# Physical-Layer Security in Power-Domain NOMA Based on Different Chaotic Maps
## 3 The Proposed C-DL-NOMA System Model
### Hyperchaotic Map  
A hyperchaotic system is a chaotic system with a positive Lyapunov exponent that exceeds 1, indicating that its chaotic dynamics are provided simultaneously in multiple directions [26]. Hyperchaos occurs in high-dimensional, nonlinear schemes with at least four dimensions. A high-dimensional chaotic system has a larger key space and more complex and unpredictable nonlinear behavior due to the increased number of state variables in a hyperchaotic system [27]. DNA (deoxyribonucleic acid) computing has recently been utilized in chaos-based image-encryption schemes because of its many advantages: enormous parallelism, massive storage, and ultralow power consumption [28]. As illustrated in Figure 7, the authors in [28] combined DNA sequencing and hyperchaotic sequencing for encryption, and we followed the same approach for our proposed C-DL-NOMA system. Figure 7: Schematic diagram of encryption using DNA sequencing and hyperchaotic sequencing. The following nonlinear equations govern the hyperchaotic system [29], which they adopted [28]:  
\[\dot{y}_{1}=\alpha(y_{2}-y_{1})+\lambda_{1}y_{4} \tag{13}\]  
\[\dot{y}_{2}=\xi y_{1}-y_{1}y_{3}+\lambda_{2}y_{4} \tag{14}\]  
\[\dot{y}_{3}=-\beta y_{3}+y_{1}y_{2}+\lambda_{3}y_{4} \tag{15}\]  
\[\dot{y}_{4}=-\tau y_{1} \tag{16}\]  
where the system's control parameters are \(\alpha,\ \xi,\ \beta,\ \tau,\ \lambda_{1},\ \lambda_{2},\ \text{and}\ \lambda_{3}\). This system describes hyperchaotic performance when \(\alpha=35\), \(\xi=35\), \(\beta=3\), \(\tau=5\), \(\lambda_{1}=1\), \(\lambda_{2}=0.2\), and \(\lambda_{1}=0.3\). The DNA sequence includes four bases of nucleic acid, which are permanently denoted by the letters G (Guanine), C (Cytosine), A (Adenine), and T (Thymine). \"G\" and \"C\" are complementary, as are \"A\" and \"T\". We utilized two-bit binary digits to represent a DNA base, since the binary digits \"0\" and \"1\" are complementary. For the representation shown in [30], there are twenty-four different types of rule, and only eight of them satisfy the Watson-Crick complement rule [31]. DNA computing follows conventional binary addition and subtraction rules [32]. Four Steps to Generate the Hyperchaotic Sequence, \(i\):  
1. To avoid negative effects and to increase security, the hyperchaotic system is pre-iterated \(N_{0}\) times. 2. After iteration of \(N_{0}\) times, the process is repeated \(m\times n\) times. We utilized \(\hat{1}\) to indicate the iteration index. For every iteration, \(\hat{1}\), four state values, \(\{\hat{y}_{1}^{\hat{1}},\hat{y}_{2}^{\hat{1}},\hat{y}_{3}^{\hat{1}},\hat{y}_{4}^ {\hat{1}}\}\) are saved. 3. Every state value, \(\hat{y}_{i}^{\hat{1}}\), is utilized to produce two various key values--\(\left(v_{i}^{\text{e}}\right)^{\hat{1}}\in[0,255]\), (\(\hat{1}=1,\,2,\,3,\,4\)) and \(v_{1}^{\hat{1}}\in[0,255]\)--through the iteration. They are determined as follows [28]: \[\left(v_{1}^{\text{e}}\right)^{\hat{1}}=mod\left\{\left[\frac{\left[\left[ \hat{y}_{1}^{\hat{1}}\right]-\left[\hat{y}_{1}^{\hat{1}}\right]\right]\ast 10^{ 15}\right]}{10^{8}},255\right\},\qquad\text{i}=\ 1,2,3,4\] (17)  
\[\left(v_{1}^{\hat{1}}\right)^{\hat{1}}=mod\left(\left|mod\ \left\{\frac{\left[\left(\left|\hat{y}_{1}^{\hat{1}} \right|-\left|\hat{y}_{1}^{\hat{1}}\right|\right)\ast 10^{15}\right]}{10^{8}} \right\}\right|,255\right),\qquad\text{i}=\ 1,2,3,4 \tag{18}\]  
Here, \(mod(. )\) indicates the modulo operation and \(\left[.\right]\) indicates the flooring process, which rounds the element to the closest integer to negative infinity. These key values are connected with Equation (19) to become a vector, \(v^{\hat{1}}\):  
\[v^{\hat{1}}=\ v_{1}^{\text{e}}+\ v_{2}^{\text{e}}+v_{3}^{\text{e}}+v_{4}^{ \text{e}}+v_{1}^{6}+v_{2}^{6}+v_{3}^{6}+v_{4}^{6} \tag{19}\]
4. After the entire iteration, these sequences are connected with the next equation to obtain \(k\), \(k=\left[v^{1},\,v^{2},\,\dots,v^{m\times n}\right]\). \(k_{\hat{1}}\) can be used to represent one element in \(k\), \(\hat{1}\in[1,8 mn]\).