# Physical-Layer Security in Power-Domain NOMA Based on Different Chaotic Maps
## 4 Simulation Results
### Security Analysis
#### 4.1.1 Statistical Analysis  
Because the efficiency of an encryption system is determined from its ability to reject statistical attacks, statistical analyses are utilized to distinguish original data from encrypted data. Such statistical analyses are discussed below.  
Histogram Analysis  
A histogram distribution is a graphical representation of data distribution. It comprises the frequency of the data group. Several fields can use histogram analysis. Encryption is effective if the histogram's distribution of the ciphered data has similar values. The closer the values of a data set are to each other, the more difficult it is to decode encrypted data. The histogram distribution in the image data is from left to right, that is, from dark to light colors. On the left side, dark colors prevailing indicate much distribution, whereas on the right side, light colors prevailing indicate much distribution. For this reason, the more uniform the distribution is, the more difficult it is to form a thought of the image or decode the ciphered data. Notably, hyperchaotic DNA has a better histogram distribution than do the logistic, Henon, and Arnold's cat maps because its ciphered data has similar values (Figure 8).  
Correlation analysis is commonly used in image-data security analysis. For correlation analysis to be useful, the relationship among variables must be linear. Testing the relationship among adjacent pixels can be used to execute correlation analysis.  
The relationship among the variables on the plain image is linear, whereas the relation after analysis is nonlinear (scattered) with a highly complex distribution, and a correlation value close to zero indicates that the analysis result is good and there is no relation between the two images. An adjacent pair of pixels in any number is chosen at random from the image; the correlation coefficient of each pair is calculated using the following formulas [18]:  
\[E(\mathbb{X})=\frac{1}{\mathbb{N}}\sum_{i=1}^{N}\mathbb{X}_{i} \tag{22}\]  
\[E(\mathbb{X})=\frac{1}{\mathbb{N}}\sum_{i=1}^{N}\mathbb{X}_{i} \tag{23}\]  
\[\text{cov}(\mathbb{X},\mathbb{Y})=\frac{1}{\mathbb{N}}\sum_{i=1}^{N}\left( \mathbb{X}_{i}-E(\mathbb{X})\right)\left(\mathbb{Y}_{i}-E(\mathbb{Y})\right) \tag{24}\]  
\[r_{\mathbb{xy}}=\frac{\text{cov}(\mathbb{X},\mathbb{Y})}{\sqrt{D(\mathbb{X})D (\mathbb{Y})}} \tag{25}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Logistic** & **Hénon** & **Arnold’s Cat** &
\begin{tabular}{c} **Hyperchaotic** \\ **DNA** \\ \end{tabular} \\ \hline \(U_{1}\)**’s Data** & 7.9974 & 7.9963 & 7.9977 & 7.9964 \\ \hline \(U_{2}\)**’s Data** & 7.9967 & 7.9975 & 7.9969 & 7.9979 \\ \hline \(U_{3}\)**’s Data** & 7.9982 & 7.9984 & 7.9987 & 7.9985 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Information entropy of encrypted images.  
\(\mathfrak{X}\) and \(\mathfrak{Y}\) are the grayscale values of the two adjacent pixels in the image, and \(\bar{\mathfrak{N}}\) refers to the total number of pixels chosen from the image. In the horizontal direction, 3000 pairs of adjacent pixels are chosen at random from the images to display their adjacent pixel distribution maps. This indicates that input images have a strong correlation effect, whereas encrypted images have a weak correlation impact, confirming that robust correlation is absent among adjacent pixels in encrypted images. As shown in Figure 8, through the four chaotic encryption schemes, the hyperchaotic DNA algorithm has a good encryption effect that damages correlation and achieves the weakest correlation impact.