# Continuous Variable Quantum Secret Sharing with Fairness
## 3 Security Analysis
### No Attack  
At first, receiver prepares the initial two-mode squeezed states \((a_{1},a_{2})\), then, send mode \(a_{2}\) to sender through the quantum channel with the additive white Gaussian noise (AWGN), so, \(a_{3}\) can be described as  
\[x_{3}=\sqrt{\eta_{1}}x_{2}+\sqrt{1-\eta_{1}}x_{N1},\ \ p_{3}=\sqrt{\eta_{1}}p_{2}+ \sqrt{1-\eta_{1}}p_{N1}. \tag{7}\]  
Here, \(\eta_{1}\) is the channel transmission efficiency and \(x_{N1}\), \(p_{N1}\sim N(0,\Sigma_{1}^{2})\) presents the channel noise from receiver to sender. Next, \(a_{3}\) is modulated by displacement operation according to message \(X^{A/B}\) and turns to \(a_{4}\) expressed as  
\[x_{4}=x_{3}+y,\ \ p_{4}=p_{3}+y. \tag{8}\]  
Then, \(a_{4}\) is sent back to receiver and becomes \(a_{5}\),  
\[x_{5}=\sqrt{\eta_{2}}x_{4}+\sqrt{1-\eta_{2}}x_{N2},\ \ p_{5}=\sqrt{\eta_{2}}p_{4}+ \sqrt{1-\eta_{2}}p_{N2}, \tag{9}\]  
where \(x_{N2}\), \(p_{N2}\sim N(0,\Sigma_{2}^{2})\) and \(\eta_{2}\) stands for the parameter of the quantum channel from sender to receiver. In order to compensate \(a_{5}\) for lossy in quantum channel, \(a_{5}\) has to be amplified with gain \(g\) before Bell Measurement, so,  
\[x_{6}=gx_{5}. \tag{10}\]  
At last, receiver plays a measurement on \(a_{1}\) and \(a_{6}\) for capturing message \(X^{A/B}\) and the results \(a_{7}\), \(a_{8}\) are  
\[ \begin{split} x_{7}&=\frac{1}{\sqrt{2}}(x_{6}+x_{1} ),p_{7}=\frac{1}{\sqrt{2}}(p_{6}+p_{1}),\\ x_{8}&=\frac{1}{\sqrt{2}}(x_{6}-x_{1}),p_{8}=\frac{ 1}{\sqrt{2}}(p_{6}-p_{1}).\end{split}  \tag{11}\]  
If \(r>0\), using Equations (5), (7)-(11) and setting \(g=\sqrt{\frac{1}{\eta_{1}\eta_{2}}}\), we obtain  
\[x_{8}=\frac{y}{\sqrt{2\eta_{1}}}+\frac{\sqrt{1-\eta_{1}}x_{N1}}{\sqrt{2\eta_{ 1}}}+\frac{\sqrt{1-\eta_{2}}x_{N2}}{\sqrt{2\eta_{1}\eta_{2}}}-e^{-r}x_{2}^{0}. \tag{12}\]  
Obviously, \(x_{8}\) obeys a Gaussian distribution, so, the variance of signal distribution is  
\[V_{s}=\frac{\sigma^{2}}{2\eta_{1}}, \tag{13}\]  
and the variance of noise is  
\[N_{s}=\frac{1-\eta_{1}}{2\eta_{1}}\Sigma_{1}^{2}+\frac{1-\eta_{2}}{2\eta_{1} \eta_{2}}\Sigma_{2}^{2}-e^{-2r}. \tag{14}\]The signal-noise-ratio (SNR) between sender and receiver is  
\[\gamma=\frac{V_{s}}{N_{s}}. \tag{15}\]  
According to the [PERSON] information theory [25], the mutual information is expressed as  
\[I(S,R)=\frac{1}{2}\log_{2}(1+\gamma). \tag{16}\]  
As for the two-mode squeezed state, the amplitude and phase are symmetric and both can be used to transfer message, so, the total mutual information is \(2I(S,R)\) and the channel capacity is also \(2I(S,R)\), when there is no attack. But, for describing more briefly, the messages carried on amplitude and phase are the same in this paper, so, the security is discussed only based on the amplitude, also the channel capacity is seen as \(C=I(S,R)\) accordingly. For satisfying the message transferring successfully, the channel capacity cannot be less than the information of \(X_{i}^{A/B}\), expressed as \(C\geq H(X_{i}^{A/B})\). According to the (2,2)-threshold CVQSS proposed above, suppose \(X_{i}\) is equally distributed in \(\{0,1,\ldots,5\}\), so \(P(X_{i}=0)=P(X_{i}=1)=\cdots=P(X_{i}=5)=\frac{1}{6},i=(1,2,...,L+L_{1}+L_{2})\), From definition of CRT or Table 1, the share of Alice is \(X_{i}^{A}\in\{0,1\}\) and its probability function is  
\[ \begin{cases}P(X_{i}^{A}=0)=P(X_{i}=0)+P(X_{i}=2)+P(X_{i}=4)=\frac{1}{2},\\ P(X_{i}^{A}=1)=P(X_{i}=1)+P(X_{i}=3)+P(X_{i}=5)=\frac{1}{2}.\end{cases}  \tag{17}\]  
Similarly, the share of Bob is \(X_{i}^{B}\in\{0,1,2\}\) and its probability function is  
\[ \begin{cases}P(X_{i}^{A}=0)=P(X_{i}=0)+P(X_{i}=3)=\frac{1}{3},\\ P(X_{i}^{A}=1)=P(X_{i}=1)+P(X_{i}=4)=\frac{1}{3},\\ P(X_{i}^{A}=2)=P(X_{i}=2)+P(X_{i}=5)=\frac{1}{3}.\end{cases}  \tag{18}\]  
Therefore, the information entropy of \(X_{i},X_{i}^{A}\) and \(X^{B}\) are \(H(X_{i})=log_{2}6\)_bit_, \(H(X_{i}^{A})=1\)_bit_ and \(H(X_{i}^{B})=\log_{2}3\)_bit_. Hence, the channel can succeed to send \(X^{A}\) and \(X^{B}\), when \(C=I(S,R)\geq\log_{2}3\)_bit_\(\approx\) 1.6 _bit_.