## 4 The Security of QOT
### Privacy for [PERSON]
#### 4.2.1 The Attack that [PERSON] Sends Only One State Dishonestly in R-OT Protocol  
In Step (3) of Protocol 1, an honest [PERSON] is supposed to send the state \(|\Psi_{0}\rangle\) or \(|\Psi_{1}\rangle\) only. However, a malicious [PERSON] may replace one of the pulse (\(|\Psi_{r_{c}}\rangle\)) with \(|\Psi_{0}\rangle^{\perp}\) instead. This attack makes [PERSON] more likely to accept this pulse as a conclusive results and [PERSON] has a higher probability to distinguish which set is Set \(I\). Then, we analyze whether this attack is effective both in R-OT and \(OT_{1}^{2}\) protocols. In Protocol 1, when [PERSON] replaces one of the pulses (\(|\Psi_{r_{c}}\rangle\)) with \(|\Psi_{0}\rangle^{\perp}\), the number of the photons in the pulse follows Poisson distribution. Let \(|n_{\frac{\pi}{2}}\rangle\) denotes an n-photon state with the polarization \(\frac{\pi}{2}\). For an honest [PERSON], he chooses the measurement basis \(B_{1}\) with the probability of \(1/2\). When he measures \(|1_{\frac{\pi}{2}}\rangle\) with \(B_{1}\), the probability that the state collapse to \(|1_{\frac{\pi}{2}}\rangle\) is \(\langle\frac{\pi}{2}|\frac{2\pi}{3}\rangle\langle\frac{2\pi}{3}|\frac{2}{2} \rangle=3/4\). When \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\mu\) & \(p_{con}(\mu)\) & \(p_{con}^{\prime}(\mu)\) & \(I_{obt}\) & \(k\) & \(\varepsilon_{2}\) & \(p_{1t}\) & \(p_{1}\) & \(p_{2}\) \\ \hline
2 & 0.197 & 0.285 & 143 & 131 & 0.0944 & 0.117 & 0.107 & \(3.25\times 10^{-6}\) \\
3 & 0.264 & 0.395 & 190 & 172 & 0.122 & 0.0887 & 0.0484 & \(1.85\times 10^{-6}\) \\
4 & 0.316 & 0.488 & 228 & 210 & 0.147 & 0.0621 & 0.0312 & \(1.53\times 10^{-6}\) \\
5 & 0.357 & 0.567 & 260 & 236 & 0.164 & 0.0435 & 0.0324 & \(7.45\times 10^{-7}\) \\
6 & 0.388 & 0.634 & 283 & 259 & 0.178 & 0.0267 & 0.0236 & \(4.73\times 10^{-6}\) \\ \hline \end{tabular}
\end{table} 
Table 1: When \(p=20\%\), \(N=800\), \(p_{2}\) is controlled to be a magnitude of \(10^{-6}\), the values of parameters with different \(\mu\). he measures \(|n_{\frac{\pi}{2}}\rangle\) with \(B_{1}\), the probability that at least one of the photons of \(|n_{\frac{\pi}{2}}\rangle\) collapse to the state with polarization of \(\frac{2\pi}{3}\) is \(1-(\frac{1}{4})^{n}\). According to Equation (5), the probability of choosing the basis \(B_{1}\) and getting a conclusive bit in a pulse is  
\[p_{B_{1}con}(\mu)=\sum_{n=1}\left\{\frac{1}{2}\left[1-\left(\frac{1}{4}\right) ^{n}\right]\frac{e^{-\mu}\mu^{n}}{n!}\right\}=\frac{1}{2}\left(1-e^{-\frac{3 \mu}{4}}\right). \tag{12}\]  
When [PERSON] chooses the measurement basis \(B_{0}\) to detect the fake pulse, if there is only one photon in the pulse, the probability that he accepts it as conclusive pulse is 100%. The probability of choosing the basis \(B_{0}\) and getting a conclusive bit in a pulse is  
\[p_{B_{0}con}(\mu)=\frac{1}{2}(1-e^{-\mu}). \tag{13}\]  
Therefore, when [PERSON] replaces one of the pulses with \(|\Psi_{0}\rangle^{\perp}\), the average probability of [PERSON] getting a conclusive result is  
\[p_{(i)}=p_{B_{0}con}(\mu)+p_{B_{1}con}(\mu)=1-\frac{1}{2}e^{-\mu}-\frac{1}{2} e^{-\frac{3\mu}{4}}. \tag{14}\]  
Consequently, [PERSON] accepts the fake pulse as a conclusive result with a larger probability of \(p_{(i)}\) than the situation where [PERSON] is honest. In the following, we will analyze that although the cheating [PERSON] has a larger probability to know [PERSON]'s choice \(m\), she still has no idea what is got by [PERSON]. In standard \(OT_{1}^{2}\), if [PERSON] has a probability larger than the legal threshold of knowing [PERSON]'s choice, she breaks [PERSON]'s privacy. In this paper, Protocol 2 is the block of constructing QBC. The security that requests [PERSON] cannot know what is obtained by [PERSON] is enough. It can be seen that the security is weaker than the standard \(OT_{1}^{2}\). Therefore, we call it weak quantum \(OT_{1}^{2}\), and [PERSON] attacking the weak \(OT_{1}^{2}\) successfully means that she knows the content of the message obtained by [PERSON].