# Efficient Conjunctive Keywords Search over Encrypted E-Mail Data in Public Key Setting
## 4 Proposed Spe-Cks Scheme
### Construction
* For an index \(I_{W}=(C_{0},C_{1},\ldots,C_{n},C_{W},D_{W})\) and a trapdoor \(T_{Q}=(T_{0},T_{1}\ldots,T_{n},T)\), the computation process of \(\hat{e}(C_{i},T_{i})\) is as follows: \[\hat{e}(C_{i},T_{i}) =\hat{e}(g^{r(a_{i}+a_{i})},g^{\frac{H_{1}(q_{1}|\phi|^{j}+H_{1}( q_{2}|\phi|^{j}+\ldots+H_{1}(q_{m}|\phi)^{j}}{u^{\beta+\sum_{j=0}^{n} \alpha_{j}(H_{1}(q_{1}|\phi)^{j}+H_{1}(q_{2}|\phi)^{j}+\ldots+H_{1}(q_{m}|\phi) ^{j})}}})\] \[=\hat{e}(g,g)^{\frac{ra_{i}(H_{1}(q_{1}|\phi|^{j}+H_{1}(q_{2}| \phi)^{j}+\ldots+H_{1}(q_{m}|\phi)^{j})+ra_{i}(H_{1}(q_{1}|\phi)^{j}+H_{1}(q_{ 2}|\phi)^{j}+\ldots+H_{1}(q_{m}|\phi)^{j})}{u^{\beta+\sum_{j=0}^{n}\alpha_{j}(H_ {1}(q_{1}|\phi)^{j}+H_{1}(q_{2}|\phi)^{j}+\ldots+H_{1}(q_{m}|\phi)^{j})}}}\]According to the above, the result of \(\theta_{1}\) is:  
\[\theta_{1} =\prod_{i=0}^{n}\hat{e}(C_{i},T_{i})\] \[=\prod_{i=0}^{n}\hat{e}(g,g)^{\frac{\frac{\tau_{i}(H_{1}(q_{1}| \hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{ \varphi}^{j})+\tau_{m}(H_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi }^{j})+\ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))}}{\imath\beta+\sum_{j=0}^{n}a_{j} (H_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_ {m}|\hat{\varphi}^{j}))}\] \[=\hat{e}(g,g)^{\frac{\tau_{i}^{\sum_{i=0}^{n}a_{i}(H_{1}(q_{1}| \hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{ \varphi}^{j})+\tau_{m}(q_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi }^{j})+\ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))}}{\imath\beta+\sum_{j=0}^{n}a_{j} (H_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_ {m}|\hat{\varphi}^{j}))}\]  
Moreover, the result of \(\theta_{2}\) is:  
\[\theta_{2} =\hat{e}(C_{W},T)=\hat{e}(g^{\tau\beta},g^{\frac{\tau_{i}^{\sum_{j =0}^{n}a_{j}(H_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+ \ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))}}{\imath\beta+\sum_{j=0}^{n}a_{j}(H_{1} (q_{1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}| \hat{\varphi}^{j}))}}\] \[=\hat{e}(g,g)^{\frac{n\beta}{\beta+\sum_{j=0}^{n}a_{j}(H_{1}(q_{1 }|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{ \varphi}^{j}))}}\]  
Then, the result of \(H_{2}(\theta_{1}\times\theta_{2})\) is:  
\[H_{2}(\theta_{1}\times\theta_{2}) =H_{2}(\hat{e}(g,g)^{\frac{\tau_{i}^{\sum_{j=0}^{n}a_{j}(H_{1}(q_{ 1}|\hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{ \varphi}^{j}))+\tau_{m}^{\sum_{j=0}^{n}a_{j}(H_{1}(q_{1}|\hat{\varphi}^{j})+H_ {1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))+\tau_{m} \beta}}{\imath\beta+\sum_{j=0}^{n}a_{j}(H_{1}(q_{1}|\hat{\varphi}^{j})+H_{1}( q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))}})\] \[=\hat{e}(g,g)^{\frac{\tau_{i}^{\sum_{j=0}^{n}a_{j}(H_{1}(q_{1}| \hat{\varphi}^{j})+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{ \varphi}^{j})}}{\imath\beta+\sum_{j=0}^{n}a_{j}(H_{1}(q_{1}|\hat{\varphi}^{j} )+H_{1}(q_{2}|\hat{\varphi}^{j})+\ldots+H_{1}(q_{m}|\hat{\varphi}^{j}))}}+r\]  
Note that, according to the keyword conversion method introduced in Section 4.1, if \(Q\subseteq W\), it must be \(\sum_{i=0}^{n}a_{i}(H_{1}(q_{1}|\hat{\varphi})^{i}+H_{1}(q_{2}|\hat{\varphi} )^{i}+\ldots+H_{1}(q_{m}|\hat{\varphi})^{i})=0\), which means that \(H_{2}(\theta_{1}\times\theta_{2})=H_{2}(\hat{e}(g,g)^{r})=D_{W}\). Therefore, according the equations above, we can argue that our scheme is correct.