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When NTT Meets Karatsuba: Preprocess-then-NTT Technique Revisited

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Information and Communications Security (ICICS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12919))

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Abstract

The Number Theoretic Transform (NTT) technique is widely used to implement cryptographic schemes based on the Ring Learning With Errors problem(RLWE), since it provides efficient algorithm for multiplication of polynomials over the finite field. However, the module in NTT must be big enough such that the finite field has some special root of unity, which makes the corresponding schemes a bit less efficient. At Inscrypt 2018, Zhou et al. proposed a technique called Preprocess-then-NTT to relax the constraint for the modulus while keeping NTT work, at the cost of time complexity. In this paper, we improve the Preprocess-then-NTT technique by mixing it with Karatsuba multiplication such that the time complexity is better than the original NTT algorithm asymptotically. The claim was also verified in our experiments. As a result, our new algorithm not only relaxes the constraint for modulus but also improves efficiency in practice. In addition, we also present some advantages and applications of our new algorithm.

This work is supported by National Key Research and Development Program of China (No. 2018YFA0704705, 2020YFA0712300), National Natural Science Foundation of China (No. 62032009).

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Acknowledgements

Thank the anonymous referees very much for their helpful comments. Thank Prof. Yunlei Zhao very much for his valuable suggestions on improving this paper and directing us to Nussbaumer’s trick [17]. Thank Renzhang Liu for his help on the programming.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yiming Zhu, Zhen Liu & Yanbin Pan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yiming Zhu, Zhen Liu & Yanbin Pan

  3. Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, 430062, China

    Zhen Liu

Authors
  1. Yiming Zhu
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  2. Zhen Liu
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  3. Yanbin Pan
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Correspondence to Yanbin Pan .

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Editors and Affiliations

  1. Singapore Management University, Singapore, Singapore

    Debin Gao

  2. Tsinghua University, Beijing, China

    Qi Li

  3. Xi'an Jiaotong University, Xi'an, China

    Xiaohong Guan

  4. Chongqing University, Chongqing, China

    Xiaofeng Liao

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Zhu, Y., Liu, Z., Pan, Y. (2021). When NTT Meets Karatsuba: Preprocess-then-NTT Technique Revisited. In: Gao, D., Li, Q., Guan, X., Liao, X. (eds) Information and Communications Security. ICICS 2021. Lecture Notes in Computer Science(), vol 12919. Springer, Cham. https://doi.org/10.1007/978-3-030-88052-1_15

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  • DOI: https://doi.org/10.1007/978-3-030-88052-1_15

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  • Print ISBN: 978-3-030-88051-4

  • Online ISBN: 978-3-030-88052-1

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