# Quantum Dual Signature with Coherent States Based on Chained Phase-Controlled Operations
## 2 Methods
### The Chained Phase-Controlled Operations  
As we know, the CNOT operator is a two-qubit gate and its function is given by  
\[|c\rangle|t\rangle\rightarrow|c\rangle|t\oplus c\rangle, \tag{1}\]  
where the two qubits \(|c\rangle\) and \(|t\rangle\) are the control qubit and the target qubit, respectively. The notation \(\oplus\) stands for summation mod 2. The target bit flips when \(|c\rangle=|1\rangle\), otherwise, the target bit stays unchanged. We draw inspiration from CNOT operation and put forward a new encryption algorithm, namely the CPCO. A phase-controlled operation (PCO) can be defined as \(PCO(|M\rangle_{j_{r}}|M\rangle_{K_{j}})\), where \(K_{j}\) is the \(j\)-th value of the final key, \(|M\rangle_{j}\) is the control coherent state and \(|M\rangle_{K_{j}}\) is the target coherent state, and \(|M\rangle_{j},|M\rangle_{K_{j}}\in\{|\alpha\rangle,|-\alpha\rangle,|i\alpha \rangle,|-i\alpha\rangle\}\)[54]. In addition, the coherent state \(|M\rangle_{j}\) satisfies \(|M\rangle_{j}=|x_{j}+ip_{j}\rangle\), where \(x_{j}\in\mathbb{R}\) and \(p_{j}\in\mathbb{R}\) are separately the amplitude and phase of \(|M\rangle_{j}\). The results of PCO operations applied to message \(|M\rangle_{j}\) can be derived as follows. \[PCO(|M\rangle_{j_{r}}|M\rangle_{K_{j}})|M\rangle_{K_{j}}= \begin{cases}P(\frac{ \pi}{2})|M\rangle_{K_{j}},&K_{j}\
eq j,x_{j}\
eq 0\,and\,p_{j}=0,\\ P(\pi)|M\rangle_{K_{j}},&K_{j}\
eq j,x_{j}=0\,and\,p_{j}\
eq 0,\\ P(0)|M\rangle_{K_{j}},&K_{j}=j,\end{cases}  \tag{2}\]  
where \(P(\theta)\) is a phase shift operator which fulfils \(P(\theta)\ =\ \exp\ (i\theta a^{\dagger}\hat{a})\), and \(\theta\in\{\frac{\pi}{2},\pi\}\) is angle of phase changed, \(\hat{a}^{\dagger}\) and \(\hat{a}\) are bosonic creation and annihilation operators, respectively. The 50/50 beam splitter can be used to achieve the measurements of amplitude and phase. Two coherent states \(|\mu\rangle\) and \(|v\rangle\) are input into the beam splitter, then \(|\frac{k+v}{\sqrt{2}}\rangle\) and \(|\frac{\mu-v}{\sqrt{2}}\rangle\) are acquired from modes 1 and 2, respectively. Hence given coherent state \(|\alpha\rangle_{s}\) and \(|M\rangle_{j}\) are input into the 50/50 beam splitter, vacuum states can be acquired from one of the two output modes according to the judgement \(|M\rangle_{j}\) is equal to \(|\alpha\rangle\) or \(|-\alpha\rangle\). Otherwise, both output of modes are not vacuum states. Suppose the message which is composed of coherent states is \(|M\rangle=\otimes_{j=1}^{N}|M\rangle_{j}\), where \(N\) stands for the message length and the secret key is \(K\). Then the encryption process based on the CPCO is deduced as:  
\[ \begin{split}|C\rangle&=E_{K}|M\rangle\\ &=PCO(|M\rangle_{n},|M\rangle_{K_{n}})PCO(|M\rangle_{n-1},|M \rangle_{K_{n-1}})\ldots PCO(|M\rangle_{1},|M\rangle_{K_{1}})|M\rangle,\end{split}  \tag{3}\]  
where \(E_{K}\) means the encryption algorithm. Meanwhile, the corresponding decryption process is  
\[ \begin{split}|M\rangle&=D_{K}|C\rangle\\ &=DPCO(|M\rangle_{1},|M\rangle_{K_{1}})DPCO(|M\rangle_{2},|M \rangle_{K_{2}})\ldots DPCO(|M\rangle_{n},|M\rangle_{K_{n}})|C\rangle,\end{split}  \tag{4}\]  
where \(D_{K}\) and \(DPCO\), stands for decryption algorithm and the phase-controlled operation for decryption respectively. The detailed functions of \(DPCO(|M\rangle_{j},|M\rangle_{K_{j}})\) are  
\[DPCO(|M\rangle_{j},|M\rangle_{K_{j}})|M\rangle_{K_{j}}= \begin{cases}P(-\frac{ \pi}{2})|M\rangle_{K_{j}},&K_{j}\
eq j,x_{j}\
eq 0\,and\,p_{j}=0,\\ P(-\pi)|M\rangle_{K_{j}},&K_{j}\
eq j,x_{j}=0\,and\,p_{j}\
eq 0,\\ P(0)|M\rangle_{K_{j}},&K_{j}=j.\end{cases}  \tag{5}\]  
For example, suppose \(n=4\), the key \(K=\{1,4,2,3\}\) and the message \(|M\rangle=|\alpha\rangle_{1}|-\alpha\rangle_{2}|-i\alpha\rangle_{3}|i\alpha \rangle_{4}\). Then the encoded message is  
\[