusion attack analysis.

Figure 14: High occlusion values.

clipping attacks, ensuring that the decrypted images retain sufficient quality for human interpretation, even under conditions of increased clipping.

### \(\chi^{2}\) Test

A crucial technique used to evaluate the independence of variables and the regularity of data distribution in image cryptosystems is the Chi-square (\(\chi^{2}\)) test. This test, which compares the frequency distribution of pixel values to a theoretically uniform distribution, is crucial for assessing the randomness and security of encrypted images. For encryption approaches like chaos-based and DNA-based systems, where attaining a high degree of unpredictability and efficient pixel value diffusion is essential for guaranteeing strong

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Clipping Range** & **MSE** & **PSNR** & **SSIM** & **NPCR** & **UACI** \\ \hline
50–200 & 336.5 & 52.6 & 0.74 & 0.41 & 0.04 \\
25–225 & 42.92 & 73.23 & 0.95 & 0.2146 & 0.01 \\
15–240 & 5.41 & 93.94 & 0.99 & 0.1169 & 0.003 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Clipping attack analysis.

Figure 15: Different dynamic range values.

security, such an analysis is essential. Greater randomness and better encryption quality are indicated by lower Chi-square values, which show how well the encryption process can mask the original image's statistical properties and maintain robust security.

Table 15 illustrates the results of the Chi-square test, offering a quantitative evaluation of the encryption method by comparing pixel intensity distributions between the original and encrypted images. The table reveals a marked difference, with encrypted images exhibiting significantly smaller Chi-square values than the original images. This result provides compelling evidence that the encryption method effectively secures images, underscoring its reliability and instilling confidence in its performance.

## 5 Conclusions

In conclusion, this study significantly contributes to image encryption by developing and validating an integrated chaos-based algorithm. This algorithm, which combines the chaotic Henon map and chaotic logistic map (CLM), represents a unique approach to enhancing the security of digital image communication. The method's innovation lies in using chaos theory to generate cryptographic keys, utilizing 1D and 2D key generators. This approach proves highly effective in producing complex and unpredictable keys essential for robust encryption. The algorithm's resilience against various potential attacks, including chosen-plaintext, noise, clipping, occlusion, and known-plaintext attacks, is demonstrated through extensive experimentation and rigorous performance evaluations. The comprehensive analysis of encryption time, throughput, and various image quality metrics further substantiates the effectiveness and efficiency of this integrated approach. The chaotic Henon-logistic map integration emerges as a powerful and secure method for safeguarding digital images during transmission and storage, offering significant potential for real-world applications in secure image sharing, cloud storage, and digital forensics. As the demand for robust image encryption continues to grow in our increasingly digital world, this research contributes a valuable tool to the field of information security, paving the way for more secure and reliable image communication systems. Future work could explore the adaptation of this algorithm to other types of digital media and its integration into existing cybersecurity frameworks to further enhance its practical applicability and impact.

Conceptualization, C.O., A.A.-H., and M.A.-F.; Formal analysis, C.O., A.A.-H., M.A., and M.A.-F.; Methodology, A.A.-H., M.A., and C.O.; software, A.A.-H., and M.A.-F.; Supervision, C.O., and M.A.-F.; Writing--original draft, M.A.-F., M.A., and A.A.-H.; Writing--review and editing, M.A.-F., C.O., and A.A.-H. All authors have read and agreed to the published version of the manuscript.

\begin{table}
\begin{tabular}{c c c} \hline \hline
**Image Index** & **Plain Image** & **Cipher Image** \\ \hline
1 & \(5.18\times 10^{5}\) & 268.72 \\
3 & \(5.67\times 10^{5}\) & 244.15 \\
4 & \(5.02\times 10^{5}\) & 278.41 \\
6 & \(7.34\times 10^{5}\) & 240.21 \\
7 & \(5.16\times 10^{5}\) & 225.04 \\
8 & \(2.85\times 10^{5}\) & 254.68 \\
11 & \(7.67\times 10^{5}\) & 302.48 \\
13 & \(3.09\times 10^{5}\) & 246.36 \\
16 & \(4.34\times 10^{5}\) & 232.94 \\
18 & \(3.93\times 10^{5}\) & 239.016 \\
20 & \(2.44\times 10^{7}\) & 620.039 \\
24 & \(8.18\times 10^{5}\) & 291.33 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Chi-square test results.

**Funding:** The open access fees for this article were funded by the Qatar National Library.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The original data presented in the study included a paragraph to describe the dataset and the link where we can download as: [[https://r0k.us/graphics/kodak/](https://r0k.us/graphics/kodak/)]([https://r0k.us/graphics/kodak/](https://r0k.us/graphics/kodak/)) (accessed on 17 January 2025).

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Abbreviations

The following abbreviations are used in this manuscript:

AESAdvanced Encryption Standard

CCCorrelation Coefficient

CLMChaotic Logistic Map

CLMKChaotic Logistic Map Key

KSKey Space

MSEMean Square Error

NPCRNumber of Pixels Change Rate

PKPrivate Key

PSNRPeak Signal-to-Noise Ratio

SSIMStructural Similarity Index Measure

UACIUnified Average Changing Intensity

## Appendix A

In this appendix, the key generation process is explained in detail.

### Generation of 1D Key

The chaotic logistic map (CLM), as described in Equation (1), generates a 1D key. This key permutes the image row- and column-wise, so the number of generated values should equal the number of rows and columns in the image to be secured. Let us assume that the dimensions of the image are (5). Given that the initial conditions for Equation (1) are \(x=0.8990\) and \(r=3.7458\), the generated values will be as follows:

\(0.3401\) \(0.8407\) \(0.5017\) \(0.9364\) \(0.2230\)

After sorting these values in ascending order and taking their corresponding indices, they will be as follows:

\(5\) \(1\) \(3\) \(2\) \(4\)

Thus, we obtain new indices that will be used to permute the original pixel locations into these new locations. There is no need for any normalization and truncation as long as the number of generated random numbers equals the dimension of the image.

### Generation of 2D Key

This section outlines the generation of a two-dimensional key using the chaotic Henon map for the encryption process. The chaotic Henon map, as described in Equation (4), is used to generate a 2D key that substitutes the original pixel values in the diffusion stage. The initial parameters for the chaotic Henon map are typically \(a=1.4\) and \(b=0.3\), with initial conditions \(X_{0}=0.1\) and \(Y_{0}=0.1\).

For a \(2\times 2\) 2D key to be used to encrypt an image of the same size, the \(x\) and \(y\) values generated according to the equation are as follows:

\(\begin{array}{ccccc}\text{x}&=&1.0860&-0.6212&0.7856&-0.0505\\ \text{y}&=&0.0300&0.3258&-0.1863&0.2357\end{array}\)Then, each value of vector \(x\) is multiplied by the corresponding value from vector \(y\), and the resulting numbers are normalized to be in the range of [0, 255], corresponding to the possible pixel values. The obtained values are as follows:

0 & 248 \\
118 & 240 \\ This 2D map is XORed with the corresponding pixel values in the image to be encrypted. \\ \end{tabular}