## 7 Conclusions  
This work considered the problem of computing statistical tests securely in a federated setting using MHE. To that end, we presented a new method for secure division based on MHE that is communication efficient when performing large batches of divisions simultaneously. We applied this method to two common statistical tests--[PERSON]'s \(t\)-test and \(\chi^{2}\)-test--in the vertical and horizontal federated settings, respectively. We evaluated our methods against equivalent methods over secret shared data using the MP-SPDZ framework and found that our methods are a lot more communication-efficient. Even when the parties are communicating over a high speed LAN network, this communication efficiency is leveraged to achieve a maximum of 5\(\times\) and 10\(\times\) improvement in runtime when computing the [PERSON]'s \(t\)-test and \(\chi^{2}\)-test, respectively. Moreover, we designed and implemented novel protocols to switch between MHE and SMPC for the CKKS scheme and to perform a table lookup using a secret shared index. We used the protocols to compute the \(p\)-value for the [PERSON]'s \(t\)-test and found that our method was 1.5\(\times\) faster than the pure SMPC-based method. We believe that our method opens up the possibility to perform large scale complicated analyses on encrypted data efficiently by switching between MHE and SMPC for approximate arithmetic, which was not previously possible with integer-based arithmetic. As for future work, we plan to expand our work and apply our methods to more applications with larger scale settings.  
Conceptualization, M.S.M.S.A., C.J. and K.M.M.A.; methodology, M.S.M.S.A. and C.J.; Writing--original draft, M.S.M.S.A., C.J. and K.M.M.A. All authors have read and agreed to the published version of the manuscript.  
This research is supported by A*STAR, Singapore under its RIE2020 Advanced Manufacturing and Engineering (AME) Programmatic Programme (Award A19E3b0099).  
Not applicable.  
Patient consent was waived due to the fact that only publicly available data was used.  
1000 Genome Dataset: [[https://www.internationalgenome.org/data](https://www.internationalgenome.org/data)]([https://www.internationalgenome.org/data](https://www.internationalgenome.org/data)); Mushroom Dataset: [[https://archive.ics.uci.edu/ml/datasets/mushroom](https://archive.ics.uci.edu/ml/datasets/mushroom)]([https://archive.ics.uci.edu/ml/datasets/mushroom](https://archive.ics.uci.edu/ml/datasets/mushroom)); Credit Dataset: [[https://www.kaggle.com/datasets/praveengovi/credit-risk-classification-dataset](https://www.kaggle.com/datasets/praveengovi/credit-risk-classification-dataset)]([https://www.kaggle.com/datasets/praveengovi/credit-risk-classification-dataset](https://www.kaggle.com/datasets/praveengovi/credit-risk-classification-dataset)); Adult Dataset: [[https://archive.ics.uci.edu/ml/datasets/adult](https://archive.ics.uci.edu/ml/datasets/adult)]([https://archive.ics.uci.edu/ml/datasets/adult](https://archive.ics.uci.edu/ml/datasets/adult)).  
\begin{table}
\begin{tabular}{c c c} \hline
**ID** & **Hybrid** & **SMPC** \\ \hline
1 & \(5.45\times 10^{-3}\) & \(5.47\times 10^{-3}\) \\
2 & \(1.68\times 10^{-9}\) & \(1.53\times 10^{-12}\) \\
3 & \(2.52\times 10^{-2}\) & \(2.52\times 10^{-2}\) \\ \hline \end{tabular}
\end{table}
Table 12: Absolute error of calculating the \(p\)-value of the \(t\) statistic and degree of freedom previously calculated in Table 6.