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On the Hardness of the NTRU Problem

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Advances in Cryptology – ASIACRYPT 2021 (ASIACRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13090))

Abstract

The 25 year-old NTRU problem is an important computational assumption in public-key cryptography. However, from a reduction perspective, its relative hardness compared to other problems on Euclidean lattices is not well-understood. Its decision version reduces to the search Ring-LWE problem, but this only provides a hardness upper bound.

We provide two answers to the long-standing open problem of providing reduction-based evidence of the hardness of the NTRU problem. First, we reduce the worst-case approximate Shortest Vector Problem over ideal lattices to an average-case search variant of the NTRU problem. Second, we reduce another average-case search variant of the NTRU problem to the decision NTRU problem.

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Notes

  1. 1.

    Note that in order to avoid having absolute values everywhere in the rest of the article, we define \(\varDelta _K\) as the absolute value of the discriminant of K.

  2. 2.

    The scaling by a factor \(1/\sqrt{2}\) in the complex case ensures that the norm of \(\psi (t)\) is still equal to |t|, which allows simpler expressions.

  3. 3.

    The term “absolute precision” refers here to \(|\tilde{x}-x| \le 2^{-\ell }\), as opposed to the “relative precision” which would be \(\frac{|\tilde{x}-x|}{|x|} \le 2^{-\ell }\).

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Acknowledgment

The authors thank Koen de Boer, Léo Ducas, Guillaume Hanrot, Miruna Rosca aux Adeline Roux-Langlois for insightful discussions. The first author was supported in part by CyberSecurity Research Flanders with reference number VR20- 192203 and by the Research Council KU Leuven grant C14/18/067 on Cryptanalysis of Post-quantum Cryptography. The second author was supported in part by European Union Horizon 2020 Research and Innovation Program Grant 780701 and BPI-France in the context of the national project RISQ (P141580).

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Authors and Affiliations

  1. CNRS, Inria and Université de Bordeaux, Bordeaux, France

    Alice Pellet-Mary

  2. ENS de Lyon, Lyon, France

    Damien Stehlé

  3. Institut Universitaire de France, Paris, France

    Damien Stehlé

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  1. Alice Pellet-Mary
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  2. Damien Stehlé
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Correspondence to Alice Pellet-Mary or Damien Stehlé .

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Editors and Affiliations

  1. NTT Corporation, Tokyo, Japan

    Mehdi Tibouchi

  2. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Pellet-Mary, A., Stehlé, D. (2021). On the Hardness of the NTRU Problem. In: Tibouchi, M., Wang, H. (eds) Advances in Cryptology – ASIACRYPT 2021. ASIACRYPT 2021. Lecture Notes in Computer Science(), vol 13090. Springer, Cham. https://doi.org/10.1007/978-3-030-92062-3_1

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  • DOI: https://doi.org/10.1007/978-3-030-92062-3_1

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