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Faster BGV Bootstrapping for Power-of-Two Cyclotomics Through Homomorphic NTT

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Advances in Cryptology – ASIACRYPT 2024 (ASIACRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15484))

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Abstract

Power-of-two cyclotomics is a popular choice when instantiating the BGV scheme because of its efficiency and compliance with the FHE standard. However, in power-of-two cyclotomics, the linear transformations in BGV bootstrapping cannot be decomposed into sub-transformations for acceleration with existing techniques. Thus, they can be highly time-consuming when the number of slots is large, degrading the advantage brought by the SIMD property of the plaintext space. By exploiting the algebraic structure of power-of-two cyclotomics, this paper derives explicit decomposition of the linear transformations in BGV bootstrapping into NTT-like sub-transformations, which are highly efficient to compute homomorphically. Moreover, multiple optimizations are made to evaluate homomorphic linear transformations, including modified BSGS algorithms, trade-offs between level and time, and specific simplifications for thin and general bootstrapping. We implement our method on HElib. With the number of slots ranging from 4096 to 32768, we obtain a 2.4x–55.1x improvement in bootstrapping throughput, compared to previous works or the naive approach.

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Notes

  1. 1.

    A non-constant monic polynomial h(X) over \(\textsf {GR}(p^r;m)\) is a monic basic primitive polynomial if \(\bar{h}(X)\) is a primitive polynomial over \(\textsf {GF}(p^m)\).

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Acknowledgments

We thank Mr. Robin Geelen at KU Leuven for identifying the issue with the capacity consumption in Table 5. We also thank the anonymous reviewers for their insightful comments that greatly improved this manuscript. The study is supported by the National Key R&D Program of China (2018YFA0704701, 2020YFA0309705), Shandong Key Research and Development Program (2020ZLYS09), the Major Scientific and Technological Innovation Project of Shandong, China (2019JZZY010133), the Major Program of Guangdong Basic and Applied Research (2019B030302008), Tsinghua University Dushi Program, and the Key Laboratory of Data Protection and Intelligent Management, Ministry of Education, Sichuan University.

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Authors and Affiliations

  1. Institute for Network Sciences and Cyberspace, Tsinghua University, Beijing, China

    Shihe Ma

  2. Institute for Advanced Study, BNRist, Tsinghua University, Beijing, China

    Tairong Huang, Anyu Wang & Xiaoyun Wang

  3. Zhongguancun Laboratory, Beijing, China

    Anyu Wang & Xiaoyun Wang

  4. National Financial Cryptography Research Center, Beijing, China

    Anyu Wang & Xiaoyun Wang

  5. Shandong Institute of Blockchain, Jinan, Shandong, China

    Xiaoyun Wang

  6. Key Laboratory of Cryptologic Technology and Information Security (Ministry of Education), School of Cyber Science and Technology, Shandong University, Qingdao, Shandong, China

    Xiaoyun Wang

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  1. Shihe Ma
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Correspondence to Anyu Wang .

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  1. Academia Sinica, Taipei, Taiwan

    Kai-Min Chung

  2. NTT Social Informatics Laboratories, Tokyo, Japan

    Yu Sasaki

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Ma, S., Huang, T., Wang, A., Wang, X. (2025). Faster BGV Bootstrapping for Power-of-Two Cyclotomics Through Homomorphic NTT. In: Chung, KM., Sasaki, Y. (eds) Advances in Cryptology – ASIACRYPT 2024. ASIACRYPT 2024. Lecture Notes in Computer Science, vol 15484. Springer, Singapore. https://doi.org/10.1007/978-981-96-0875-1_5

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