## 4 Numerical Simulation  
In numerical simulations of practical implementations, groups of \(t\) legitimate participants from all participants are designated with different values and geographical positions. We assume that the distance of the dealer and the farthest legitimate participant is \(\hbar\) and all the other \(t-1\) participants are randomly distributed. The minimum secure key rate is seen as the secure key rate of the proposed CVQSS system. Because there exists the excess noise \(\epsilon\) for every participant, the farthest legitimate participant is the one that achieves the minimum secure key rate of the dealer. The secret rate can be derived as [31]  
\[K=\kappa I_{AB}-\chi_{BE}, \tag{7}\]where \(\kappa\) is the reconciliation efficiency, \(I_{AB}\) expresses the [PERSON] mutual information shared between [PERSON] and [PERSON], and \(\chi_{BE}\) represents upper bound information accessible to [PERSON] (including the other \(t-1\) participants and eavesdropper) on [PERSON]'s secret key. The channel transmittance of the \(l^{th}\) participant can be defined as  
\[T_{l}=10^{-\frac{7p_{B}h}{10}}, \tag{8}\]  
where \(\gamma\) is the attenuation coefficient of the quantum channel, \(\rho_{l}\) represents the fiber length ratio of the \(l^{th}\) participant and the farthest participant to the dealer. In addition, the excess noise of all legitimate \(t\) participants is given by [29; 39]  
\[\varepsilon_{t}=\sum_{l=1}^{t}\frac{T_{l}}{T_{h}}\varepsilon, \tag{9}\]  
with \(T_{h}=10^{-\frac{7h}{10}}\). Using the channel input from [PERSON] as reference, \(\frac{T_{l}}{T_{h}}\varepsilon\) represents the \(l^{th}\) participant's excess noise. In this case, the excess noise is connected with the channel transmittance. The channel-added noise can be described as  
\[\chi_{f}=\frac{1}{T_{h}}-1+\varepsilon_{t}. \tag{10}\]  
Consequently, the total noise referred to the channel input between [PERSON] and [PERSON] can be given by  
\[\chi_{tot}=\chi_{f}+\frac{\chi_{h}}{T_{h}}, \tag{11}\]  
with \(\chi_{h}=[1+(1-\eta)+2v_{el}]/\eta\), where \(\eta\) is an efficiency and \(v\) denotes the noise owing to detector electronics. Moreover, the mutual information of [PERSON] and [PERSON] is given by  
\[I_{AB}=log_{2}\frac{V+\chi_{tot}}{1+\chi_{tot}}, \tag{12}\]  
where \(V=V_{A}+1\) and \(V_{A}\) is the modulation variance of [PERSON]. The maximum information accessible to [PERSON] on [PERSON]'s secret key is the Holevo quantity, which can be derived as  
\[\chi_{BE}=\sum_{i=2}^{2}G(\frac{\lambda_{i}-1}{2})-\sum_{i=3}^{5}G(\frac{ \lambda_{i}-1}{2}), \tag{13}\]  
where \(G(x)=(x+1)log_{2}(x+1)-xlog_{2}(x)\). The symplectic eigenvalues \(\lambda_{1,2}\) can be calculated as  
\[\lambda_{1,2}=\sqrt{\frac{\Delta\pm\sqrt{\Delta^{2}-4D}}{2}}, \tag{14}\]  
where \(\Delta=V^{2}(1-2T_{h})+2T_{h}+T_{h}^{2}(V+\chi_{f})^{2}\) and \(D=T_{h}^{2}(V\chi_{f}+1)^{2}\). Furthermore, the symplectic eigenvalues \(\lambda_{3,4}\) can be calculated as  
\[\lambda_{3,4}=\sqrt{\frac{A\pm\sqrt{A^{2}-4B}}{2}}, \tag{15}\]  
with the notations  
\[ \begin{split} A=\left\{\Delta\chi_{h}^{2}+D+1+2\chi_{h}[V\sqrt{ D}+T_{h}(V+\chi_{f})]+2T_{h}(V^{2}-1)\right\}/[T_{h}(V+\chi_{tot})]^{2},\\ B=(\frac{V+\sqrt{D}\chi_{h}}{T_{h}(V+\chi_{tot})})^{2}.\end{split}  \tag{16}\]The last symplectic eigenvalue \(\lambda_{5}\) is equal to 1. As mentioned above, the channel transmittance of the \(l^{th}\) legitimate participant changes with different fiber length to the dealer. Since the secret key rate is relevant to distance between each legitimate participant and the dealer, different kinds of distribution have a great effect on calculating the secure key rate. Because \(\rho_{l}\) represents the fiber length ratio of the \(l^{th}\) participant and the farthest participant to the dealer, different kinds of distribution can be described by different kinds of \(\rho_{l}\). As shown in Figure 3, we achieve the relation of the secure key rate and the fiber length at different kinds of distribution and identical number of legitimate participants \(t=20\).