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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2019: Advances in Cryptology – ASIACRYPT 2019 pp 59–90Cite as

An LLL Algorithm for Module Lattices

Part of the Lecture Notes in Computer Science book series (LNSC,volume 11922)

Abstract

The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors. We provide a generalization to R-modules contained in \(K^n\) for arbitrary number fields K and dimension n, with R denoting the ring of integers of K. Concretely, we introduce an algorithm that efficiently finds short vectors in rank-n modules when given access to an oracle that finds short vectors in rank-2 modules, and an algorithm that efficiently finds short vectors in rank-2 modules given access to a Closest Vector Problem oracle for a lattice that depends only on K. The second algorithm relies on quantum computations and its analysis is heuristic.

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Notes

  1. 1.

    See https://csrc.nist.gov/projects/post-quantum-cryptography.

  2. 2.

    Observe that even if complex conjugation might not be well defined over K (i.e., the element \(\bar{x}\) might not be in K even if x is), it is however always defined over \(K_\mathbb {R}\). In this article, complex conjugation will only be used on elements of \(K_\mathbb {R}\), and we make no assumption that K should be stable by conjugation.

  3. 3.

    The vectors \(\mathbf {b}_j\)’s are said to be \(K_\mathbb {R}\)-linearly independent if and only if there is no non-trivial ways to write the zero vector as a \(K_\mathbb {R}\)-linear combination of the \(\mathbf {b}_j\)’s. Because \(K_\mathbb {R}\) is a ring and not a field, this definition is stronger than requiring that none of the \(\mathbf {b}_j\)’s is in the span of the others.

  4. 4.

    Note that ideal scaling and size-reduction have been suggested in [FS10, Se. 4.1], but without a complexity analysis (polynomial complexity was claimed but not proved).

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Acknowledgments

We thank Léo Ducas for helpful discussions. This work was supported in part by BPI-France in the context of the national project RISQ (P141580), by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701) and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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Authors and Affiliations

  1. Univ. Lyon, EnsL, UCBL, CNRS, Inria, LIP, 69342, Lyon Cedex 07, France

    Changmin Lee, Alice Pellet-Mary & Damien Stehlé

  2. NTT Secure Platform Laboratories, Tokyo, Japan

    Alexandre Wallet

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  1. Changmin Lee
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  2. Alice Pellet-Mary
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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Lee, C., Pellet-Mary, A., Stehlé, D., Wallet, A. (2019). An LLL Algorithm for Module Lattices. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_3

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