Minimax Approximation of Sign Function by Composite Polynomial for Homomorphic Comparison
- Authors
- Lee, E.; Lee, J.-W.; No, J.-S.; Kim, Y.-S.
- Issue Date
- Nov-2022
- Publisher
- Institute of Electrical and Electronics Engineers Inc.
- Keywords
- Cheon-Kim-Kim-Song (CKKS) scheme; fully homomorphic encryption; homomorphic comparison operation; minimax approximation polynomial; Remez algorithm; sign function
- Citation
- IEEE Transactions on Dependable and Secure Computing, v.19, no.6, pp 3711 - 3727
- Pages
- 17
- Journal Title
- IEEE Transactions on Dependable and Secure Computing
- Volume
- 19
- Number
- 6
- Start Page
- 3711
- End Page
- 3727
- URI
- https://scholarworks.bwise.kr/cau/handle/2019.sw.cau/61150
- DOI
- 10.1109/TDSC.2021.3105111
- ISSN
- 1545-5971
1941-0018
- Abstract
- The comparison operation for two numbers is one of the most frequently used operations in several applications, including deep learning. As such, lots of research has been conducted with the goal of efficiently evaluating the comparison operation in homomorphic encryption schemes. Recently, Cheon et al. (Asiacrypt 2020) proposed new comparison methods that approximated the sign function on homomorphically encrypted data using composite polynomials and proved that these methods had optimal asymptotic complexity. In this article, we propose a practically optimal method that approximates the sign function using compositions of minimax approximation polynomials. We prove that this approximation method is optimal with respect to depth consumption and the number of non-scalar multiplications. In addition, we propose a polynomial-time algorithm that determines the optimal composition of minimax approximation polynomials for the proposed homomorphic comparison operation using dynamic programming. The numerical analysis demonstrates that when minimizing runtime, the proposed comparison operation reduces the runtime by approximately 45 percent on average when compared to the previous algorithm. Likewise, when minimizing depth consumption, the proposed algorithm reduces the runtime by approximately 41 percent on average. In addition, when high precision in the comparison operation is required, the previous algorithm does not achieve 128-bit security, while the proposed algorithm does due to its small depth consumption.
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