**Numerical Methods for Comparison on Homomorphically Encrypted Numbers**

*Jung Hee Cheon and Dongwoo Kim and Duhyeong Kim and Hun Hee Lee and Keewoo Lee*

**Abstract: **We propose a new method to compare numbers which are encrypted by Homomorphic Encryption (HE).
Previously, comparison and min/max functions were evaluated using Boolean functions where input numbers are encrypted bit-wisely. However, the bit-wise encryption methods require relatively expensive computation of basic arithmetic operations such as addition and multiplication.

In this paper, we introduce iterative algorithms that approximately compute the min/max and comparison operations of several numbers which are encrypted word-wisely. From the concrete error analyses, we show that our min/max and comparison algorithms have $\Theta(\alpha)$ and $\Theta(\alpha\log\alpha)$ computational complexity to obtain approximate values within an error rate $2^{-\alpha}$, while the previous minimax polynomial approximation method requires the exponential complexity $\Theta(2^{\alpha/2})$ and $\Theta(\sqrt{\alpha}\cdot 2^{\alpha/2})$, respectively. We also show the (sub-)optimality of our min/max and comparison algorithms in terms of asymptotic computational complexity among polynomial evaluations to obtain approximate min/max and comparison results. Our comparison algorithm is extended to several applications such as computing the top-$k$ elements and counting numbers over the threshold in encrypted state.

Our new method enables word-wise HEs to enjoy comparable performance in practice with bit-wise HEs for comparison operations while showing much better performance on polynomial operations. Computing an approximate maximum value of any two $\ell$-bit integers encrypted by HEAAN, up to error $2^{\ell-10}$, takes only $1.14$ milliseconds in amortized running time, which is comparable to the result based on bit-wise HEs.

**Category / Keywords: **applications / Homomorphic Encryption, Comparison, Min/Max, Iterative Method

**Date: **received 22 Apr 2019, last revised 25 Apr 2019

**Contact author: **doodoo1204 at snu ac kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20190425:073431 (All versions of this report)

**Short URL: **ia.cr/2019/417

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