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On the complexity of the BKW algorithm on LWE

Abstract

This work presents a study of the complexity of the Blum–Kalai–Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension \(n \approx 250\) when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples.

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Notes

  1. It is common in the literature on LWE to parameterise discrete Gaussian distributions by \(s = \sigma \sqrt{2\pi }\) instead of \(\sigma \). Since we are mainly interested in the “size” of the noise, we deviate from this standard in this work.

  2. However, a detailed study of the algorithm to the LPN case was provided [14], which in fact heavily inspired this work. The authors of [14] conducted a detailed analysis of the BKW algorithm as applied to LPN, while also giving revised security estimates for some HB-type authentication protocols relying on the hardness of LPN.

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Acknowledgments

We are grateful to Frederik Johansson for advice on numerical integration. We are also grateful to anonymous referees whose feedback substantially improved this work. The work described in this paper has been partially supported by the Royal Society Grant JP090728 and by the Commission of the European Communities through the ICT program under contract ICT-2007-216676 (ECRYPT-II). Jean-Charles Faugère, and Ludovic Perret are also supported by the Computer Algebra and Cryptography (CAC) project (ANR-09-JCJCJ-0064-01) and the HPAC grant of the French National Research Agency. Carlos Cid is supported in part by the US Army Research Laboratory and the UK Ministry of Defence under Agreement Number W911NF-06-3-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the US Army Research Laboratory, the U.S. Government, the UK Ministry of Defense, or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.

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Correspondence to Robert Fitzpatrick.

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Communicated by R. Steinwandt.

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Albrecht, M.R., Cid, C., Faugère, JC. et al. On the complexity of the BKW algorithm on LWE. Des. Codes Cryptogr. 74, 325–354 (2015). https://doi.org/10.1007/s10623-013-9864-x

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  • DOI: https://doi.org/10.1007/s10623-013-9864-x

Keywords

  • Learning with errors
  • BKW
  • LPN
  • FHE

Mathematics Subject Classification

  • 94A60