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International Conference on Information Security and Cryptology

ICISC 2018: Information Security and Cryptology – ICISC 2018 pp 196–214Cite as

On the Complexity of the LWR-Solving BKW Algorithm

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11396))

Abstract

Duc et al. applied the Blum-Kalai-Wasserman (BKW) algorithm to the learning with rounding (LWR) problem. The number of blocks is a parameter of the BKW algorithm. By optimizing the number of blocks, we can minimize the time complexity of the BKW algorithm. However, Duc et al. did not derive the optimal number of blocks theoretically, but they searched it for numerically. In this paper, we theoretically derive the asymptotically optimal number of blocks and show the minimum time complexity of the algorithm. Furthermore, we derive an equation that relates the Gaussian parameter \(\sigma \) of the LWE problem and the modulus p of the LWR problem. When \(\sigma \) and p satisfy the equation, the asymptotic time complexity of the BKW algorithm to solve the LWE and LWR problems are the same.

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Notes

  1. 1.

    Let \(\tilde{a}\) satisfies \(\exp (\pi \alpha _\mathrm {lwr}^22^{\tilde{a}})= q^{n/{\tilde{a}}}\), and Let \(t_{\tilde{a}}\) be the time complexity with \(a=\tilde{a}\), namely \(t_{\tilde{a}}=O(\exp (\pi \alpha _\mathrm {lwr}^22^{\tilde{a}} )(n/a)\ln q)\). If we set \(a > \tilde{a}\), then we obtain \(t_a=O(\exp (\pi \alpha _\mathrm {lwr}^22^a ) (n/a)\ln q)\), and \(t_a>t_{\tilde{a}}\) since \(\exp (\pi \alpha _\mathrm {lwr}^22^{a})>\exp (\pi \alpha _\mathrm {lwr}^22^{\tilde{a}})\). If we set \(a < \tilde{a}\), then we obtain \(t=O(q^{n/a}(n/a)\ln q)\), and \(t_a>t_{\tilde{a}}\) since \(q^{n/a} > q^{n/{\tilde{a}}}\). Therefore, \(\tilde{a}\) is asymptotically optimal.

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Authors and Affiliations

  1. KDDI Research, Inc., Saitama, Japan

    Hiroki Okada, Kazuhide Fukushima & Shinsaku Kiyomoto

  2. The University of Tokyo, Tokyo, Japan

    Atsushi Takayasu & Tsuyoshi Takagi

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  1. Hiroki Okada
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  2. Atsushi Takayasu
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  3. Kazuhide Fukushima
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  4. Shinsaku Kiyomoto
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  1. Sejong University, Seoul, Korea (Republic of)

    Kwangsu Lee

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Okada, H., Takayasu, A., Fukushima, K., Kiyomoto, S., Takagi, T. (2019). On the Complexity of the LWR-Solving BKW Algorithm. In: Lee, K. (eds) Information Security and Cryptology – ICISC 2018. ICISC 2018. Lecture Notes in Computer Science(), vol 11396. Springer, Cham. https://doi.org/10.1007/978-3-030-12146-4_13

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