## 2 Methods
### Quantum Proxy Signature Scheme
#### 2.3.3 Signing Phase
The second motivation originates from the existing quantum signature schemes, such as chain-based CNOT [49], key-controlled chained CNOT [50], which make the encrypted qubit related to not only the qubit and the key of the current position but also other qubits and keys of other positions. We thus design the improved QOTP encryption algorithm by introducing assistant CNOT gates. In addition, considering the storage space and usage efficiency of the secret key, we circularly use the key to accomplish the encryption task, the length of which is reduced into one third of the key required in the QOTP algorithm. Therefore, in terms of both security and efficiency, we put forward the QOTP-CNOT encryption algorithm, in which we write the required \(K_{AT}\) as  
\[K_{AT}=\{K_{AT}^{1},K_{AT}^{2},\ldots,K_{AT}^{i},\ldots,K_{AT}^{n}\}. \tag{22}\]  
The length of \(K_{AT}\) is assumed to be the same with the length of \(|\varphi\rangle_{M^{\prime\prime}}\), i.e., \(n\). [PERSON] executes the corresponding operation on the qubit \(|\varphi_{i}\rangle_{M^{\prime\prime}}\) in \(|\varphi\rangle_{M^{\prime\prime}}\) according to the values of \((K_{AT}^{i},K_{AT}^{i+1},K_{AT}^{i+2})\), where \(K_{AT}^{i}\) decides whether \(\sigma_{x}\) is operated on the corresponding qubit with \((\sigma_{x})^{i}_{AT}\) and \(K_{AT}^{i+1}\) controls the operation of \(\sigma_{z}\) with \((\sigma_{z})^{K_{AT}^{i+1}}\) and \(K_{AT}^{i+2}\) determines whether the CNOT operation is applied on \(|\varphi_{i}\rangle_{M^{\prime\prime}}\) with \(|S_{A}\rangle_{i-1}\) acting as the control qubit. The encryption process can be expressed as follows,  
\[|S_{A}\rangle_{i}= \begin{cases}|\varphi_{i}\rangle_{M^{\prime\prime}}\otimes( \sigma_{x})^{K_{AT}^{i}}\otimes(\sigma_{z})^{K_{AT}^{i+1}}_{AT}\otimes(\sigma _{x})^{K_{AT}^{i+2}}_{AT}|S_{A}\rangle_{i-1}&i\
eq 1,n-1,n.\\ |\varphi_{i}\rangle_{M^{\prime\prime}}\otimes(\sigma_{x})^{K_{AT}^{i}}\otimes( \sigma_{z})^{K_{AT}^{i+1}}_{AT}\otimes(\sigma_{x})^{K_{AT}^{i+2}}_{AT}|\varphi _{n}\rangle_{M^{\prime\prime}}&i=1.\\ |\varphi_{i}\rangle_{M^{\prime\prime}}\otimes(\sigma_{x})^{K_{AT}^{i}}\otimes( \sigma_{z})^{K_{AT}^{i+1}}_{AT}\stackrel{{\text{mod}}}{{ {}_{\sim}}}\otimes(\sigma_{x})^{|S_{A}\rangle_{i-1}}_{AT}K_{AT}^{(i+2)} \stackrel{{\text{mod}}}{{{}_{\sim}}}&i=n-1,n.\end{cases}  \tag{23}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \((S_{i},S_{i+1})\) & **Unitary Operator** & **Matrix Representation** \\ \hline
00 & \(I\) & \(\left(\begin{array}{cc}1&0\\ 0&1\\ \end{array}\right)\) \\
01 & \(\sigma_{x}\) & \(\left(\begin{array}{cc}0&1\\ 1&0\\ \end{array}\right)\) \\
10 & \(\sigma_{y}\) & \(\left(\begin{array}{cc}0&-i\\ i&0\\ \end{array}\right)\) \\
11 & \(\sigma_{z}\) & \(\left(\begin{array}{cc}1&0\\ 0&-1\\ \end{array}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The relationship of \((S_{i},S_{i+1})\) in the sequence \(S\) and the corresponding unitary operation. Here let us consider an instance to expound on it. Given \(n=8\), \(|\varphi\rangle_{M^{\prime\prime}}=|01011101\rangle\) and \(K_{AT}=01011001\), in line with Equation (23), the encryption result of \(|\varphi\rangle_{M^{\prime\prime}}\) should be \(|00000100\rangle\), where the implementation of \(|\varphi_{2}\rangle_{M^{\prime\prime}}=|1\rangle\) can be demonstrated as follows. According to the associated key \((K_{AT}^{2},K_{AT}^{3},K_{AT}^{4})=(1,0,1)\), we can obtain the qubit \(|S_{A}\rangle_{2}\) with the form of  
\[|S_{A}\rangle_{2}=|\varphi_{2}\rangle_{M^{\prime\prime}}\otimes(\sigma_{x})^{ K_{AT}^{2}}\otimes(\sigma_{z})^{K_{AT}^{3}}\otimes(\sigma_{x})^{K_{AT}^{4} \cdot|S_{A}\rangle_{1}}=|\varphi_{2}\rangle_{M^{\prime\prime}}\otimes\sigma_{ x}\otimes(\sigma_{x})^{|S_{A}\rangle_{1}}=|0\rangle, \tag{24}\]  
which coincides with the presented encryption and decryption processes of \(|\varphi\rangle_{M^{\prime\prime}}\) shown in Figures 4 and 5.