# Development of Quantum Private Queries Protocol on Collective-Dephasing Noise Channel
## 4 Security Analysis  
### Database Security Analysis  
When [PERSON] is dishonest, [PERSON] will be considered to be the honest one automatically. Next, we will focus on database security and analyze what the impact of [PERSON]'s attack has on database security. A defect was found in reference [25], i.e., the parity check information of two consecutive key bits \(O_{j}\) and \(O_{j+l}\) can be derived from the generated key bits \(q_{j}\) and \(q_{j+k}\). If \(q_{j}\) and \(q_{j+k}\) have been tested successfully, then the parity check information of \(O_{j}\) and \(O_{j+l}\) will be derived. The scheme mentioned in this article provides more protection for parity between key bits, \(2l+1\) raw key bit addition operation is mainly used in this post-processing process. After the decimal to binary conversion and the subsequent XOR addition operation, [PERSON]'s threat to the database is greatly reduced and the security of the database is increased [26,27]. The final key bit \(O_{j}\) in reference [28] is generated by calculating \(O_{j}=XOR(q_{j},q_{j+l},\ldots,q_{j+k-l})\), the parity information of the \(k\) qubits is represented by the final key. Because of the special way that the final key bit is generated, odd-even check will not damage the \(2l+1\) qubits in this scheme. Following the method in step (6) to generate the final key, we calculate \(\sum_{m=0}^{m=l}(q_{j-m}+q_{j+m})\) first, and convert it to binary form \(p_{1}^{j},p_{2}^{j},\ldots,p_{\lceil log(2l+1)\rceil}^{j},j=1,2,\ldots,N\), then get the final key bit \(O_{j}\). To verify, we assume that \(l=2\), the possible values of \(q_{j-2},q_{j-1},q_{j},q_{j+1},q_{j+2}\) are \(\{00000,0001,00010,00011,00100,00101,001010,00110,00111,01000,01001,01010,01110,01110,01111,\)\(10000,10001,10010,10010,1001,10010,10110,10111,\)\(10000,10001,1001,1010,10110,10111,\)\(11000,1101,11100,11110,11111\). The possible decimal set is \(\{0,1,2,3,4,5\}\), the corresponding binary number is \(\{000,001,010,011,100,101\}\). Performing formula(3) operation on the resulting binary number, then the corresponding XOR of \(000,001,010,011,100,101\) is \(0,1,1,0,1,0\). According to the above analysis, we found that \(q_{j-2},q_{j-1},q_{j},q_{j+1},q_{j+2}\) have no linear correlation to the final key bit. Therefore, it can be proved that the joint measurement is invalid. In case of that [PERSON]'s registers have quantum memory, it is possible for [PERSON] to delay [PERSON]'s measurements after he announces his state during the quantum oblivious key distribution phase. If [PERSON] is going to make a joint quantum measurement for \(2l+1\) qubits, as the analysis we have mentioned above, the final key bit \(0\) corresponds to the possible initial value of \(\{00000,0011,01011,01101,01110,10011,\)\(10101,\)\(10101,\)\(10110,11001,\)\(111001,\)\(111001,\)\(11100,11111\}\) and \(1\) corresponds to a possible initial value of \(\{00001,00010,\)\(000011,00010,\)\(000011,0010,01010,\)\(010100,\)\(011010,\)\(01100,\)\(011111,\)\(10000,\)\(10010,\)\(10111,\)\(111000,\)\(11011,\)\(11110\}\). We can assume that all the five states [PERSON] publishes are logical Bell states \(\{|\Phi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\), [PERSON] can make the unambiguous state discrimination (USD) [29,30] on \(p_{0}^{O_{j}}\) and \(p_{1}^{O_{j}}\) to clearly distinguish the two mixed states corresponding to parity. Reference [30] points out that the probability \(Q_{F}\) of clearly distinguishing the minimum failure is at least twice the probability \(P_{E}\) of the minimum error of two arbitrarily mixed quantum states in the fuzzy discrimination. Therefore, we should use a fuzzy test to get the most information. The conclusion of reference [20] is used to distinguish the two mixed states \(p_{1}\) and \(p_{2}\) under the prior probability of \(\rho_{1}\) and \(\rho_{2}\), thereinto, \(p_{1}+p_{2}=1\), \(P_{E}=\frac{1}{2}-\frac{1}{2}Tr|p_{2}\rho_{2}-p_{1}\rho_{1}|\). For the other operand \(\sigma\), it has \(|\sigma|=\sqrt{\sigma+\sigma}\).