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Recent Progress on Coppersmith’s Lattice-Based Method: A Survey

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Mathematical Modelling for Next-Generation Cryptography

Part of the book series: Mathematics for Industry ((MFI,volume 29))

Abstract

In 1996, Coppersmith proposed a lattice-based method to solve the small roots of a univariate modular equation in polynomial time. Since its invention, Coppersmith’s method has become an important tool in the cryptanalysis of RSA crypto algorithm and its variants. In 2006, Jochemsz and May introduced a general strategy to solve small roots of any form of multivariate modular equations in polynomial time. Based on Jochemsz–May’s strategy, for any given multivariate equations one can easily construct the desired lattices with triangular matrix basis. However, for some attacks, Jochemsz–May’s general strategy could not fully capture the algebraic structure of the target polynomials. Thus, some sophisticated techniques that can deeply exploit the algebraic relations have been proposed. In this paper, we give a survey of these recent approaches for lattice constructions, and also give small examples to show how these approaches work.

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Author information

Authors and Affiliations

  1. School of Frontier Sciences, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8561, Japan

    Yao Lu & Noboru Kunihiro

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Liqiang Peng

Authors
  1. Yao Lu
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  2. Liqiang Peng
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  3. Noboru Kunihiro
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Correspondence to Yao Lu .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

  2. Kyushu University, Fukuoka, Japan

    Masato Wakayama

  3. Tokyo Institute of Technology, Tokyo, Japan

    Keisuke Tanaka

  4. The University of Tokyo, Kashiwa, Japan

    Noboru Kunihiro

  5. University of the Ryukyus, Nakagami-gun, Japan

    Kazufumi Kimoto

  6. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Dung Hoang Duong

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Lu, Y., Peng, L., Kunihiro, N. (2018). Recent Progress on Coppersmith’s Lattice-Based Method: A Survey. In: Takagi, T., Wakayama, M., Tanaka, K., Kunihiro, N., Kimoto, K., Duong, D. (eds) Mathematical Modelling for Next-Generation Cryptography. Mathematics for Industry, vol 29. Springer, Singapore. https://doi.org/10.1007/978-981-10-5065-7_16

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  • DOI: https://doi.org/10.1007/978-981-10-5065-7_16

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