## Security Analysis
### Holevo Bound of the Stolen Information  
Since the results of the \(i\)th Bob's mode are utilized as the reference, the eavesdropper [PERSON]'s stolen information is defined by the Holevo information \(H(E:B_{i})\) between [PERSON] and the result of the \(i\)th Bob. Since [PERSON] has the ability to purify the entire system, the state \(\rho_{E\rho_{2}}\) that describes the system after transmission through the quantum links is pure, where \(E\) denotes the measurement results of [PERSON]'s restored modes \(\hat{E}_{1,\cdots,N}^{\prime}\). After the users' homodyne measurement, the conditional state \(\rho_{E\rho^{\prime}|B_{i}}\) is still pure, where \(B^{\prime}\) denotes the measurement results of all the users. Hence, the Holevo bound of the stolen information is given by  
\[H(E:B_{i})=S(\rho_{E})-S(\rho_{E|B_{i}})=S(\rho_{2})-S(\rho_{B^{\prime}|B_{i}}), \tag{33}\]  
where \(S(\rho)\) denotes the Holevo entropy of state \(\rho\). In order to derive the Holevo entropy \(S(\rho_{2})\), we calculate the symplectic eigenvalues of the CM \(\mathbf{V}_{4}\), which is the eigenvalues of the [PERSON] normal form \(|I\Omega\mathbf{V}_{4}|\)[25; 26], where \(I\) is the imaginary unit and \(\Omega\) is the symplectic form  
\[\Omega=\bigoplus_{i=1}^{N} \begin{pmatrix}0&1\\ -1&0\end{pmatrix} . \tag{34}\]  
We derive the single N-degenerate symplectic eigenvalues of \(\mathbf{V}_{4}\), which is described by  
\[\
u=\sqrt{[\eta e^{-r}+(1-\eta)e^{r}\omega][\eta e^{r}+(1-\eta)e^{-r}\omega]}. \tag{35}\]  
The von Neumann entropy \(S(\rho_{2})\) is generated from the \(N\) symplectic eigenvalues \(\
u\) by the following equation  
\[S(\rho_{2})=Nh(\
u), \tag{36}\]  
where  
\[h(x)=\frac{x+1}{2}log_{2}\frac{x+1}{2}-\frac{x-1}{2}log_{2}\frac{x-1}{2}. \tag{37}\]The conditional CM \(\mathbf{V}_{B^{\prime}|B_{i}}\) describes the received modes of the Bobs after the \(i\)th [PERSON] performs homodyne detection on his mode. We reconstruct the CM \(\mathbf{V}_{4}\) in the following matrix:  
\[\mathbf{V}_{4}^{\prime}= \begin{pmatrix}\mathbf{A}_{1}&\mathbf{C}_{1}\\ \mathbf{C}_{1}&\mathbf{B}_{1}\end{pmatrix} , \tag{38}\]  
where \(\mathbf{B}_{1}\) describes the mode of the \(i\)th Bob, \(\mathbf{A}_{1}\) describes all the other Bobs' modes and \(\mathbf{C}_{1}\) describes the relations between the above two blocks. After the homodyne detection on the \(p\)-quadrature of the \(i\)th [PERSON]'s mode, the conditional CM \(\mathbf{V}_{B^{\prime}|B_{i}}\) is given by  
\[\mathbf{V}_{B^{\prime}|B_{i}}=\mathbf{A}_{1}-\mathbf{C}_{1}\left(\Pi\mathbf{B }_{1}\Pi\right)\mathbf{C}_{1}^{T}. \tag{39}\]  
Similarly, we derive the symplectic eigenvalues of \(\mathbf{V}_{B^{\prime}|B_{i}}\), which contain \(N-1\) identical symplectic eigenvalues given by Equation (36). The conditional von Neumann entropy \(S(\rho_{B}|B_{i})\) is given by  
\[S(\rho_{B}|B_{i})=(N-1)h(\
u). \tag{40}\]  
Then, the Holevo bound of [PERSON]'s stolen information is derived from the above equations  
\[H([PERSON]:B_{i})=h(\
u). \tag{41}\]  
In addition, we consider the Holevo bound when the users perform heterodyne detection on their received modes. The symplectic eigenvalues of \(\mathbf{V}_{B^{\prime}|B_{i}}\) after the heterodyne detecion include \(N-2\) symplectic eigenvalues given by Equation (36), and one given by  
\[\
u_{N}=\sqrt{\frac{\tau_{1}\tau_{2}}{\lambda_{1}\lambda_{2}}}, \tag{42}\]  
where  
\[ \begin{split}&\tau_{1}=N(1-\eta)\omega\left(\eta+\eta e^{4\tau}+e^{2 \tau}\right)+\eta[e^{4\tau}+N\eta e^{2\tau}+(N-1)]+N(1-\eta)^{2}e^{2\tau} \omega^{2},\\ &\tau_{2}=N(1-\eta)\omega\left(\eta+\eta e^{4\tau}+e^{2\tau} \right)+\eta[(N-1)e^{4\tau}+N\eta e^{2\tau}+1]+N(1-\eta)^{2}e^{2\tau}\omega^{2 },\\ &\lambda_{1}=(N-1)\eta+e^{2\tau}[N(1-\eta)\omega+\eta e^{2\tau}+N],\\ &\lambda_{2}=\eta+e^{2\tau}[N(1-\eta)\omega+(N-1)\eta e^{2\tau}+N].\end{split}