**Efficient Homomorphic Comparison Methods with Optimal Complexity**

*Jung Hee Cheon and Dongwoo Kim and Duhyeong Kim*

**Abstract: **Comparison of two numbers is one of the most frequently used operations, but it has been a challenging task to efficiently compute the comparison function in homomorphic encryption (HE) which basically support addition and multiplication.
Recently, Cheon et al. (Asiacrypt 2019) introduced a new approximate representation of the comparison function with a rational function, and showed that this rational function can be evaluated by an iterative algorithm. Due to this iterative feature, their method achieves a logarithmic computational complexity compared to previous polynomial approximation methods; however, the computational complexity is still not optimal, and the algorithm is quite slow for large-bit inputs in HE implementation.

In this work, we propose new comparison methods with optimal asymptotic complexity based on composite polynomial approximation. The main idea is to systematically design a constant-degree polynomial $f$ by identifying the \emph{core properties} to make a composite polynomial $f\circ f \circ \cdots \circ f$ get close to the sign function (equivalent to the comparison function) as the number of compositions increases. We additionally introduce an acceleration method applying a mixed polynomial composition $f\circ \cdots \circ f\circ g \circ \cdots \circ g$ for some other polynomial $g$ with different properties instead of $f\circ f \circ \cdots \circ f$. Utilizing the devised polynomials $f$ and $g$, our new comparison algorithms only require $\Theta(\log(1/\epsilon)) + \Theta(\log\alpha)$ computational complexity to obtain an approximate comparison result of $a,b\in[0,1]$ satisfying $|a-b|\ge \epsilon$ within $2^{-\alpha}$ error.

The asymptotic optimality results in substantial performance enhancement: our comparison algorithm on encrypted $20$-bit integers for $\alpha = 20$ takes $1.43$ milliseconds in amortized running time, which is $30$ times faster than the previous work.

**Category / Keywords: **applications / Homomorphic Encryption, Comparison, Min/Max, Composite polynomial approximation

**Original Publication**** (with minor differences): **IACR-ASIACRYPT-2020

**Date: **received 21 Oct 2019, last revised 17 Aug 2020

**Contact author: **doodoo1204 at snu ac kr, dwkim606 at snu ac kr, jhcheon at snu ac kr

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

**Version: **20200818:024859 (All versions of this report)

**Short URL: **ia.cr/2019/1234

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