## 4 Numerical Simulation
In Figure 3, the secret key rates of different kinds distribution of legitimate participants are the same when the fiber length is short. The longer of the fiber length means more influence of different kinds distribution of legitimate participants. Different of secure key rates increases for the same fiber length. In practical implementations, various tasks need different number of legitimate participants to accomplish for the secret sharing. The number of legitimate participants of one task is determined by the dealer, and the possibility of each participant to participate the task is equal. In this case, the relation of the secure key rate and the fiber length at different number of legitimate participants \(t\) out of identical all participants \(n=25\) is shown in Figure 4. For \(t=5\), the secret key rate decreases with the fiber length increasing. With the number of legitimate participants increasing at the same for \(n=25\), the secret key rate reduces gradually at the same fiber length. When the number of legitimate participants reaches the upper limit for \(t=n=25\), the secret key rate is the smallest at the same fiber length. In this case, it becomes the \((n,n)\)-threshold CVQSS scheme. This scheme can be conducted at 4 km with 25 legitimate participants for \(\varepsilon=0.01\). Figure 5 shows the relations of the secret key rate and the fiber length for different numbers of legitimate participants \(t=5,50,100\) and \(200\) out of identical all participants \(n=200\). For \(\varepsilon=0.001\), the number of legitimate participants can be determined randomly between 2 and 200. We find that this scheme can be conducted at 10 km with 200 legitimate participants. The reconciliation efficiency ranges from 0 when no information was extracted to 1 for a perfect reconciliation scheme. It is connected with the Shannon mutual information shared between [PERSON] and [PERSON]. The relation of the secure key rate and the fiber length at different number of legitimate participants \(t\) and reconciliation efficiency out of identical all participants \(n=50\) is shown in Figure 6. Here, we choose \(t=\{15,30,45\}\), and values of \(\kappa\) are 0.98 and 0.9. The numeral relation can be displayed in Figure 6. It is obviously that by using the higher reconciliation efficiency the secure key rate under the same number of legitimate participants \(t\) is increased in comparison to the lower  
Figure 4: The secure rates of different number of legitimate participants for \(n=25\). The parameters are given by \(V_{A}=6\), \(\gamma=0.2\) dB/km, \(\varepsilon=0.01\), \(\kappa=0.98\), \(\eta=0.6\), and \(v_{el}=0.01\). Figure 5: The secret rates of different numbers of legitimate participants for \(n=200\). The parameters are given by \(V_{A}=6\), \(\gamma=0.2\) dB/km, \(\varepsilon=0.001\), \(\kappa=0.98\), \(\eta=0.6\), and \(v_{el}=0.01\). The attenuation coefficient has influence on the channel transmittance of all participants. We use a similar analytical approach to illustrate the effect of the attenuation coefficient \(\gamma\) on the secure key rate with different number of legitimate participants \(t\). Figure 7 shows that the higher secure key rate and longer maximum transmission distance can be achieved by tuning the value of parameter \(\gamma\) to be smaller. Figure 6: The secret rates of different numbers of legitimate participants and reconciliation efficiency for \(n=50\). The parameters are given by \(V_{A}=6\), \(\gamma=0.2\) dB/km, \(\varepsilon=0.005\), \(\eta=0.6\), and \(v_{el}=0.01\). Figure 7: The secret rates of different numbers of legitimate participants and attenuation coefficient for \(n=50\). The parameters are given by \(V_{A}=6\), \(\varepsilon=0.005\), \(\kappa=0.98\), \(\eta=0.6\), and \(v_{el}=0.01\).