International Association for Cryptologic Research

International Association
for Cryptologic Research


Numerical Method for Comparison on Homomorphically Encrypted Numbers

Jung Hee Cheon
Dongwoo Kim
Duhyeong Kim
Hun Hee Lee
Keewoo Lee
DOI: 10.1007/978-3-030-34621-8_15
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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-wise. However, the bit-wise encryption methods require relatively expensive computations for 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-wise. From the concrete error analyses, we show that our min/max and comparison algorithms have $$\varTheta (\alpha )$$ and $$\varTheta (\alpha \log \alpha )$$ computational complexity to obtain approximate values within an error rate $$2^{-\alpha }$$, while the previous minimax polynomial approximation method requires the exponential complexity $$\varTheta (2^{\alpha /2})$$ and $$\varTheta (\sqrt{\alpha }\cdot 2^{\alpha /2})$$, respectively. Our algorithms achieve (quasi-)optimality in terms of asymptotic computational complexity among polynomial approximations for min/max and comparison operations. The comparison algorithm is extended to several applications such as computing the top-k elements and counting numbers over the threshold in encrypted state.Our 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 ms in amortized running time, which is comparable to the result based on bit-wise HEs.
  title={Numerical Method for Comparison on Homomorphically Encrypted Numbers},
  booktitle={Advances in Cryptology – ASIACRYPT 2019},
  series={Advances in Cryptology – ASIACRYPT 2019},
  author={Jung Hee Cheon and Dongwoo Kim and Duhyeong Kim and Hun Hee Lee and Keewoo Lee},