# Registered Keyword Searchable Encryption Based on SM9
## 2 Preliminaries
### Baver Triple
### Experimental Results

Experimental simulation of the above three schemes are evaluated under the following environment: Windows 10 operating system, 1.0 GHz 4-core 64-bit Intel(R) Core(TM) i5-1035G1 CPU, 16 GB RAM, with Intellij IDEA 2021 as the experimental platform, using Java as the programming language, and using the JPBC package (Java pairing-based cryptography library2.0), wherein the Type-F curve in the library is used to construct an asymmetric prime order bilinear group, and the order of the curve group is 256 bits with embedding degree \(k=12\). Since the schemes [13, 15, 17, 18] have different systems from our scheme, the scheme in this paper mainly compares it with the schemes in the studies [8, 30] through experiments. SA-PEKS [8] is the first server-aided searchable encryption scheme with registered keywords, but its scheme is not based on the standard SM9. The scheme SM9-PEKS [30] is a searchable encryption scheme based on SM9, but it is not a registration keyword searchable encryption scheme, and it cannot resist KGA. First, we increase the number of keywords to experiment with each algorithm and test the execution time of the same algorithm in different schemes. Then, the running time of the original registration keyword generation algorithm proposed in this paper and the improved registration keyword algorithm is compared. During each collection of data, the results will be analyzed and summarized. All the experimental results are as follows. Figure 4 shows the time consumed by PEKS Ciphertext in each scheme, with the number of keywords on the x-axis and time on the y-axis. The PEKS Ciphertext of our scheme is almost the same as the scheme of [30], and is better than the scheme of [8]. Specifically, when generating 1000 keyword ciphertexts, the time for our scheme is 19 s, the time for [8] is 119 s, and the scheme of [30] takes the same time as our scheme. It can be seen from Table 3 that in the PEKS Ciphertext, our scheme and [30] do not require the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Schemes** & **Public Key Size** & **PEKS Ciphertext Size** & **Trapdoor Size** \\ \hline
[8] & \(3|G_{1}|+|G_{T}|\) & \(|G_{1}|+|G_{T}|\) & \(|G_{1}|\) \\
[13] & \(|G_{1}|\) & \(|G_{1}|\) & \(2|G_{1}|\) \\
[15] & \(|G_{1}|\) & \(2|G_{1}|\) & \(|G_{T}|\) \\
[17] & \(\backslash\) & \(2|G_{1}|+|G_{T}|\) & \(2|G_{1}|\) \\
[18] & \(|G_{1}|\) & \(|G_{1}|\) & \(|G_{1}|\) \\
[30] & \(|G_{1}|\) & \(|G_{1}|\) & \(|G_{2}|\) \\ Our Scheme & \(|G_{1}|\) & \(|G_{1}|\) & \(|G_{2}|\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Communication overhead. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Operations** & \(\mathbf{T_{\text{expt}}}\) & \(\mathbf{T_{\text{m1}}}\) & \(\mathbf{T_{\text{m2}}}\) & \(\mathbf{T_{\text{bp}}}\) & \(\mathbf{H_{G}}\) & \(\mathbf{T_{H}}\) \\ \hline Execution time & 14.3 & 1.78 & 3.43 & 68.7 & 41.2 & \(\approx 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time consumption of basic operations (ms). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Schemes** & **PEKS Ciphertext** & **Trapdoor** & **Testing** \\ \hline
[8] & \(T_{\text{bp}}+2T_{\text{m1}}+H_{\text{G}_{T}}\) & \(T_{\text{m1}}\) & \(T_{\text{bp}}+H_{\text{G}_{T}}\) \\
[13] & \(T_{\text{bp}}+2T_{\text{m1}}+T_{H}\) & \(3T_{\text{m1}}+2T_{H}\) & \(T_{\text{bp}}+2T_{\text{m1}}+T_{H}\) \\
[15] & \(T_{\text{bp}}+2T_{\text{m1}}+2T_{H}\) & \(T_{\text{m1}}+T_{H}\) & \(T_{\text{bp}}+T_{H}\) \\
[17] & \(2T_{\text{bp}}+3T_{\text{m1}}+2T_{H}\) & \(T_{\text{bp}}+2T_{\text{m1}}+2T_{H}\) & \(T_{\text{bp}}\) \\
[18] & \(T_{\text{bp}}+3T_{\text{m1}}+2T_{H}\) & \(T_{\text{bp}}+2T_{\text{m1}}+T_{H}\) & \(T_{\text{bp}}+T_{H}\) \\
[30] & \(2T_{\text{m1}}+T_{\text{expt}}+T_{H}\) & \(T_{\text{m2}}+T_{H}\) & \(T_{\text{bp}}\) \\ Our Scheme & \(2T_{\text{m1}}+T_{\text{expt}}+T_{H}\) & \(T_{\text{m2}}+T_{H}\) & \(T_{\text{bp}}+T_{H}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computational overhead. time-consuming operations of \(T_{\text{bp}}\) and \(T_{G_{T}}\).