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Paper: Numerical Method for Comparison on Homomorphically Encrypted Numbers

Authors:
Jung Hee Cheon
Dongwoo Kim
Duhyeong Kim
Hun Hee Lee
Keewoo Lee
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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.
BibTeX
@article{asiacrypt-2019-30046,
  title={Numerical Method for Comparison on Homomorphically Encrypted Numbers},
  booktitle={Advances in Cryptology – ASIACRYPT 2019},
  series={Advances in Cryptology – ASIACRYPT 2019},
  publisher={Springer},
  volume={11922},
  pages={415-445},
  doi={10.1007/978-3-030-34621-8_15},
  author={Jung Hee Cheon and Dongwoo Kim and Duhyeong Kim and Hun Hee Lee and Keewoo Lee},
  year=2019
}