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Discretization Error Reduction for High Precision Torus Fully Homomorphic Encryption

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

In recent history of fully homomorphic encryption, bootstrapping has been actively studied throughout many HE schemes. As bootstrapping is an essential process to transform somewhat homomorphic encryption schemes into fully homomorphic, enhancing its performance is one of the key factors of improving the utility of homomorphic encryption.

In this paper, we propose an extended bootstrapping for TFHE, which we name it by \(\textsf{EBS}\). One of the main drawback of TFHE bootstrapping was that the precision of bootstrapping is mainly decided by the polynomial dimension N. Thus if one wants to bootstrap with high precision, one must enlarge N, or take alternative method. Our \(\textsf{EBS}\) enables to use small N for parameter selection, but to bootstrap in higher dimension to keep high precision. Moreover, it can be easily parallelized for faster computation. Also, the \(\textsf{EBS}\) can be easily adapted to other known variants of TFHE bootstrappings based on the original bootstrapping algorithm.

We implement our \(\textsf{EBS}\) along with the full domain bootstrapping methods known (\(\textsf{FDFB}\), \(\textsf{TOTA}\), \(\textsf{Comp}\)), and show how much our \(\textsf{EBS}\) can improve the precision for those bootstrapping methods. We provide experimental results and thorough analysis with our \(\textsf{EBS}\), and show that \(\textsf{EBS}\) is capable of bootstrapping with high precision even with small N, thus small key size, and small complexity than selecting large N by birth.

© IACR 2023. This article is the final version submitted by the authors to the IACR and to Springer-Verlag on 17th February 2023. The version published by Springer-Verlag is available at https://doi.org/00.00000/0000000000.

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2022R1F1A1074291).

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Authors and Affiliations

  1. School of CyberSecurity, Korea University, Seoul, South Korea

    Kang Hoon Lee & Ji Won Yoon

Authors
  1. Kang Hoon Lee
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  2. Ji Won Yoon
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Corresponding author

Correspondence to Ji Won Yoon .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

A Appendix

A Appendix

1.1 A.1 Noise Analysis

We present detailed analysis of noise for the \(\textsf{BlindRotate}\), \(\textsf{KeySwitch}\), and \(\textsf{FDFB}\text {-}\textsf{ACC}\) in Table 5. Due to the error added during polynomial multiplication (with FFT), the experimental error standard deviation for \(N \ge 2048\) is larger than estimated results.

Table 5. Estimated noise standard deviation (with label \(^{(c)}\)) and experimental noise standard deviation (with label \(^{(E)}\)). The standard deviations are presented in the form of \(\log _{2}\).
Full size table

1.2 A.2 Benchmarks

In this section, we present the benchmark results for our parallelized and non-parallelized \(\textsf{EBS}\) along with the benchmarks for three full-domain bootstrapping methods. Note that none of the operations except the \(\textsf{EBS}\) were parallelized for fair comparison (Table 6).

Table 6. Benchmark results for \(\textsf{EBS}\) and three full-domain bootstrapping methods. The NP is for non-parallelized, and P denotes parallelized results. All results are presented in milliseconds (ms).
Full size table

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Lee, K.H., Yoon, J.W. (2023). Discretization Error Reduction for High Precision Torus Fully Homomorphic Encryption. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_2

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