# Quantum Dual Signature with Coherent States Based on Chained Phase-Controlled Operations
## 4 Discussion  
Quantum dual signature which involves three parties, [PERSON], [PERSON] and [PERSON], is suitable for e-commerce. With QKD protocols, communicators can share secret keys with each other for signature or de-signaturing massage. Assume that the process of QKD is secure, the quantum dual signature scheme may mainly face three threats, i.e., forging, disavowing, denying [PERSON]'s signature. For forging signature, we analyze the situation and discover that when the attackers are strongly powerful, the success probability of forging signature [PERSON]'s is also small. Actually, three methods can be introduced for reducing \(P_{f_{A}}\): one is increasing the length of randomly choosing states in the step 1 of the verification phase, the second is increasing the length of message, and the last is improving the security of QKD. In fact, with the current quantum technologies, attackers find it more difficult to forge the signature since they cannot capture transmitted information, such as \(|S\rangle_{D}\) and \(|M\rangle_{O}^{\prime}\), without any error.  
Denial may lead to delaying or even stoping communication, which should be avoided in practice. In the scheme, a situation that [PERSON] claims he receives nothing but he has already received [PERSON] may occur and it can be alleviated as follows: [PERSON] waits for a moment then she sends another signature to him again, which may lengthen the communication time, while the scheme is still secure because \(K_{RO}\) is only known by [PERSON]. In other words, [PERSON] never sends original keys to [PERSON] unless she accepts these verifications from [PERSON] and [PERSON]. i.e., [PERSON] cannot deny [PERSON]'s signature when he has sent verifications, since the loyal [PERSON] knows the communication process.  
In addition, the simulations of the CPCOs are implemented on Strawberry Fields [49] interactive web app to valid the feasibility of the scheme. The related parameters for the experiment are shown in Table 4, and the schematic is shown in Figure 4. In particular, four input coherent states are designed as \(|0+2i\rangle\), \(|-1+0i\rangle\), \(|4+0i\rangle\) and \(|0+3i\rangle\), and \(K_{final}=\{2,4,1,3\}\). According to the CPCO rule, the encryption process can be stated as  
\[PCO(M_{3},M_{4})PCO(M_{1},M_{3})PCO(M_{4},M_{2})PCO(M_{2},M_{1})| 0+2i\rangle|-1+0i\rangle|4+0i\rangle|0+3i\rangle \tag{22}\] \[=|-2+0i\rangle|1+0i\rangle|0+4i\rangle|0-3i\rangle.\]  
In \(PCO(M_{2},M_{1})|0+2i\rangle|-1+0i\rangle\), for example, the output phase of the first port is shifted by \(\pi\). As a consequence, the control-phase gate \(P_{21}\) in Figure 4 can be set to achieve the goal in the simulation.  
Simulated results are presented in Figure 5 where the subfigures (a), (b), (c), and (d) separately represent the quadrature outputs of the first, the second, the third and the forth port and \(x,p\in(-5,5)\), i.e., the cutoff dimension is 5. It can be seen that the correct quantum coherent states can be acquired with the 80% fidelity. For example, in Figure 5a, the probability of \(x=-2\) is 0.8 and the probability of \(p=0\) is also 0.8. Therefore, the coherent state \(|-2+0i\rangle\) will be output with the high probability 0.8.  
Figure 4: Design schematic on the CPCOs with four input ports whose values are \(|0+2i\rangle\), \(|-1+0i\rangle\), \(|4+0i\rangle\) and \(|0+3i\rangle\) respectively. \(P_{hk}\) represents the PCO where the \(h\)-th coherent state controls the \(k\)-th coherent state. Square dots \(\blacksquare\) in the gray virtual box represents multiple judgement conditions referred to Equation (2). “HD” in the red box is the heterodyne detection [56].