Quantile estimation for encrypted data

Abstract

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.

Appendix

The pseudo-code of the NAG method is described in Algorithm 1.

Pseudo code of NAG method where $\mu$, $\eta$, # of maximum iterations T

The pseudo-code of the Newton method is described in Algorithm 2.

Pseudo-code of Newton method where $\epsilon$, # of maximum iterations T

Fig. 3

The epoch absolute errors across multiple epochs for NAG (blue, dashed line) and Newton (red, solid line). The data set consists of $32,768 \times s$ random numbers sampling from the standard normal distribution

1.1 Choosing the best epoch

As it can be challenging to monitor the objective value in HE, the number of epochs should be predefined appropriately. To determine the epoch number and examine the convergence of objective values, we utilized two methods with randomly generated data in plaintext. In Fig. 3, we present the trajectories of the epoch absolute errors at $\tau =0.25$ for the NAG method (blue, dashed line) with $\alpha =0.1$ and $\eta =0.4$, and the Newton method (red, solid line) with $\alpha =0.1$. Across multiple s values, the NAG method converges within 10 epochs, while the Newton method converges within 5 epochs, indicating that Newton converges faster than NAG. Based on this observation, we used 20 and 10 epochs for NAG and Newton, respectively, in our numerical study to ensure convergence. We find that this strategy works well.