# Continuous Variable Quantum Secret Sharing with Fairness
## 2 CVQSS Scheme with Fairness
### \((2,2)\)-CVQSS Scheme with Fairness
#### 2.3.2 Distribution  
Dealer distributes \(X^{A}\), \(X^{B}\) to the participants [PERSON], [PERSON] respectively, via continuous variable quantum deterministic key distribution based on two-mode squeezed states protocol [21]. In another words, each communication is from one Sender (Dealer) to one Receiver ([PERSON] or [PERSON]). The communication is briefly described as follow. **D1** Every Receiver prepares \(L+L_{1}+L_{2}\) two-mode squeezed vacuum states \(a_{1}=x_{1}+ip_{1}\) and \(a_{2}=x_{2}+ip_{2}\) as Figure 2. Here  
\[x_{1} = [e^{r}\hat{x}_{1}^{(0)}+e^{-r}\hat{x}_{2}^{(0)}]/\sqrt{2},\] \[p_{1} = [e^{-r}\hat{p}_{1}^{(0)}+e^{r}\hat{p}_{2}^{(0)}]/\sqrt{2},\] \[x_{2} = [e^{r}\hat{x}_{1}^{(0)}-e^{-r}\hat{x}_{2}^{(0)}]/\sqrt{2},\] \[p_{2} = [e^{-r}\hat{p}_{1}^{(0)}-e^{r}\hat{p}_{2}^{(0)}]/\sqrt{2}, \tag{5}\]  
where \(a_{in1}=\hat{x}_{1}^{(0)}+i\hat{p}_{1}^{(0)}\) and \(a_{in2}=\hat{x}_{2}^{(0)}+i\hat{p}_{2}^{(0)}\) are two vacuum states and \(\hat{x}_{1,2}^{(0)},\hat{p}_{1,2}^{(0)}\sim N(0,1)\) follow Gaussian distribution and \([\hat{x}_{1,2}^{(0)},\hat{p}_{1,2}^{(0)}]=2i\). As the squeezed parameter \(|r|\) increases, the correlation between \(a_{1}\) and \(a_{2}\) becomes increasingly perfect, i.e.,  
\[\lim_{r\rightarrow+\infty}x_{1}=x_{2},\quad\lim_{r\rightarrow+\infty}p_{1}=-p _{2}. \tag{6}\]  
**D2** Receiver keeps \(a_{1}\) at home and sends Sender \(a_{2}\) with some coherent states \(c=|x^{c}+ip^{c}\rangle\) for checking eavesdropping. After receiving the whole state \(a_{3}\), Sender sends back an acknowledge. Following Receiver's instructions, Sender accurately selects out and measures the coherent states, so as to check eavesdropping. If the error rate exceeds certain threshold, receiver goes back to **D1**. In this paper, the strategy for checking eavesdropping is the same as above, so, it will be written as eavesdropper detection for short. **D3** According to the message \(X^{A/B}\), Sender modulates \(a_{3}\) by \(D(\alpha_{j})\) to obtain \(a_{4}\). Here \(\alpha=y+iy\), \(y\sim N(X^{A/B},\sigma^{2})\) follows the Gaussian distribution and \(\sigma^{2}\) is the variance of message. **D4** Sender sends back \(a_{4}\) to Receiver with some coherent states. After receiving \(a_{5}\), Receiver does eavesdropper detection under Sender's help. If the channel is insecure, they give up this communication and go back to **D1**. **D5** Receiver plays a gain on \(a_{6}\) before joint Bell Measurement on \(a_{1}\) and \(a_{6}\) to obtain the message \(X^{A/B}\). From Figure 2, it is shown that the joint Bell measurement consists one balanced beamsplitter (BS) and two detectors using homodyne measurement. Figure 1: The generating of message \(X\) using secret \(S\), checking sequence \(R\) and determine pointer \(P*\). 2.3. Recovery  
When [PERSON] and [PERSON] wants to rebuild secret \(S\), they exchange their own shares. Dishonest one may refuse to send her or his correct share, after receiving the other one's. In order to avoid this situation, this protocol applies some strategies to achieve fairness, in another words, all participants can or cannot acquire the secret simultaneously. In this part, firstly, [PERSON] and [PERSON] generate random number sequences \(A,B\in\{0,1,...,M\}^{(L+L_{1}+L_{2})}\) to encrypt their shares as \((X_{j}^{A/B}+A_{j}/B_{j})\) mod \((M+1),j\in[1,L+L_{1}+L_{2}]\), respectively. Then [PERSON] and [PERSON] exchange their encrypted messages as steps (D1) to (D5) and obtain the measurement results \(M_{e}^{A}\) and \(M_{e}^{B}\). At last, they decrypt \(M_{e}^{A}\) and \(M_{e}^{B}\) to reconstruct \(X\) and verify them one by one number, which is described below in detail. **V1** Define \(j\) is the round of secret reconstruction and the initial value j = 0. **V2** j = j + 1.