# Multi-Party Controlled Semi-Quantum Dialogue Protocol Based on Hyperentangled Bell States
## 2 MCSQD Protocol
### Protocol Process  
Step 1: [PERSON]\({}_{1}\) prepares \(N\) identical hyperentangled photon pairs in the polarization-spatial mode, denoted as \(\left|\,\Phi\right\rangle_{AB}^{PS}=\left.\left|\,\varphi^{+}\right\rangle^{P} \otimes\left.\left|\,\varphi^{+}\right\rangle^{S}\right.\). The particles A and B are separated to form sequences \(S_{A}\) and \(S_{B}\), respectively. Subsequently, [PERSON]\({}_{1}\) randomly applies unitary operations \(R=\{I,X,H\}\) for the first n(n \(<N\)) particles in \(S_{A}\) at each DOF. The sequence \(S_{A}\) is then sequentially transmitted to [PERSON]\({}_{2}\), [PERSON]\({}_{3}\), \(\ldots\), [PERSON]\({}_{n}\), while recording the operations performed. [PERSON]\({}_{n}\) sends \(S_{A}\) to the quantum party [PERSON] and retains \(S_{B}\). Step 2: To prevent Trojan horse attacks [31; 32; 33], [PERSON] and [PERSON] place a photon number splitter (PNS) and a wavelength filter [20] in front of their devices. Upon confirming the receipt of the sequence by [PERSON], she randomly selects a sufficient number of particles from the remaining \(N-n\) in the \(S_{A}\) to serve as decoy particles for security checks. [PERSON] measures these particles using either the Z basis or X basis across the two DOF and collaborates with [PERSON]\({}_{n}\) to calculate the quantum bit error rate (QBER). If the error rate is below a pre-defined threshold, [PERSON] removes the decoy particles. Subsequently, the controllers execute hierarchical authorization, requiring all independent controllers [PERSON]\({}_{i}(i=1,\ldots,n)\) to collectively authorize permission for [PERSON] and [PERSON] to communicate. [PERSON] applies the same unitary operation \(R\) to recover the initial hyperentangled state \(\left|\,\Phi\right\rangle_{AB}^{PS}\). Finally, [PERSON]\({}_{n}\) transmits \(S_{B}\) to [PERSON]. [PERSON]\({}_{i}(i=1,\ldots,n)\) will not publish or send any information if communication is not permitted. Step 3: After confirming that [PERSON] has received the sequence, [PERSON] initiates measurements on the first \(n\) particles of \(S_{B}\) using the measurement basis \(Z_{M}=Z_{P}\otimes Z_{S}\). Subsequently, [PERSON] employs Huffman compression coding to compress the source information \(m_{B}\) into a binary sequence \(M_{B}\). Based on the value of each bit \(M_{B}^{i}\), [PERSON] prepares corresponding qubits in the two DOF. If \(M_{B}^{i}=0\), [PERSON] prepares the same quantum state in the respective DOF; otherwise, he prepares the opposite quantum state. For example, if \(M_{B}=001011\), [PERSON] will sequentially prepare separable single-photon states: \(\left|\,\mathrm{H}\right\rangle^{P}\left|\,\mathrm{b}_{1}\right\rangle^{S}\), \(\left|\,\mathrm{V}\right\rangle^{P}\left|\,\mathrm{b}_{1}\right\rangle^{S}\), and \(\left|\,\mathrm{V}\right\rangle^{P}\left|\,\mathrm{b}_{2}\right\rangle^{S}\), which are then sent to [PERSON] as Code particles. From the remaining \(N-n\) particles, [PERSON] selects decoy particles for the subsequent security check and reflects them directly to [PERSON] as Reflect particles. Once ready, [PERSON] reorders and records the Reflect and Code particles before transmitting them to [PERSON]. Step 4: [PERSON] first announces the correct order of the Reflect particles. [PERSON] performs measurements on these particles and collaborates with [PERSON]\({}_{n}\) to calculate the QBER. If the error rate is below a pre-defined threshold, [PERSON] removes the Reflect particles. Step 5: [PERSON] then announces the correct order of the Code particles. [PERSON] restores the order and measures the Code particles along with her retained \(S_{A}\) using the measurement basis \(Z_{M}=Z_{P}\otimes Z_{S}\) to obtain \(M_{B}\). Finally, [PERSON] applies Huffman decompression to recover the original source information \(m_{B}\). To facilitate the conversation, [PERSON] also encodes her secret message \(m_{A}\) using Huffman compression into a binary string \(M_{A}\).