# Experimental Quantum Message Authentication with Single Qubit Unitary Operation
## 2 Quantum Message Authentication Protocol
### Verification Phase  
The verification phase is divided into five stages: \"quantum decryption\", \"[PERSON]'s encryption\", \"QMAC recovery\", \"[PERSON]'s decryption\", and \"swap test\". In Stage 1, for quantum decryption, [PERSON] uses secret key sequences \(K_{AB}\) and \(K_{H}\), which were pre-shared with [PERSON] to decrypt the encrypted QMAC state pair \(\left|\,M\right\rangle_{u}\Big{[}\otimes_{i=1}^{N}\sigma_{i_{AB}}^{\mu}H^{k_{ H}^{i}}\left|\,S\right\rangle_{d}\Big{]}\) in Equation (5), received from [PERSON] to obtain the QMAC state pair \(\left|\,M\right\rangle_{u}\left|S\right\rangle_{d}\) of Equation (4). In Stage 2, [PERSON]'s encryption, [PERSON] generates his own private key sequence \(B=(b_{1},\ b_{2},\ \ldots,\ b_{N})\) and re-encrypts quantum state \(\left|\,S\right\rangle_{d}=\otimes_{i=1}^{N}R_{y}(s_{i})\left|\,M\right\rangle _{d}\) with it to obtain quantum state \(\left|\,S^{\prime}\right\rangle_{d}=\otimes_{i=1}^{N}R_{y}(b_{i})\left|\,S \right\rangle_{d}\). Then, he sends \(\left|\,S^{\prime}\right\rangle_{d}\) to [PERSON] while keeping the other quantum message state \(\left|\,[PERSON]\right\rangle_{u}\). In Stage 3, QMAC recovery, [PERSON] uses her own private key sequence \(S\) to apply rotation operator \(\otimes_{i=1}^{N}R_{y}(-s_{i})\) to quantum state \(\left|\,[PERSON]^{\prime}\right\rangle_{d}\) and sends quantum state \(\left|\,[PERSON]^{\prime\prime}\right\rangle_{d}=\otimes_{i=1}^{N}R_{y}(-s_{i})\left| \,[PERSON]^{\prime}\right\rangle_{d}\) to [PERSON]. In Stage 4, [PERSON]'s decryption, [PERSON] uses his own private key sequence \(B\) and applies rotation operator \(\otimes_{i=1}^{N}R_{y}(-b_{i})\) to quantum state \(\left|\,S^{\prime\prime}\right\rangle_{d}\) to obtain quantum message state \(\left|\,M^{\prime}\right\rangle_{d}=\otimes_{i=1}^{N}R_{y}(-b_{i})\left|\,S^{ \prime\prime}\right\rangle_{d}\). Because the proposed quantum message authentication based on the quantum three-pass protocol operates [PERSON]'s private key \(s_{i}\), there is a need for a method to verify the encrypted QMAC pair described thus far. This is an important element that the proposed protocol can guarantee the origin of quantum message. In addition, to avoid counterfeiting, it is assumed that quantum encryption such as \(\sigma_{i_{AB}^{\prime}}H^{k_{H}^{i}}\) in Equation (5) is applied to [PERSON] and [PERSON] in every process of exchanging quantum states. In the final stage, [PERSON] performs the swap test [42; 45] twice to verify the QMAC state pair. In the first swap test, [PERSON] verifies whether quantum message state \(\left|\,M\right\rangle_{u}\) and quantum message state \(\left|\,M^{\prime}\right\rangle_{d}\) are the same. If the test result reveals that \(\left|\,M\right\rangle_{u}\) and \(\left|\,M^{\prime}\right\rangle_{d}\) agree, [PERSON] accepts QMAC state pair \(\left|\,M\right\rangle_{u}\left|\,S\right\rangle_{d}\) sent by [PERSON]. Otherwise, he does not accept it. In the second swap test, [PERSON] generates quantum state \(\left|\,M^{\prime\prime}\right\rangle\) corresponding to the public bit message sequence \(M\) and verifies that it matches quantum message state \(\left|\,M\right\rangle_{u}\) or \(\left|\,M^{\prime}\right\rangle_{d}\). If the test result reveals that (\(\left|M^{\prime\prime}\right\rangle\), \(\left|M\right\rangle_{u}\)) or (\(\left|M^{\prime\prime}\right\rangle\), \(\left|M^{\prime}\right\rangle_{d}\)) agree, then the integrity of QMAC state pair \(\left|M\right\rangle_{u}\left|S\right\rangle_{d}\) is verified completely. For the second swap test, it is noted that the first swap test requires a non-demolition swap test.