## Security Analysis
\begin{cases}-\frac{1}{N}e^{-2r}+\frac{1}{N}e^{2r}&\text{for }i\
eq j,\\ \frac{1}{N}e^{-2r}+\frac{N-1}{N}e^{2r}&\text{for }i=j,\end{cases}  \tag{15}\]  
and  
\[ \begin{pmatrix}\mathbf{V}_{p}^{\prime}\end{pmatrix} _{ij}= \begin{cases}\frac{1}{N}e^{-2r}-\frac{1}{N}e^{2r}&\text{for }i\
eq j,\\ \frac{N-1}{N}e^{-2r}+\frac{1}{N}e^{2r}&\text{for }i=j.\end{cases}  \tag{16}\]  
The eavesdropper [PERSON] prepares an ancillary state \(\rho_{E}\) to perform entangled cloner attacks. Each entangled mode of \(\rho_{E}\) is injected onto a travelling mode \(\hat{\mathbf{a}}_{k}^{\prime}\) of \(\rho_{1}\), respectively. The CM of \(\rho_{E}\) that corresponds to \(\rho_{1}\) is given by  
\[\mathbf{V}_{E}= \begin{pmatrix}\omega\mathbf{I}_{N}&\mathbf{0}\\ \mathbf{0}&\omega\mathbf{I}_{N}\end{pmatrix} , \tag{17}\]  
where \(\omega=1+2n\) denotes the variance of [PERSON]'s ancillary modes. The CM of the whole system before the transmission through the quantum channels is defined by  
\[\mathbf{V}_{2}=\mathbf{V}_{1}\bigoplus\mathbf{V}_{E}. \tag{18}\]  
In order to implement eavesdropping, [PERSON] replaces the noisy channels from the source to all the legitimate parties with noisy-free quantum links and beam splitters, respectively. The transmittance \(\eta\) of each beam splitter of [PERSON] is the same as the original noisy channel it has replaced. The symplectic matrices of [PERSON]'s beam splitters are denoted by \[\mathbf{S}_{k}= \begin{pmatrix}\sqrt{\eta}\mathbf{I}&-\sqrt{1-\eta}\mathbf{I}\\ \sqrt{1-\eta}\mathbf{I}&\sqrt{\eta}\mathbf{I}\end{pmatrix} ,\quad k=1,\cdots,N. \tag{19}\]  
We use the symplectic matrix \(\mathbf{S}_{E}\) to describe the transmission action of the entire system of both the entangled modes from the source and Eve's ancillary modes by Eve's beam splitters  
\[\mathbf{S}_{E}= \begin{pmatrix}\sqrt{\eta}\mathbf{I}_{2N}&-\sqrt{1-\eta} \mathbf{I}_{2N}\\ \sqrt{1-\eta}\mathbf{I}_{2N}&\sqrt{\eta}\mathbf{I}_{2N}\end{pmatrix} . \tag{20}\]  
After the transmission of the legitimate users' modes through Eve's thermal-loss channels and beam splitters, the CM of the whole system is described by  
\[\mathbf{V}_{3}=\mathbf{S}_{E}\cdot\mathbf{V}_{2}\cdot\mathbf{S}_{E}^{\mathrm{ T}}. \tag{21}\]  
We distill the partial matrix \(\mathbf{V}_{4}\) from \(\mathbf{V}_{3}\), which describes the entangled modes of the users' received state \(\rho_{2}\). The form of \(\mathbf{V}_{4}\) is described by  
\[\mathbf{V}_{4}= \begin{pmatrix}\mathbf{\Lambda}&\mathbf{\Gamma}&\cdots& \mathbf{\Gamma}\\ \mathbf{\Gamma}&\mathbf{\Lambda}&\ddots&\mathbf{\Gamma}\\ \vdots&\ddots&\ddots&\vdots\\ \mathbf{\Gamma}&\mathbf{\Gamma}&\cdots&\mathbf{\Lambda}\end{pmatrix} , \tag{22}\]  
with  
\[\mathbf{\Lambda}= \begin{pmatrix}\eta\left(\frac{N-1}{N}e^{2r}+\frac{1}{N}e^{- 2r}\right)+(1-\eta)\omega&0\\ 0&\eta\left(\frac{1}{N}e^{2r}+\frac{N-1}{N}e^{-2r}\right)+(1-\eta)\omega \end{pmatrix} , \tag{23}\]  
and  
\[\mathbf{\Gamma}= \begin{pmatrix}\eta\left(\frac{1}{N}e^{2r}-\frac{1}{N}e^{2r} \right)&0\\ 0&\eta\left(-\frac{1}{N}e^{2r}+\frac{1}{N}e^{2r}\right)\end{pmatrix} . \tag{24}\]  
As a result, the CM of any \(i\)th and \(j\)th Bobs' modes is described by  
\[\mathbf{V}_{5}= \begin{pmatrix}\mathbf{\Lambda}&\mathbf{\Gamma}\\ \mathbf{\Gamma}&\mathbf{\Lambda}\end{pmatrix} . \tag{25}\]