## 2 PWC-Based \(m\)CAP-DM with 2D Chaotic Encryption
### DM with 2D Chaotic Encryption  
The principle of DM with 2D chaotic encryption is described as follows. As can be seen from Figure 1a, the proposed 2D chaotic encryption scheme is performed with respect to each \(m\)CAP subblock, which mainly consists of the following two chaotic scrambling processes: one is the chaotic constellation scrambling (CCS) and the other is the chaotic mode scrambling (CMS). In each subblock, CCS and CMS are performed via the chaotic constellation mapper and the chaotic index selector, respectively. For the \(m\)CAP-DM system with two distinguishable constellation sets, i.e., \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\), the CCS process is conducted with respect to each constellation set. Letting \(\mathbf{p}_{g}^{1}=[p_{g,1}^{1},p_{g,2}^{1},\cdots,p_{g,M_{1}}^{1}]^{T}\) and \(\mathbf{p}_{g}^{2}=[p_{g,1}^{2},p_{g,2}^{2},\cdots,p_{g,M_{2}}^{2}]^{T}\) denote, respectively, the corresponding permutation vectors for \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) in the \(g\)th subblock with \(g=1,\cdots,G\), the resultant constellation sets corresponding to \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) after performing CCS are expressed as follows:  
\[\mathcal{M}_{1}^{\text{CCS}}=\text{src}\{\mathcal{M}_{1},\mathbf{p}_{g}^{1} \}=[S_{p_{g,1}^{1}}^{1},S_{p_{g,2}^{1}}^{1},\cdots,S_{p_{g,M_{1}}}^{1}]^{T}, \tag{8}\]  
\[\mathcal{M}_{2}^{\text{CCS}}=\text{src}\{\mathcal{M}_{2},\mathbf{p}_{g}^{2} \}=[S_{p_{g,1}^{2}}^{2},S_{p_{g,2}^{2}}^{2},\cdots,S_{p_{g,M_{2}}}^{2}]^{T}, \tag{9}\]  
where \(\text{src}\{\cdot,\cdot\}\) denotes the scrambling function that scrambles \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) according to their respective permutation vectors \(\mathbf{p}_{g}^{1}\) and \(\mathbf{p}_{g}^{2}\), where the permutation vector contains the new positions of the constellation points in the corresponding constellation set. Moreover, the CMS process is performed with respect to the two constellation modes during the DM within each subblock. In the DM without applying CMS, each constellation mode corresponds to a fixed constellation set. However, in the CMS-encrypted DM, the corresponding relationship between two constellation modes, i.e., mode 1 and mode 2, and two encrypted constellation sets \(\mathcal{M}_{1}^{\text{CCS}}\) and \(\mathcal{M}_{2}^{\text{CCS}}\) is scrambled according to a binary scrambling vector. Letting \(\mathbf{s}=[s_{1},s_{2},\cdots,s_{C}]^{T}\) denote the binary scrambling vector for the \(m\)CAP block with a total of \(G\) subblocks, the corresponding relationship between two constellation modes and two encrypted constellation sets in the \(g\)th (\(g=1,\cdots,G\)) subblock is given in Table 1. As we can see, when \(s_{g}=0\), the two encrypted constellation sets corresponding to mode 1 and mode 2 are \(\mathcal{M}_{1}^{\text{CCS}}\) and \(\mathcal{M}_{2}^{\text{CCS}}\), respectively. However, when \(s_{g}=1\), the corresponding relationship becomes reversed, i.e., the two encrypted constellation sets corresponding to mode 1 and mode 2 become \(\mathcal{M}_{2}^{\text{CCS}}\) and \(\mathcal{M}_{1}^{\text{CCS}}\), respectively. By performing DM with 2D chaotic encryption, including CCS and CMS, the physical-layer security of \(m\)CAP-DM can be greatly enhanced. In order to successfully perform CCS and CMS, the permutation vectors \(\mathbf{p}_{g}^{1}\) and \(\mathbf{p}_{g}^{2}\) with respect to the \(g\)th subblock and the binary scrambling vector \(\mathbf{s}\) with respect to the \(m\)CAP block with a total of \(G\) subblocks \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{s_{g}=0}\) & \(\mathbf{s_{g}=1}\) \\ \hline Mode 1 & \(\mathcal{M}_{1}^{\text{CCS}}\) & \(\mathcal{M}_{2}^{\text{CCS}}\) \\ Mode 2 & \(\mathcal{M}_{2}^{\text{CCS}}\) & \(\mathcal{M}_{1}^{\text{CCS}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Corresponding relationship of CMS in the \(g\)th subblock. should be obtained in advance.