# Continuous Variable Quantum Secret Sharing with Fairness
## 3 Security Analysis
### Internal Attack
At last, [PERSON] measures \(a_{E1}^{\prime}\) and \(a_{E2}^{\prime}\) and gets  
\[a_{E}=\frac{1}{\sqrt{2}}(a_{E1}^{\prime}-a_{E2}^{\prime}). \tag{22}\]  
According to the equations before, \(x_{E}\) can be expressed as  
\[x_{E}=\frac{1}{\sqrt{2}}(\frac{\sqrt{\eta_{1}}}{\sqrt{1-\eta_{2}}}x_{N1}- \frac{\sqrt{\eta_{2}}}{\sqrt{\eta_{1}(1-\eta_{2})}}x_{N2}-\frac{\sqrt{1-\eta_ {1}}}{\sqrt{\eta_{1}}}x_{N2}+\frac{1}{\eta_{1}}y). \tag{23}\]  
So, the signal variance of \(x_{E}\) is  
\[V_{E}=\frac{\sigma^{2}}{2\eta_{1}} \tag{24}\]  
and the noise variance of \(x_{E}\) is  
\[N_{E}=\frac{1}{2}(\frac{\eta_{1}}{1-\eta_{2}}\Sigma_{1}^{2}+\frac{\eta_{2}}{ \eta_{1}(1-\eta_{2})}\Sigma_{2}^{2}+\frac{1-\eta_{1}}{\eta_{1}})\Sigma_{2}^{2}. \tag{25}\]  
So, the SNR of \(x_{E}\) is  
\[\gamma_{E}=\frac{V_{E}}{N_{E}} \tag{26}\]  
and the mutual information between sender and Eve is  
\[I(S,E)=\frac{1}{2}\log_{2}(1+\gamma_{E}). \tag{27}\]  
Therefore, the channel capacity \(C\) or the information rate \(\Delta I\) is  
\[x_{E1}=\sqrt{\eta_{1}}x_{N1}-\sqrt{1-\eta_{1}}x_{2},~{}~{}p_{E1}=\sqrt{\eta_{ 1}}p_{N1}-\sqrt{1-\eta_{1}}p_{2}, \tag{19}\]0  
Suppose the two quantum channels are the same with \(\eta_{1}=\eta_{2}=\eta\) and \(\Sigma_{1}=\Sigma_{2}=\Sigma\). According to the analysis above, the mutual information between sender and [PERSON] is draw in \[x_{E1}=\sqrt{\eta_{1}}x_{N1}-\sqrt{1-\eta_{1}}x_{2},~{}~{}p_{E1}=\sqrt{\eta_{ 1}}p_{N1}-\sqrt{1-\eta_{1}}p_{2}, \tag{19}\]1 It is obvious that setting lower noise variance of \(a_{N1}\) and \(a_{N2}\), \(\Sigma_{2}^{2}\), [PERSON] can obtain more information. Moreover, the growth of \(\sigma^{2}\) also can increase \(I(S,E)\), but \(r\) has little effect on it. Then, when \(\Sigma^{2}=1\), the relationship between information rate \(\Delta I\) and transmission efficiency \(\eta\) is exhibited in \[x_{E1}=\sqrt{\eta_{1}}x_{N1}-\sqrt{1-\eta_{1}}x_{2},~{}~{}p_{E1}=\sqrt{\eta_{ 1}}p_{N1}-\sqrt{1-\eta_{1}}p_{2}, \tag{19}\]2 Under different conditions, such as \(\sigma^{2}=1,r=1,\sigma^{2}=1,r=5,\sigma^{2}=28,r=1\) and \(\sigma^{2}=28,r=5\), it is clear that the positions of \(\Delta I=0\) are all very closed to \((0.43,0)\). When \(\eta<0.43\), \(\Delta I<0\) and [PERSON] can acquire more information than legal communicators, so, the communication is insecure. When \(\eta>0.43\), \(\Delta I>0\) and \(\Delta I\) increases with the growth of \(\sigma^{2}\) or \(r\), especially \(\sigma^{2}\) can invite lots of improvement on \(\Delta I\). In order to transfer the secret shares successfully (\(\Delta I\geq\log_{2}3\approx 1.6\)), the requirement is rather more stringent, for example, when \(\sigma^{2}=28,r=5\), \(\eta\) need be greater than 0.79. Furthermore, Figure 7 depicts the relation between the squeezed parameter \(r\) and the variance of message \(\sigma^{2}\) for satisfying \(\Delta I=\log_{2}3\), under conditions \(\eta=\{1,0.9,0.8\}\). Obviously, when \(r>3\), the demand of \(\sigma^{2}\) approaches to a stable value, which is seriously related with \(\eta\) of BSs. Compared with Figure 3, the requirement of \(\sigma^{2}\) an \(r\) is much rougher. Because the loss of information is collected and used by [PERSON], rather than lost merely. Although the loss of channel is inevitable and [PERSON] always exists, some strategies can be adopted to avoid [PERSON] utilizing the lost or stolen information to infer the real secret. For example, encrypt the secret with one time pad generated by CVQKD [9], which is discussed in Ref. [26]. In this way, the \(I(S,E)\) would be the same, but [PERSON] can not exact any useful information about secret, which means no secret information leakage. So, the protocol would be feasible and secure under the low quantum channel transmission efficiency, only if the requirement of \(\sigma^{2}\) an \(r\) can be achieved. \[x_{E1}=\sqrt{\eta_{1}}x_{N1}-\sqrt{1-\eta_{1}}x_{2},~{}~{}p_{E1}=\sqrt{\eta_{ 1}}p_{N1}-\sqrt{1-\eta_{1}}p_{2}, \tag{19}\]3 The mutual information of [PERSON].