## 6 Benchmarks
### p-Value Computation  
We tested our hybrid protocol for \(p\)-value computation that starts from encrypted \(\hat{t}^{2}\) and degree of freedom values and performs the lookup using our MHE method against an equivalent SMPC protocol, which starts from the secret shared \(\hat{t}^{2}\) and degree of freedom values and performs the lookup using SMPC. The size of the lookup table used was 16,320. For the SMPC table lookup, we use the semi-bin-party from the MP-SPDZ library that implements a semi-honest SMPC protocol over the binary domain instead of the SPDZZK protocol previously used for other benchmarks so as to present a fair comparison as lookup operations are far more efficiently implemented in the binary domain as opposed to an arithmetic domain. The runtime (s) and absolute error of computing the \(p\)-value across the three test cases from the 1000 Genome dataset previously used in Section 6.3 can be found in Tables 11 and 12, respectively. The absolute error has been reported here instead of the relative error as some of the \(p\)-values are very close to 0.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & \(|\mathcal{D}|\) & **MHE** & **SMPC** \\ \hline Mushroom & 90 & \(1.06\times 10^{-6}\) & \(1.60\times 10^{-5}\) \\ Credit & 100 & \(2.03\times 10^{-6}\) & \(4.50\times 10^{-5}\) \\ ADULT & 574 & \(3.19\times 10^{-3}\) & \(1.51\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Relative error (%) of calculating \(\chi^{2}\)-statistic in the secure federated \(\chi^{2}\)-test on various datasets.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & \(|\mathcal{D}|\) & **MHE** & **SMPC** \\ \hline Mushroom & 90 & 196 & 7814 \\ Credit & 100 & 196 & 8680 \\ ADULT & 574 & 196 & 49,631 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Communication overhead (MB) of running the secure federated \(\chi^{2}\)-test on various datasets.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & \(|\mathcal{D}|\) & **MHE** & **SMPC** \\ \hline Mushroom & 90 & \(19.0\) & \(41.8\) \\ Credit & 100 & \(19.3\) & \(46.9\) \\ ADULT & 574 & \(19.3\) & \(267\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Runtime (s) of running the secure federated \(\chi^{2}\)-test on various datasets.  
We notice that our hybrid method is much faster compared to the pure SMPC method. This is mainly due to the number of rounds of communication taken up by the SMPC lookup, which was 33,245. Compared to the SMPC lookup, our hybrid method only took 1150 rounds, which is 30\(\times\) less. In fact, the number of rounds of communication made such an impact that even though our hybrid method resulted in 236 MB of communication compared to the SMPC lookup, which only took 126 MB, our hybrid method was up to 1.5\(\times\) faster than the SMPC method in the LAN environment. The increase in communication for the MHE lookup is mainly due to the extra MHE to SMPC protocol that had to be run before the lookup could be performed, with the actual lookup protocol only taking about 36 MB of communication. This clearly shows that our hybrid method is far more efficient than the pure SMPC method for \(p\)-value calculation. We see that the table lookup method is fairly accurate with very small errors across the three experiments.