# Multi-Party Controlled Semi-Quantum Dialogue Protocol Based on Hyperentangled Bell States
## 3 Security Analysis
### The Entangle-Measure Attack
The state is then updated to:  
\[\hat{F}\left|\mathrm{\varphi}^{\prime}\right\rangle_{ABe}^{PS}\left|\delta \right\rangle_{e}=\left|\mathrm{\varphi}^{\prime\prime}\right\rangle_{ABe}^{PS} \tag{10}\]  
Specifically:  
\[=\left[\alpha_{1}\left|\mathrm{H}\right\rangle_{A}\left|\mathrm{H}\right\rangle _{B}\left|\delta_{00}\right\rangle_{e}+\beta_{2}\left|\mathrm{V}\right\rangle _{A}\left|\mathrm{H}\right\rangle_{B}\left|\delta_{10}\right\rangle_{e} \right]^{P}\]  
\[\otimes\left[\alpha_{1}{}^{\prime}\left|a_{1}\right\rangle_{A}\left|b_{1} \right\rangle_{B}\left|\delta_{00}\right\rangle_{e}+\beta_{2}{}^{\prime}\left| a_{2}\right\rangle_{A}\left|b_{1}\right\rangle_{B}\left|\delta_{10}\right\rangle_{e} \right]^{S} \tag{11}\]  
If [PERSON] aims to execute a perfect entangle-measure attack without being detected by the legitimate parties, the state in Equation (11) will collapse into the following form:  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]0  
with the conditions:  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]1  
Similarly, for the remaining three cases in Equation (9), analogous results can be obtained, such as:  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]2  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]3  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]4  
\(\hat{\hat{\varepsilon}}\) If [PERSON] intercepts the Reflect particles and applies two unitary operations, \(\hat{E}\) and \(\hat{F}\), the state of the system will undergo transformations as follows:  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]5  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]6  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]7  
It also satisfies Equations (8), (13), and (14):  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]8  
\[\otimes\frac{1}{\sqrt{2}}\left[\left|a_{1}\right\rangle_{A}\left(\alpha_{1^{ \prime}}\left|b_{1}\right\rangle_{B}\left|\varepsilon_{00}\right\rangle_{e}+ \beta_{1^{\prime}}\left|b_{2}\right\rangle_{B}\left|\varepsilon_{01}\right\rangle _{e}\right)+\left|a_{2}\right\rangle_{A}\left(\beta_{2^{\prime}}\left|b_{1} \right\rangle_{B}\left|\varepsilon_{10}\right\rangle_{e}+\alpha_{2^{\prime}} \left|b_{2}\right\rangle_{B}\left|\varepsilon_{11}\right\rangle_{e}\right) \right]^{S}\]9  
From the transformation of Equation (2), we know that:  
\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]0  
\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]1\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]2 \[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]3  
Substituting Equation (17) into Equation (16), we obtain:  
\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]4  
\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]5  
Similarly, for [PERSON] to avoid detection during this entangle-measure attack, Equation (18) must satisfy the following conditions:  
\[=\frac{1}{\sqrt{2}}\left[\left|\mathrm{H}\right\rangle_{B}\left(\alpha_{1} \right|\mathrm{H}\right)_{A}\left|\varepsilon_{00}\right\rangle_{e}+\beta_{2} \left|\mathrm{V}\right\rangle_{A}\left|\varepsilon_{10}\right\rangle_{e}+ \left|\mathrm{V}\right\rangle_{B}\left(\beta_{1}\left|\mathrm{H}\right\rangle_ {A}\left|\varepsilon_{01}\right\rangle_{e}+\alpha_{2}\left|\mathrm{V}\right\rangle _{A}\left|\varepsilon_{11}\right\rangle_{e}\right)\right]^{P}\]6  
In summary, it is not difficult to conclude that, regardless of whether the attack occurs in the polarization DOF or the spatial-mode DOF, [PERSON] cannot distinguish between the states \(\left|\epsilon_{ij}\right\rangle^{P/S}\).