## 2 Materials and Methods
### Physical-Layer Security Analysis: Colluding Eavesdroppers
\begin{c}\mu_{e}^{(k)}\\ -\left|z_{e}^{2}\prod_{j=1}^{n_{e}^{(k)}}\mu_{ej}^{(k)}\left(1+\kappa_{ej}^{( k)}\right)\right.\end{subarray}, \tag{5}\]  
\[\text{where }\beta_{e}^{(k)}=\mu_{e1}^{(k)}-\mu_{e2}^{(k)}+r_{1}^{(k)}-r_{2}^ {(k)}+1,\cdots,\mu_{e1}^{(k)}-\mu_{en_{e}}^{(k)}+r_{1}^{(k)}-r_{ne}^{(k)}+1,1 \text{ and }\]  
\[a_{2}^{(k)}=2\prod_{j=1}^{n_{e}^{(k)}}\left[\frac{\left[\mu_{ej}^{(k)}\left(1 +\kappa_{ej}^{(k)}\right)\right]^{\mu_{e1}^{(k)}-p_{ej}^{(k)}+r_{1}^{(k)}-r_{ j}^{(k)}}h_{ej}^{(k)}}{\kappa_{ej}^{(k)}\frac{p_{ej}^{(k)}-1}{2}\exp\left( \kappa_{ej}^{(k)}\mu_{ej}^{(k)}\right)\Gamma\left(r_{j}^{(k)}+\mu_{ej}^{(k)} \right)}\right]\prod_{j=1}^{n_{e}^{(k)}}\left[\frac{\left[2\mu_{ej}^{(k)} \sqrt{\kappa_{ej}^{(k)}\left(1+\kappa_{ej}^{(k)}\right)}\right]^{2\gamma_{j} ^{(k)}+p_{ej}^{(k)}-1}}{\left(r_{j}^{(k)}\right)!2^{2^{\gamma_{j}}+p_{ej}^{( k)}-1}}\right]\]  
\[\times\prod_{j=1}^{n_{e}^{(k)}}\left[\left(1+\kappa_{ej}^{(k)}\right)^{\frac{p_ {ej}^{(k)}+1}{2}}\right].\]  
The CDF of the RV \(Z_{E,k}\) is expressed as  
\[F_{Z_{E,k}}(z_{e})=\sum_{r_{1}^{(k)}=0}^{\infty}\sum_{r_{2}^{(k)}=0}^{\infty} \cdots\sum_{r_{ne}^{(k)}=0}^{\infty}\frac{a_{2}^{(k)}}{2}z_{e}^{2\left(\mu_{e1} ^{(k)}+r_{1}^{(k)}\right)}G_{1}^{\ n_{e}^{(k)}+1} \begin{subarray}{c}1\\ 1\end{subarray} \ \begin{subarray}{c}1-\mu_{e1}^{(k)}-r_{1}^{(k)}\\ s^{(k)}\end{subarray} \left|z_{e}^{2}\prod_{j=1}^{n_{e}^{(k)}}\mu_{ej}^{(k)} \left(1+\kappa_{ej}^{(k)}\right)\right.\end{subarray}, \tag{6}\]  
\[\text{where }s^{(k)}=-\mu_{e1}^{(k)}+\mu_{e2}^{(k)}-r_{1}^{(k)}+r_{2}^{(k)}, \cdots,-\mu_{e1}^{(k)}+\mu_{en_{e}}^{(k)}-r_{1}^{(k)}+r_{n_{e}}^{(k)},0,-\mu_{e1 }-r_{1}^{(k)}\ \text{. The SNR at Bob is given by }\gamma_{B}=|Z_{N}|^{2}\frac{P}{N_{0}}\text{ with the following being the PDF and the CDF of }\gamma_{B}\]  
\[f_{\gamma_{B}}(\gamma)=\sum_{v_{1}=0}^{\infty}\sum_{v_{2}=0}^{\infty}\cdots \sum_{v_{N}=0}^{\infty}\frac{a_{1}}{2}\left(\frac{\prod_{i=1}^{N}E\left[X_{i} ^{2}\right]}{\tilde{\gamma_{B}}}\right)^{\mu_{1}+v_{1}}G_{N}^{\ \ 0} \begin{subarray}{c} 0\\ 0\end{subarray} \begin{c} N\\ 0\end{subarray}\begin{c}\epsilon\\ \gamma\prod_{i=1}^{N}E\left[X_{i}^{2}\right]\mu_{i}(1+\kappa_{i})\end{subarray} \times\gamma^{\mu_{1}+v_{1}-1}, \tag{7}\]  
\[F_{\gamma_{B}}(\gamma)=\sum_{v_{1}=0}^{\infty}\sum_{v_{2}=0}^{\infty}\cdots \sum_{v_{N}=0}^{\infty}\frac{a_{1}}{2}\left(\gamma\frac{\prod_{i=1}^{N}E\left[X_ {i}^{2}\right]}{\tilde{\gamma_{B}}}\right)^{\mu_{1}+v_{1}}\]  
\[\times G_{1}^{\ N} \begin{subarray}{c}1\\ N+1\end{subarray}\begin{subarray}{c}1\\ 1\end{subarray}\begin{subarray}{c}1-\mu_{1}-\
u_{1}\\ \rho\end{subarray} \left[\frac{\gamma\prod_{i=1}^{N}E\left[X_{i}^{2}\right]\mu_{i} (1+\kappa_{i})}{\tilde{\gamma_{B}}}\right), \tag{8}\]where \(\tilde{\gamma}_{B}\) is the average received SNR at Bob. The eavesdropper utilizes the MRC on the received messages. Hence, the received SNR at E is given by \(\gamma_{E}=\sum_{i=1}^{K}\gamma_{E,i}=\sum_{i=1}^{K}\lvert Z_{E,i}\rvert^{2}\frac{ p}{N_{0}}\). Using [27] and (5), the PDF of \(\gamma_{E}\) is given by  
\[f_{\gamma_{E}}(\gamma_{e})=\sum_{r_{1}=0}^{\infty}\sum_{r_{2}=0}^ {\infty}\cdots\sum_{r_{n_{e}}=0}^{\infty}\frac{c_{x,e}}{2}G\left(\frac{\prod_{ j=1}^{n_{e}}E\left[X_{j}^{2}\right]}{\gamma_{E}K}\right)^{\mu_{\alpha}K+r_{1}} \gamma_{e}^{\mu_{\alpha}K+r_{1}-1}\] \[\times G \begin{subarray}{c}0&n_{e}\\ n_{e}&0\end{subarray}