Quantile estimation for encrypted data

As data-based studies continue to increase, the need for privacy protection has become a crucial issue. One proposed solution to address this obstacle is homomorphic encryption (HE); however, the complexity of handling ciphertexts used in HE poses a serious challenge due to the extended calculation time of elementary operations. As a result, it has much more complex than handling plaintexts, limiting various subsequent data analyses. This paper proposes a quantile estimation method for encrypted data, where quantiles are core statistics for understanding the data distribution in statistical analysis. We developed an HE-friendly method for large homomorphic encrypted data using an approximate quantile loss function. Numerical studies show that the proposed method significantly improves the calculation time for simulated and real homomorphically encrypted data. Specifically, the proposed method takes approximately 26 minutes for calculating a dataset of four million, which is about 14 times faster than the sorting method. Furthermore, we applied the proposed method to construct boxplots for homomorphically encrypted data.