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How to Find Ternary LWE Keys Using Locality Sensitive Hashing

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Cryptography and Coding (IMACC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13129))

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Abstract

Let \(As = b + e \bmod q\) be an LWE-instance with ternary keys \(s,e \in \{0, \pm 1\}^n\). Let s be taken from a search space of size \(\mathcal {S}\). A standard Meet-in-the-Middle attack recovers s in time \(\mathcal {S}^{0.5}\). Using the representation technique, a recent improvement of May shows that this can be lowered to approximately \(\mathcal {S}^{0.25}\) by guessing a sub-linear number of \(\varTheta (\frac{n}{\log n})\) coordinates from e. While guessing such an amount of e can asymptotically be neglected, for concrete instantiations of e.g. NTRU, BLISS or GLP the additional cost of guessing leads to complexities around \(\mathcal {S}^{0.3}\).

We introduce a locality sensitive hashing (LSH) technique based on Odlyzko’s work that avoids any guessing of e’s coordinates. This LSH technique involves a comparably small cost such that we can significantly improve on previous results, pushing complexities towards the asymptotic bound \(\mathcal {S}^{0.25}\). Concretely, using LSH we lower the MitM complexity estimates for the currently suggested NTRU and NTRU Prime instantiations by a factor in the range \(2^{20}-2^{49}\), and for BLISS and GLP parameters by a factor in the range \(2^{18}-2^{41}\).

Elena and Kirshanova: Supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2021-1748) and the “Young Russian Mathematics” grant.

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Notes

  1. 1.

    The scripts to reproduce the tables are available at https://github.com/ElenaKirshanova/ntru_with_lsh.

  2. 2.

    In fact, the ‘multiple’ labels assignment is what is done in [Ind01] to handle worst-case inputs. We could also use this algorithm but it turns out to be less memory-efficient than what we propose for the average-case setting.

  3. 3.

    We used commit 4027151 of the branch NTRU_keygen, https://github.com/lducas/leaky-LWE-Estimator/tree/NTRU_keygen.

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Authors and Affiliations

  1. Horst Görtz Institute for IT-Security, Ruhr University Bochum, Bochum, Germany

    Elena Kirshanova & Alexander May

  2. Immanuel Kant Baltic Federal University, Kaliningrad, Russia

    Elena Kirshanova

Authors
  1. Elena Kirshanova
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  2. Alexander May
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Correspondence to Alexander May .

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  1. Economics, Mathematics and Statistics, University of London Birkbeck, London, UK

    Maura B. Paterson

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Kirshanova, E., May, A. (2021). How to Find Ternary LWE Keys Using Locality Sensitive Hashing. In: Paterson, M.B. (eds) Cryptography and Coding. IMACC 2021. Lecture Notes in Computer Science(), vol 13129. Springer, Cham. https://doi.org/10.1007/978-3-030-92641-0_12

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  • DOI: https://doi.org/10.1007/978-3-030-92641-0_12

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