## 2 Methods
### Quantum Proxy Signature Scheme
#### 2.3.4 Verifying Phase  
In verifying phase, [PERSON] is required to verify the validity of the completed proxy signature and the integrality and authenticity of the conveyed messages based on chosen verifying algorithm with the assistance of [PERSON], who plays the role of the trusted third party to facilitate the interaction of [PERSON] and [PERSON].  
**Step 1** After receiving \(|\phi_{A}\rangle\), [PERSON] encrypts \(|S_{A}\rangle\) and \(|\varphi\rangle_{M^{\prime\prime}}\) using \(K_{BT}\) and gets \(|\phi_{B}\rangle\), i.e.,  
\[|\phi_{B}\rangle=E^{\prime}_{K_{BT}}(|S_{A}\rangle,|\varphi\rangle_{M^{\prime \prime}}), \tag{31}\]  
which is then transmitted to [PERSON].  
**Step 2** [PERSON] decodes \(|\phi_{B}\rangle\) to get \(|S_{A}\rangle\) and \(|\varphi\rangle_{M^{\prime\prime}}\). To begin with, with the acquired random sequence \(S\) by Steps 4-6 in the signing phase, [PERSON] can perform the same unitary operators on \(|\varphi\rangle_{M^{\prime\prime}}\) and obtains \(|\varphi\rangle_{M^{\prime}}^{\text{out}}\). He compares it with the obtained \(|\varphi\rangle_{M^{\prime}}\) from [PERSON] to verify whether the warranty delegated by [PERSON] to [PERSON] is valid. If the warranty is consilient, [PERSON] implements the associated unitary operations based on \(S\) on \(|\varphi\rangle_{M^{\prime}}\) or \(|\varphi\rangle_{M^{\prime\prime}}^{\text{out}}\) to regain \(|\varphi\rangle_{M^{\prime\prime}}\) and achieves \(|S_{T}\rangle\) with \(K_{AT}\). Afterwards, he compares \(|S_{T}\rangle\) with \(|S_{A}\rangle\) by using swap test technique [51], where the independent comparisons of \(|S_{T}\rangle_{i}\) and \(|S_{A}\rangle_{i}\) for \(n\) times are required. Thus, if the value of \(n\) is proper, for any \(\varepsilon>0\), the error probability can be reduced to \([\frac{1}{2}(1+\delta^{2})]^{n}<\varepsilon\). It acts the same when [PERSON] implements the comparison of the quantum states \(|\varphi\rangle_{M^{\prime\prime}}\) and \(|\varphi\rangle_{M^{\prime\prime}}^{\text{out}}\) to verify the completed signature in the later step. If the result \(\tau\) is negative, the communication is terminated. Otherwise, [PERSON] firstly decrypts \(|S_{T}\rangle\) to gain \(|\varphi\rangle_{M^{\prime\prime}}\) and then encrypts \(|S_{A}\rangle\), \(|\varphi\rangle_{M^{\prime\prime}}\) and \(|\tau\rangle\) to generate \(|\phi_{T}\rangle\), i.e.,  
\[|\phi_{T}\rangle=E^{\prime}_{K_{BT}}(|S_{A}\rangle,|\varphi\rangle_{M^{\prime \prime}},|\tau\rangle), \tag{32}\]  
which is delivered to [PERSON].  
**Step 3** [PERSON] decrypts the received \(|\phi_{T}\rangle\) and achieves \(|S_{A}\rangle\), \(|\varphi\rangle_{M^{\prime\prime}}\) and \(|\tau\rangle\). If \(\tau=0\), it shows that \(|S_{A}\rangle\) is disavowed or forged by some manner. That is, \(|S_{A}\rangle\) is invalid and the protocol will be abandoned. Otherwise, [PERSON] compares \(|\varphi\rangle_{M^{\prime\prime}}\) and \(|\varphi\rangle_{M^{\prime\prime}}^{\text{out}}\), which is obtained from [PERSON] via teleportation protocol based on quantum walks on circles with two coins described in Section 2.2. If \(|\varphi\rangle_{M^{\prime\prime}}^{\text{out}}\
eq|\varphi\rangle_{M^{\prime \prime}}\), the communication fails. If \(|\varphi\rangle_{M^{\prime\prime}}^{\text{out}}=|\varphi\rangle_{M^{\prime \prime}}\), [PERSON] makes a request for announcing the random sequence \(S\) from [PERSON].  
**Step 4** [PERSON] publishes \(S\) on the public channel.  
**Step 5** After receiving \(S\), [PERSON] decodes \(|\varphi\rangle_{M^{\prime}}^{\text{out}}\) or \(|\varphi\rangle_{M^{\prime\prime}}\) and obtains the whole original message \(|\varphi\rangle_{M^{\prime}}\), in which the \(i\)th qubit \(|\varphi_{i}\rangle_{M^{\prime}}\) with \(m_{i}=1\) reveals \(M_{i}=0\) and \(|\varphi_{i}\rangle_{M^{\prime}}\) with \(m_{i}=-1\) reveals \(M_{i}=1\). At this time, [PERSON] can recognize \((|S_{A}\rangle,S)\) as [PERSON]'s completed proxy signature.