## 2 Methods
### Quantum Proxy Signature Scheme
#### 2.3.3 Signing Phase  
In signing phase, [PERSON] generates the proxy signature based on chosen signing algorithm, which is expected to ensure [PERSON]'s undeniability, the integrity and authenticity of the message to be signed. **Step 1** [PERSON] randomly chooses from \(\{0,1\}\) to generate an \(n\)-bit classical sequence  
\[S=\{S_{1},S_{2},\ldots,S_{i},\ldots,S_{n}\}. \tag{16}\]  
**Step 2** [PERSON] encrypts \(|\varphi\rangle_{M^{\prime}}\) through appropriate encryption algorithm based on \(S\) and obtains \(|\varphi\rangle_{M^{\prime\prime}}\) with  
\[|\varphi\rangle_{M^{\prime\prime}}=E_{S}(|\varphi\rangle_{M^{\prime}}). \tag{17}\]  
For an arbitrary qubit \(|\varphi_{i}\rangle_{M^{\prime}}\) in \(|\varphi\rangle_{M^{\prime}}\), it can be expressed as follows,  
\[|\varphi_{i}\rangle_{M^{\prime\prime}}=E_{S}(|\varphi_{i}\rangle_{M^{\prime}} )=|\varphi_{i}\rangle_{M^{\prime}}\otimes(I)^{\widetilde{S_{i}}\cdot \widetilde{S}_{i+1}}\otimes(\sigma_{x})^{\widetilde{S_{i}}\cdot S}_{i+1} \otimes(\sigma_{y})^{S_{i}\cdot\widetilde{S}_{i+1}}\otimes(\sigma_{z})^{S_{i} \cdot S_{i+1}}, \tag{18}\]  
where \(i+1=(i+1)\mod n\). Concretely, according to \((S_{i},S_{i+1})\) in \(S\), [PERSON] performs the corresponding unitary operator on the qubit \(|\varphi_{i}\rangle_{M^{\prime}}\)[33], which is listed in Table 2. The relationship of \(|\varphi\rangle_{M^{\prime}}\), \(S\) and \(|\varphi\rangle_{M^{\prime\prime}}\) is shown in Figure 3, in which it can be seen that the operation on the last qubit \(|\varphi_{n}\rangle_{M^{\prime}}\) of \(|\varphi\rangle_{M^{\prime}}\) is controlled by \((S_{n},S_{1})\), which shows the random sequence is used circularly. Then [PERSON] needs to broadcast the value of \(n\). It should be noted that, in our scheme, three copies of \(|\varphi\rangle_{M^{\prime\prime}}\) are required. One of them is employed to create proxy signature state, the other is delivered to [PERSON] along with the proxy signature state at the last step in the signing phase and the third one is transmitted by teleportation based on quantum walks on circles with two coins described in Section 2.2. Figure 3: [PERSON] packages \(|\varphi\rangle_{M^{\prime}}\) with unitary operation \(U\) governed by the random sequence \(S\) to obtain \(|\varphi\rangle_{M^{\prime\prime}}\). **Step 3** [PERSON] generates proxy signature state \(|S_{A}\rangle\) by encrypting \(|\varphi\rangle_{M^{\prime\prime}}\) with \(K_{AT}\), i.e.,  
\[|S_{A}\rangle=E^{\prime}_{K_{AT}}(|\varphi\rangle_{M^{\prime\prime}}), \tag{19}\]  
where \(E^{\prime}_{K_{AT}}\) refers to the improved QOTP algorithm with assistant CNOT gates in terms of \(K_{AT}\). Before elaborating on the QOTP-CNOT operation, we first describe the QOTP algorithm in the following, where \(2n\) random classical bits are required for the encryption of an unknown \(n\)-qubit quantum state with the guarantee of informational security [47; 48]. Denote \(|\phi\rangle\) as the \(n\)-qubit message expected to be encrypted with \(|\phi_{i}\rangle=\alpha_{i}|0\rangle+\beta_{i}|1\rangle\) and \(K_{q}\) as the \(2n\)-bit key. The encryption process is formulated as  
\[|\phi\rangle_{C}=E_{K_{q}}(|\phi\rangle)=\mathop{\otimes}\limits_{i=1}^{n}( \sigma_{x})^{K_{q}^{2i}}(\sigma_{z})^{K_{q}^{2i-1}}_{q}|\phi_{i}\rangle, \tag{20}\]  
where \(K_{q}^{p}\) is the \(p\)th bit of \(K_{q}\). The corresponding decryption process is  
\[|\phi\rangle=D_{K_{q}}(|\phi\rangle_{C})=\mathop{\otimes}\limits_{i=1}^{n}( \sigma_{z})^{K_{q}^{2i-1}}(\sigma_{x})^{K_{q}^{2i}}_{q}|\phi_{i}\rangle_{C}. \tag{21}\]  
Then the motivation of the QOTP-CNOT operation includes the following two aspects. The first aspect is from some classical coding schemes, such as the differential encoding, where the encrypted bits correlate with the other bits in order to improve its capacity of resisting disturbance.