# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 3 The Proposed Scheme
\begin{tabular}{|c|c|c|} \hline
1 Bit & 11 bits & 52 bits \\ Sign & Exponent & Mantissa \\ \hline \end{tabular} 

The key's first half includes the CLM parameters, r and x. Starting with r, which should take a value in the period \(3.57<r<4.0\), the difference between these two values is 0.43, a double value that can be stored in the mantissa part of the double, and these are the optimal values for the CLM. Since the mantissa is 52 bits long (see image above), the number of different binary numbers representable is \(2^{52}\). Thus, the range of values of \(r\) can be computed as \(2^{52}=4.5E15\). Thus, taking into account all the range of values for r, this yields:

\(4.E15-3.57E15=0.93E15=9.3E14\approx 0.9(2^{50})\) different possible values. Following the same procedure for the second CLM parameter, x, which can take any value in the interval \(0<x<1\), and since the difference is 1, this will result in a total of \(2^{52}\) distinct possible values. As for the second half of the key, which includes the initial values for the chaotic Henon map parameters, \(x_{0}\) and \(y_{0}\), these can take values that lie in the polygon (ABCD) with points \(A\) (\(-1.33\), 0.42), \(B\) (1.32, 0.133), \(C\) (1.245, \(-0.14\)), and \(D\) (\(-1.06\), \(-0.5\)), as shown in Figure 5.

The area of the polygon (ABCD) is \(>0.5\times 1=0.5\), and the number of possible points inside the area, which can be used as initial values, is proportional to the area of the polygon (ABCD) and their representation in double. Thus, \(x_{0}\) represents the interval between \(-1\) and 1, and \(y_{0}\) represents the interval from \(-0.5\) to \(0.5\). The number of double values between \(-1\) and 1 is \(2(2^{52})\), and the number of double values between \(-0.5\) and \(0.5\) is \((2^{52})\). Thus, \((x_{0},y_{0})\) has over \((2(2^{52})\times 2^{52})\) representations. Therefore, the search space \(>0.5(2^{52})\times 2^{52}\times[2^{52}\times 2^{52}]>2^{200}\). ## 4 Results and Discussion

### Image Quality Analysis

In image processing activities, especially image encryption, metrics like the peak signal-to-noise ratio, Mean Square Error (MSE), Structural Similarity Index Measure, and Unified Average Changing Intensity are essential. Their widespread adoption underscores their paramount significance within this field. The average squared deviations between the original and encrypted images are measured using the MSE. A higher degree of similarity between the two images is indicated by a lower MSE. While the MSE alone does not fully encapsulate an image's perceptual quality,

Figure 5: Key-space analysis. combined with other metrics, it provides a more comprehensive assessment, bolstering our confidence in the evaluation process. The peak signal-to-noise ratio (PSNR) assesses the relationship between the maximum possible signal and the noise power that influences the fidelity of the image representation. It considers the root-mean-square error (RMSE) ratio between the original and encrypted images to the highest pixel intensity. A higher PSNR score indicates greater similarity between the encrypted and original images. In the context of image encryption, these metrics serve as instrumental tools in evaluating the quality of the encrypted image and assessing the efficacy of the encryption algorithm in preserving the image's content. They not only guide us in assessing the performance of an image encryption algorithm but also empower us to make informed decisions when selecting the most suitable algorithm for a specific application. Equations (5) and (6) are used to calculate these metrics [6]:

\[MSE=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}(x_{ij}-y_{ij})^{2} \tag{5}\]

where \(x_{ij}\) stands for the pixel value from the original picture, \(y_{ij}\) stands for the pixel value from the transmitted image, \(m\) represents the number of rows, and \(n\) represents the number of columns in the cover image.