# Physical-Layer Security in Power-Domain NOMA Based on Different Chaotic Maps
## 3 The Proposed C-DL-NOMA System Model
### Henon Chaotic Map  
The Henon chaotic map is a discrete-time dynamic system that depends on two parameters, \(a\) and \(b\), which are essential because they control the system's dynamic behavior. The classical Henon map uses the values \(a=1.4\) and \(b=0.3\), which results in chaotic behavior. This map can be chaotic, be intermittent, or converge to a periodic orbit for alternative values of \(a\) and \(b\)[19]. The Henon chaotic map is formed using the following equations:  
\[x_{n+1}=1-\ \mathrm{a}x_{n}^{2}+y_{n} \tag{10}\]  
\[y_{n+1}=\ \mathrm{b}x_{n} \tag{11}\]  
Figure 3: The logistic map bifurcation diagram.  
where (\(n\) = 0, 1, 2,... ); \(x_{n}\) and \(y_{n}\) are the current point positions; and \(x_{n+1}\) and \(y_{n+1}\) are the next point positions. Initial points \(x_{1}\) and \(y_{1}\)[20] act as a symmetric key for a chaotic cryptographic system that is utilized for encryption and decryption at the sender's and receiver's terminals. Since the Henon map is deterministic, decryption of the cipher image will reconstruct the original image on the receiver's end with the same initial points: \(x_{1}\) and \(y_{1}\)[21].  
Figure 4a[22] shows the \(x-y\) plane graph of the Henon map's attractor. Figure 4b[23] shows the Henon map's bifurcation behavior, with \(b=0.3\). The attractive set's \(x\) coordinate is plotted along the horizontal axis for various values in the range of \(a\) from 0 to 1.4. The repetition sequence begins to divide into a two-period oscillation at \(a=0.35\), which continues until \(a=0.85\). Periodic-doubling bifurcation is another name for this dividing mechanism. In the approximate range of \(a\) from 0.85 to 1.1, periods are successively doubled. In the dark region, periodicity switches to chaotic behavior when \(a~{}\geq~{}1.1\). The Henon map is used to generate two random sequences that permute an image's row and column positions, respectively, due to this chaotic tendency [24].  
**Encryption and Decryption Process**:  
Encryption Process, as shown in Figure 5:  
1. The number of pixels of the original image are extracted via multiplication of the image's height and width.
2. The Henon chaotic map is used to scramble the original image's pixels.
3. The logistic chaotic map is used to generate key values or pseudorandom numbers.
4. The encrypted image is obtained via the XOR operation, which is performed between the key values gained from the logistic map and the pixel values obtained from the scrambled image that resulted from the operation with the Henon chaotic map. Decryption Process, as shown in Figure 5:  
1. The encrypted image's pixels are extracted via measurement of the image's height and width.
2. The XOR operation is performed between the pixel values obtained from the encrypted image and the key values gained from the logistic map.
3. The decrypted image is obtained via use of the Henon chaotic map to scramble the pixels of the image that resulted from the XOR operation.  
Figure 4: (**a**) The strange attractor of the Hénon map for \(a\) = 1.4 and \(b\) = 0.3. (**b**) Hénon map bifurcation diagram for \(b\) = 0.3.  
2 [PERSON]'s cat chaotic map is utilized to scramble the original image's pixels.
3 The logistic chaotic map is used to generate key values or pseudorandom numbers.
4 The encrypted image is obtained via performance of the XOR operation using the key values obtained from the logistic map and the pixel values obtained from the scrambled image that resulted from the operation with [PERSON]'s cat chaotic map.  
**Decryption Process**:  
1. The encrypted image's pixels are extracted via measurement of the image's height and width.
2. The XOR operation is performed between the pixel values obtained from the encrypted image and the key values gained from the logistic map.
3. The decrypted image is obtained via use of [PERSON]'s cat chaotic map to scramble the pixels of the image that resulted from the XOR operation.