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Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Dawson, Ed / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti

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CiteScore 2017: 1.43

SCImago Journal Rank (SJR) 2017: 0.293
Source Normalized Impact per Paper (SNIP) 2017: 1.117

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Volume 9, Issue 3


On the concrete hardness of Learning with Errors

Martin R. Albrecht / Rachel Player / Sam Scott
Published Online: 2015-10-02 | DOI: https://doi.org/10.1515/jmc-2015-0016


The learning with errors (LWE) problem has become a central building block of modern cryptographic constructions. This work collects and presents hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and use a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the LWE problem.

Keywords: Learning with Errors; lattice-based cryptography; lattice reduction

MSC: 94A60; 11T71

We thank Steven Galbraith, Paul Kirchner and Cong Ling for pointing out mistakes and oversights in an earlier version of this work.

About the article

Received: 2015-03-19

Revised: 2015-09-20

Accepted: 2015-09-24

Published Online: 2015-10-02

Published in Print: 2015-10-01

Funding Source: EPSRC

Award identifier / Grant number: EP/L018543/1

Funding Source: ACE-CSR PhD grant

Funding Source: EPSRC

Award identifier / Grant number: EP/K035584/1

Citation Information: Journal of Mathematical Cryptology, Volume 9, Issue 3, Pages 169–203, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc-2015-0016.

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