International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: High-Precision Bootstrapping for Approximate Homomorphic Encryption by Error Variance Minimization

Yongwoo Lee , Samsung Advanced Institute of Technology
Joon-Woo Lee , Seoul National University
Young-Sik Kim , Chosun University
Yongjune Kim , DGIST
Jong-Seon No , Seoul National University
HyungChul Kang , Samsung Advanced Institute of Technology
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Presentation: Slides
Conference: EUROCRYPT 2022
Abstract: The Cheon-Kim-Kim-Song (CKKS) scheme (Asiacrypt'17) is one of the most promising homomorphic encryption (HE) schemes as it enables privacy-preserving computing over real (or complex) numbers. It is known that bootstrapping is the most challenging part of the CKKS scheme. Further, homomorphic evaluation of modular reduction is the core of the CKKS bootstrapping. As modular reduction is not represented by the addition and multiplication of complex numbers, approximate polynomials for modular reduction should be used. The best-known techniques (Eurocrypt'21) use a polynomial approximation for trigonometric functions and their composition. However, all the previous methods are based on an indirect approximation, and thus it requires lots of multiplicative depth to achieve high accuracy. This paper proposes a direct polynomial approximation of modular reduction for CKKS bootstrapping, which is optimal in error variance and depth. Further, we propose an efficient algorithm, namely the lazy baby-step giant-step (BSGS) algorithm, to homomorphically evaluate the approximate polynomial, utilizing the lazy relinearization/rescaling technique. The lazy-BSGS reduces the computational complexity by half compared to the ordinary BSGS algorithm. The performance improvement for the CKKS scheme by the proposed algorithm is verified by implementation using HE libraries. The implementation results show that the proposed method has a multiplicative depth of 10 for modular reduction to achieve the state-of-the-art accuracy, while the previous methods have depths of 11 to 12. Moreover, we achieve higher accuracy within a small multiplicative depth, for example, 93-bit within multiplicative depth 11.
Video from EUROCRYPT 2022
  title={High-Precision Bootstrapping for Approximate Homomorphic Encryption by Error Variance Minimization},
  author={Yongwoo Lee and Joon-Woo Lee and Young-Sik Kim and Yongjune Kim and Jong-Seon No and HyungChul Kang},