## 2 System Model  
This work considers a base station composed of \(M\) uncorrelated antennas and \(K\) uncorrelated users, as shown in Figure 1. The base station sends the same message to all users; however, it applies a normalized precoding vector for each one of the fading channels. The signal received by the uncorrelated antenna array at the user side is  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]  
where \(\mathbf{H}\in\mathbb{C}^{M\times N}\) is the channel between each one of the uncorrelated antennas at the base station and each one of the LIS reflectors, \(\mathbf{G}\in\mathbb{C}^{N\times K}\) is the channel between the LIS reflectors and the users, \(\mathbf{\Phi}\in\mathbb{C}^{N\times N}\) is the phase matrix with the phase shifts applied by the LIS in the incoming signals, and \(\mathbf{H}_{d}\in\mathbb{C}^{M\times K}\) is the direct link between the base station and each user. The eavesdropper link is \(\mathbf{w}\in\mathbb{C}^{M\times 1}\) and is only considered for the secrecy outage probability analysis. Therefore, it was neglected in the first analysis. One possible strategy to cancel the channel phase is to apply a precoding matrix at the base station; in this case, the sub-optimal solution is the normalized hermitian of the overall channel. Let  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]  
whose dimensions are \(\mathbf{Y}\in\mathbb{C}^{M\times K}\), and the elements of the matrix \(\mathbf{Y}\) are the \(\mathbf{v}_{k}\) vectors. The scalar elements of \(\mathbf{H}\) have a Nakagami distribution with parameters \(m_{H}\) and \(\Omega_{H}\), the elements of \(\mathbf{G}\) have a Nakagami distribution with parameters \(m_{G}\) and \(\Omega_{G}\), and the elements of \(\mathbf{H}_{d}\) follow the complex normal distribution with mean zero. The mean of a variable \(T\) with Nakagami-\(m\) distribution with parameters \(m\) and \(\Omega\) can be calculated by  
\[E[T]=\frac{\Gamma\left(m+\frac{1}{2}\right)}{\Gamma(m)}\left(\frac{\Omega}{m} \right)^{\frac{1}{2}}, \tag{3}\]  
and the second-order moment \(E\big{[}T^{2}\big{]}=\Omega\). The transmitted symbols after the application of the precoder are  
\[\mathbf{\
abla}=P\mathbf{A}\mathbf{s}, \tag{4}\]  
where \(P\) is the power gain and \(A\) is the precoding matrix, and, for analysis purposes, the \(\mathbf{s}\) symbols are generated as complex normal distributed \(\mathbf{s}\sim CN(0,\mathbf{I}_{K})\). The sub-optimal precoding matrix, considering complete knowledge of the Channel State Information (CSI), is given by  
\[\mathbf{A}=\sqrt{P}\frac{\mathbf{Y}}{\mathbf{Y}_{F}}, \tag{5}\]  
Figure 1: System model with an eavesdropper link. where \(._{F}\) is the Frobenius norm. The precoding vector for each user is given by  
\[\mathbf{a}_{k}=\sqrt{\frac{P}{K}}\frac{\mathbf{v}_{k}}{\mathbf{v}_{k}}, \tag{6}\]  
and each one of the \(\mathbf{v}_{k}\) has dimensions \(\mathbf{v}_{k}\in\mathbb{C}^{M\times 1}\). The scalar definition of each element of the \(\mathbf{v}_{k}\) vectors is  
\[v_{lk}=\sum_{p=1}^{N}\sum_{q=1}^{N}h_{lp}\phi_{pq}\delta_{qk}+h_{(d),kl}, \tag{7}\]  
after the application of the LIS, these coefficients can be rewritten as  
\[v_{lk}=\sum_{p=1}^{N}\sum_{q=1}^{N}\left|h_{lp}\right|\Big{|}g_{qk}\Big{|}e^{i \left(\delta_{(k),lp}+\delta_{(g),kl}-\phi_{pq}\right)}+h_{(d),kl}, \tag{8}\]  
where \(\delta_{(h),lp}=arg(h)\), \(\phi_{pq}=arg(\phi_{pq})\), and \(\delta_{(g),kk}=arg(g_{qk})\) are the phases of each channel. To nullify the overall channel phase, the phase shift applied by the LIS panel must be equal to  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]0  
where \(\delta_{pq}\) is the residual error of the LIS phase correction.