# Quantum Private Comparison Based on Four-Particle Cluster State
## 4 Security Analysis
### External Attacks
#### 4.1.2 Entangle-Measure Attack  
When executing the entangle-measure attack [51], [PERSON] intercepts the transmitted quantum sequence and prepares auxiliary particles \(|e\rangle\). She then uses a specific unitary operation \(U\) to entangle \(|e\rangle\) with the intercepted particle. When the target particles in the intercepted quantum sequence are subsequently measured, [PERSON] analyzes the auxiliary particles to deduce the target particles. This attack can be effective unless [PERSON] is detected during the eavesdropping process. When entangling \(|e\rangle\) with an intercepted particle that is in state \(|0\rangle\) or \(|1\rangle\), the resulting states are as follows:  
\[U|e\rangle|0\rangle=\lambda_{0}|0\rangle|e_{00}\rangle+\lambda_{1}|1\rangle|e_ {01}\rangle \tag{6}\]  
\[U|e\rangle|1\rangle=\lambda_{2}|0\rangle|e_{10}\rangle+\lambda_{3}|1\rangle|e_ {11}\rangle \tag{7}\]  
where \(\{|e_{00}\rangle,|e_{01}\rangle,|e_{10}\rangle,|e_{11}\rangle\}\) are pure states; the coefficients \(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) must satisfy the following condition:  
\[\left\|\lambda_{0}\right\|^{2}+\left\|\lambda_{1}\right\|^{2}=1 \tag{8}\]\[\|\lambda_{2}\|^{2}+\|\lambda_{3}\|^{2}=1 \tag{9}\]  
When entangling \(|e\rangle\) with an intercepted particle that is in state \(|+\rangle\) or \(|-\rangle\), the resulting states are as follows:  
\[ \begin{array}{l}U|e\rangle|+\rangle=\frac{1}{\sqrt{2}}(\lambda_{0}|0\rangle|e _{00}\rangle+\lambda_{1}|1\rangle|e_{01}\rangle+\lambda_{2}|0\rangle|e_{10} \rangle+\lambda_{3}|1\rangle|e_{11}\rangle)\\ =\frac{1}{\sqrt{2}}\Big{(}\lambda_{0}\frac{|+\rangle+|-\rangle}{\sqrt{2}}|e_{ 00}\rangle+\lambda_{1}\frac{|+\rangle-|-\rangle}{\sqrt{2}}|e_{01}\rangle+ \lambda_{2}\frac{|+\rangle+|-\rangle}{\sqrt{2}}|e_{10}\rangle+\lambda_{3} \frac{|+\rangle-|-\rangle}{\sqrt{2}}|e_{11}\rangle\Big{)}\\ =\frac{1}{2}|+\rangle(\lambda_{0}|e_{00}\rangle+\lambda_{1}|e_{01}\rangle+ \lambda_{2}|e_{10}\rangle+\lambda_{3}|e_{11}\rangle)\\ +\frac{1}{2}|-\rangle(\lambda_{0}|e_{00}\rangle-\lambda_{1}|e_{01}\rangle+ \lambda_{2}|e_{10}\rangle-\lambda_{3}|e_{11}\rangle)\\ \end{array}  \tag{10}\]  
\[ \begin{array}{l}U|e\rangle|-\rangle=\frac{1}{\sqrt{2}}(\lambda_{0}|0\rangle|e _{00}\rangle+\lambda_{1}|1\rangle|e_{01}\rangle-\lambda_{2}|0\rangle|e_{10} \rangle-\lambda_{3}|1\rangle|e_{11}\rangle)\\ =\frac{1}{\sqrt{2}}\Big{(}\lambda_{0}\frac{|+\rangle+|-\rangle}{\sqrt{2}}|e_{ 00}\rangle+\lambda_{1}\frac{|+\rangle-|-\rangle}{\sqrt{2}}|e_{01}\rangle- \lambda_{2}\frac{|+\rangle+|-\rangle}{\sqrt{2}}|e_{10}\rangle-\lambda_{3}\frac{ |+\rangle-|-\rangle}{\sqrt{2}}|e_{11}\rangle\Big{)}\\ =\frac{1}{2}|+\rangle(\lambda_{0}|e_{00}\rangle+\lambda_{1}|e_{01}\rangle- \lambda_{2}|e_{10}\rangle-\lambda_{3}|e_{11}\rangle)\\ +\frac{1}{2}|-\rangle(\lambda_{0}|e_{00}\rangle-\lambda_{1}|e_{01}\rangle- \lambda_{2}|e_{10}\rangle+\lambda_{3}|e_{11}\rangle)\\ \end{array}  \tag{11}\]  
If [PERSON] introduces no errors during the detection process, the following conditions must be satisfied:  
\[\lambda_{1}=\lambda_{2}=0 \tag{12}\]  
\[\lambda_{0}=\lambda_{3}=1 \tag{13}\]  
\[\lambda_{0}|e_{00}\rangle-\lambda_{1}|e_{01}\rangle+\lambda_{2}|e_{10}\rangle -\lambda_{3}|e_{11}\rangle=\mathbf{0} \tag{14}\]  
\[\lambda_{0}|e_{00}\rangle+\lambda_{1}|e_{01}\rangle-\lambda_{2}|e_{10}\rangle -\lambda_{3}|e_{11}\rangle=\mathbf{0} \tag{15}\]  
where \(\mathbf{0}\) is the column vector.