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An Extension of Kannan’s Embedding for Solving Ring-Based LWE Problems

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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

The hardness of the learning with errors (LWE) problem supports the security of modern lattice-based cryptography. Ring-based LWE is the analog of LWE over univariate polynomial rings that includes the polynomial-LWE and the ring-LWE, and it is useful to construct efficient and compact LWE-based cryptosystems. Any ring-based LWE instance can be transformed to an LWE instance, which can also be reduced to a particular case of the shortest vector problem (SVP) on a certain lattice by Kannan’s embedding. In this paper, we extend Kannan’s embedding for solving the search version of the ring-based LWE problem. Specifically, we propose a new extended lattice to include multiple short errors that are amplified by rotation operations for the coefficient vector of an error polynomial. Since multiple short errors have the same length and are embedded in our extended lattice, our extension could increase the success probability of solving the ring-based LWE problem by using the Block Korkine-Zorotarev (BKZ) algorithm that is widely used in cryptanalysis. We demonstrate the efficacy of our extension by experiments for solving various ring-based LWE instances.

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Authors and Affiliations

  1. NTT Social Informatics Laboratories, Tokyo, Japan

    Satoshi Nakamura

  2. Department of Mathematics, Rikkyo University, Tokyo, Japan

    Masaya Yasuda

Authors
  1. Satoshi Nakamura
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  2. Masaya Yasuda
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Correspondence to Masaya Yasuda .

Editor information

Editors and Affiliations

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

    Maura B. Paterson

A A Sample Code for Our Extended Kannan’s Embedding

A A Sample Code for Our Extended Kannan’s Embedding

Here we give a sample Python code in SageMath [12] of our extended Kannan’s embedding for solving a ring-based LWE instance. (We use the ring-LWE oracle generator in SageMath to generate ring-LWE samples, and also BKZ 2.0 in fpylll for BKZ. See also Subsect. 4.1 for details.)

figure b
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Nakamura, S., Yasuda, M. (2021). An Extension of Kannan’s Embedding for Solving Ring-Based LWE Problems. 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_10

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  • DOI: https://doi.org/10.1007/978-3-030-92641-0_10

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  • Online ISBN: 978-3-030-92641-0

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