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Bootstrapping Bits with CKKS

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Advances in Cryptology – EUROCRYPT 2024 (EUROCRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14652))

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Abstract

The Cheon–Kim–Kim–Song (CKKS) fully homomorphic encryption scheme is designed to efficiently perform computations on real numbers in an encrypted state. Recently, Drucker et al [J. Cryptol.] proposed an efficient strategy to use CKKS in a black-box manner to perform computations on binary data.

In this work, we introduce several CKKS bootstrapping algorithms designed specifically for ciphertexts encoding binary data. Crucially, the new CKKS bootstrapping algorithms enable to bootstrap ciphertexts containing the binary data in the most significant bits. First, this allows to decrease the moduli used in bootstrapping, saving a larger share of the modulus budget for non-bootstrapping operations. In particular, we obtain full-slot bootstrapping in ring degree \(2^{14}\) for the first time. Second, the ciphertext format is compatible with the one used in the DM/CGGI fully homomorphic encryption schemes. Interestingly, we may combine our CKKS bootstrapping algorithms for bits with the fast ring packing technique from Bae et al [CRYPTO’23]. This leads to a new bootstrapping algorithm for DM/CGGI that outperforms the state-of-the-art approaches when the number of bootstraps to be performed simultaneously is in the low hundreds.

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Notes

  1. 1.

    We note that the \(h_1\) noise-reducing function is run after every binary gate in [DMPS24], which is over-conservative as noted in [ADE+23]; a saving of a factor 3.4 can be obtained by calling the \(h_1\) function less frequently and using complex bootstrapping.

  2. 2.

    This is actually intrinsic. The number of slots s modulo 2 is exactly the number of distinct factors modulo 2 of the cyclotomic polynomial and the degrees of these factors are all equal. Assume N is the degree of the cyclotomic polynomial, and d the degree of the factors: the number of distinct factors is \(s \le N/d\). We also have \(s \le 2^d\), the number of polynomials modulo 2 of degree \(<d\). The second bound implies that \(d \ge \log _2 (s)\) and the first one then gives \(s \cdot \log _2 (s) \le N\).

  3. 3.

    As far as we are aware of, it has not been formally described so far in an article, but it is used in [EPF22, Cry22].

  4. 4.

    For the sake of comparison, \(\textsf{BinBoot}\) for \(\textsf{Param14}\) and real bootstrapping (optimized for latency), takes \(84.8\,\mu \)s per gate which is 4.82x slower than \(\textsf{Param16}\) with complex bootstrapping (optimized for throughput).

  5. 5.

    One may need \(\le 30\) bits primes for lazy modular reductions (see [Har14]). To be compatible with this technique, we may use slightly smaller primes for \(\textsf{Param14}\) and \(\textsf{Param16}\) for Tables 5 and 7.

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Author information

Authors and Affiliations

  1. CryptoLab Inc., Seoul, Republic of Korea

    Youngjin Bae, Jung Hee Cheon & Jaehyung Kim

  2. Seoul National University, Seoul, Republic of Korea

    Jung Hee Cheon

  3. CryptoLab Inc., Lyon, France

    Damien Stehlé

Authors
  1. Youngjin Bae
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  2. Jung Hee Cheon
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  3. Jaehyung Kim
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  4. Damien Stehlé
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Correspondence to Youngjin Bae .

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Editors and Affiliations

  1. Zama, Paris, France

    Marc Joye

  2. Ruhr University Bochum, Bochum, Germany

    Gregor Leander

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Bae, Y., Cheon, J.H., Kim, J., Stehlé, D. (2024). Bootstrapping Bits with CKKS. In: Joye, M., Leander, G. (eds) Advances in Cryptology – EUROCRYPT 2024. EUROCRYPT 2024. Lecture Notes in Computer Science, vol 14652. Springer, Cham. https://doi.org/10.1007/978-3-031-58723-8_4

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