# Quantum Dual Signature with Coherent States Based on Chained Phase-Controlled Operations
## 3 Results
### Impossibility of Forgery  
If the attacker [PERSON] attempts to forge [PERSON]'s signatures \(|S\rangle_{O}\) or \(|S\rangle_{P}\), she has to know the corresponding secret keys \(K_{AB}\) or \(K_{AT}\). However, the original keys are shared with unconditionally secure QKD protocol [55], it is impossible for her to eavesdrop the secret keys distributed between signatory and verifiers. Moreover, the final keys are obtained according to the rules which are consensuses among legal participants. The attacker is completely unaware of the rules, so [PERSON] cannot know the keys \(K_{AB}\) and \(K_{AT}\). Thus, we describe two theorems of our encryption algorithm which are used to valid the security of our signature scheme. **Theorem 1**. : _Several errors in the final secret key \(K\) may cause an obvious deviation in the result of encryption._  
For example, if \(n=4,K=\{3,1,2,4\}\) and \(|M\rangle=|\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3}|-\alpha \rangle_{4}\), the result of encryption is  
\[ \begin{split}|C\rangle&=PCO(M_{4},M_{4})PCO(M_{3},M_{2}PCO(M_{2},M_{1})PCO(M_{1},M_{3}))\\ &\quad(|\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3} |-\alpha\rangle_{4})\\ &=|-\alpha\rangle_{1}|\alpha\rangle_{2}|-\alpha\rangle_{3}|- \alpha\rangle_{4}.\end{split}  \tag{15}\]  
When the key is changed into \(K=\{1,3,2,4\}\), the result will become,  
\[ \begin{split}|C\rangle&=PCO(M_{4},M_{4})PCO(M_{3},M_{2})PCO(M_{2},M_{3})PCO(M_{1},M_{1})\\ &\quad(|\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3} |-\alpha\rangle_{4})\\ &=|\alpha\rangle_{1}|i\alpha\rangle_{2}|-i\alpha\rangle_{3}|- \alpha\rangle_{4}.\end{split}  \tag{16}\]  
According to Equations (15) and (16), we realize that any error in the key may lead to a chain reaction in the encryption result. In Table 3, we enumerate all the encryption results based on different keys in the case of \(|M\rangle=|\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3}|-\alpha \rangle_{4}\). In accordance with Table 3, we know that any result is different from others. So the Theorem 1 is reasonable. Figure 2: The probability that all bits of the original key are 0 or 1 as a function of the length of the original key. It shows that the larger \(n\) is, the smaller probability is. Such as \(n\) is greater than 10-bit, the probability of \(P=(\frac{1}{2})^{n}\) approaches to 0. **Theorem 2**. : _Errors in the message \(|M\rangle\) may cause an obvious deviation in the result of encryption._  
We also suppose \(n=4\) and \(K=\{3,1,2,4\}\). However, the message \(|M\rangle=|\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3}|-\alpha \rangle_{4}\) is replaced by \(|M\rangle=|i\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3}|-\alpha \rangle_{4}\). Then the result of encryption is  
\[ \begin{split}|C\rangle&=PCO(M_{4},M_{4})PCO(M_{3},M _{2})PCO(M_{2},M_{1})PCO(M_{1},M_{3})\\ &\quad(|i\alpha\rangle_{1}|-i\alpha\rangle_{2}|i\alpha\rangle_{3} |-\alpha\rangle_{4})\\ &=|-i\alpha\rangle_{1}|i\alpha\rangle_{2}|-i\alpha\rangle_{3}|- \alpha\rangle_{4}.\end{split}  \tag{17}\]  
Comparing Equation (15) with Equation (17), it is easy to prove that a discrepancy in message may cause some errors in the ciphertext. Therefore, the Theorem 2 is tenable. According to the above two Theorems, it is concluded that any changes to the message or the key may lead to a wrong signature and the attack can be definitely detected. Once the forgery behavior is discovered, the honest correspondents may terminate this communication. Even if [PERSON] intercepts part of \(K_{AB}\)\((K_{AT})\), she may fail to forge the signature of message \(M_{O}\) (\(M_{P}\)) because [PERSON] blinds her order message (payment message) based on the secret key \(K_{RO}\) (\(K_{RP}\)). Therefore, only [PERSON] knows the complete information about original messages. The operation that blinding the message is benefit to prevent the recipient or the eavesdropper forging messages in quantum signature procedures.