## 2 Materials and Methods  
### Physical-Layer Security Analysis: Colluding Eavesdroppers  
A three-node wiretap system model is shown in Figure 1. In the presence of an eavesdropper \((E)\) equipped with several antennas, it is assumed that the transmitter ([PERSON]) is communicating with a legitimate receiver ([PERSON]) over the main channel. This eavesdropper replaces the multiple colluding eavesdroppers, each equipped with a single antenna [20; 21; 25]. Moreover, the eavesdropper utilizes the maximal ratio combining (MRC) technique over the received messages to improve the received signal-to-noise ratio (SNR). Through the wiretap channel, \(E\) is attempting to capture the messages conveyed between [PERSON] and [PERSON]. Both the main and wiretap channels follow the cascaded \(\kappa\)-\(\mu\) fading model. The received signal at the legitimate receiver ([PERSON]) is given by  
\[y_{m}=\sqrt{P}Z_{N}x+w_{m}, \tag{1}\]  
where \(P\) is the transmit power. \(x\) is the transmitted symbol at Alice and \(w_{m}\) is the additive white Gaussian noise (AWGN) at the receiver with zero mean and variance \(N_{0}\). \(Z_{N}\) is the channel gain for the main link, which is defined by \(Z_{N}=\prod\limits_{i=1}^{N}X_{i}\). \(X_{i}\) is a set of independent \(\kappa\)-\(\mu\) random variables (RVs) with the parameters \(\kappa_{i}\) and \(\mu_{i}\) (\(i\in\{1,2,\cdots,N\}\)). Therefore, \(Z_{N}\) follows cascaded \(\kappa-\mu\) fading with the following probability density function (PDF) [20]  
\[f_{Z_{N}}(z) = \sum\limits_{v_{1}=0}^{\infty}\sum\limits_{v_{2}=0}^{\infty} \cdots\sum\limits_{v_{N}=0}^{\infty}a_{1}z^{2\mu_{1}+2v_{1}-1}G\overset{0}{N} \overset{N}{0}\left(\frac{\epsilon}{\varepsilon}\left|\frac{1}{z^{2}\prod_{i= 1}^{N}\mu_{i}(1+\kappa_{i})}\right.\right), \tag{2}\]  
where \(G\overset{m}{p}\overset{n}{q}\left(\frac{a_{i}}{b}\right.|z)\) is the Meijer G-function defined in [26] (Equation 9-301), \(\epsilon=\mu_{1}-\mu_{2}+v_{1}-v_{2}+1,\cdots,\mu_{1}-\mu_{N}+v_{1}-v_{N}+1,1\), and  
\[a_{1}=2\prod\limits_{i=1}^{N}\left[\frac{[\mu_{i}(1+\kappa_{i})]^{\mu_{1}-\mu _{i}+v_{1}-v_{1}}\mu_{i}(1+\kappa_{i})^{\frac{\kappa_{i}+1}{2}}}{\kappa_{i}^ {\frac{\kappa_{i}-1}{2}}\exp(\kappa_{i}\mu_{i})\Gamma(\
u_{i}+\mu_{i})}\right] \prod\limits_{i=1}^{N}\left[\frac{[2\mu_{i}\sqrt{\kappa_{i}(1+\kappa_{i})}]^{ 2\
u_{i}+\mu_{i}-1}}{(v_{i})!2^{2\
u_{i}+\mu_{i}-1}}\right].\]  
Figure 1: The system model. The cumulative distribution function (CDF) of the RV \(Z_{N}\) is given by [20]  
\[F_{Z_{N}}(z)=\sum_{v_{1}=0}^{\infty}\sum_{v_{2}=0}^{\infty}\cdots\sum_{v_{N}=0}^{ \infty}\frac{a_{1}}{2}z^{2(\mu_{1}+v_{1})}G_{1}^{\ N}\ \begin{subarray}{c}1\\ 1\end{subarray} \ \begin{subarray}{c}1\\ 1\end{subarray} \ \begin{subarray}{c}1-\mu_{1}-v_{1}\\ \rho\end{subarray} \left|z^{2}\prod_{i=1}^{N}\mu_{i}(1+\kappa_{i})\right.\right), \tag{3}\]  
where \(\rho=-\mu_{1}+\mu_{2}-v_{1}+v_{2},\cdots,-\mu_{1}+\mu_{N}-v_{1}+v_{N},0,-\mu_ {1}-v_{1}\). The intercepted message at E is given by  
\[y_{E,k}=\sqrt{P}Z_{E,k}x+w_{E,k}, \tag{4}\]  
where \(w_{E,k}\) is the AWGN at the \(k^{th}\) antenna of E with zero mean and variance \(N_{0}\). \(Z_{E,k}\) is the channel gain for the wiretap link, which is the one between Alice and the \(k^{th}\) antenna of E for \(k=1,2,\cdots,K\). \(K\) is the total number of eavesdroppers, which also represents the number of antennas at E. \(Z_{E,k}\) is defined by \(Z_{E,k}=\prod_{j=1}^{n_{e}}Y_{j}^{(k)}\). \(Y_{j}^{(k)}\) is a set of independent \(\kappa\)-\(\mu\) RVs with the parameters \(\kappa_{ej}^{(k)}\) and \(\mu_{ej}^{(k)}\) (\(j\in\{1,2,\cdots,n_{e}\}\)) for the \(k^{th}\) link. Hence, \(Z_{E,k}\) follows the cascaded \(\kappa-\mu\) fading distribution with the following PDF  
\[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}a_{2}^{(k)}z_{e}^{2\mu_{e}^{(k)}+2r_{1}^{( k)}-1}G_{n_{e}^{(k)}}^{\ \ 0} \begin{subarray}{c}0\\ 0\end{subarray}