Skip to main content
SpringerLink
Log in
Book cover

Australasian Conference on Information Security and Privacy

ACISP 2014: Information Security and Privacy pp 322–337Cite as

Lattice Decoding Attacks on Binary LWE

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8544)

Abstract

We consider the binary-LWE problem, which is the learning with errors problem when the entries of the secret vector are chosen from { 0, 1} or { − 1, 0, 1 }. Our main result is an improved lattice decoding algorithm for binary-LWE, by translating to the inhomogeneous short integer solution (ISIS) problem, and then re-scaling the lattice. We also discuss modulus switching as an approach to the problem. Our conclusion is that binary-LWE is easier than general LWE. We give experimental results, and theoretical estimates for parameters that achieve certain security levels.

Keywords

  • lattice attacks
  • learning with errors
  • closest vector problem

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albrecht, M.R., Cid, C., Faugère, J.-C., Fitzpatrick, R., Perret, L.: On the Complexity of the BKW Algorithm on LWE. In: Designs, Codes and Cryptography (Published online July 19, 2013) (to appear)

    Google Scholar 

  2. Albrecht, M.R., Fitzpatrick, R., Göpfert, F.: On the Efficacy of Solving LWE by Reduction to Unique-SVP. In: Proceedings of 2013 International Conference on Information Security and Cryptology (2013) (to appear)

    Google Scholar 

  3. Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  4. Arora, S., Ge, R.: New Algorithms for Learning in Presence of Errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  5. Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. Journal of ACM 50(4), 506–519 (2003)

    CrossRef  MathSciNet  Google Scholar 

  6. Brakerski, Z., Vaikuntanathan, V.: Efficient Fully Homomorphic Encryption from (Standard) LWE. In: Ostrovsky, R. (ed.) IEEE FOCS 2011, pp. 97–106 (2011)

    Google Scholar 

  7. Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) ACM STOC 2013, pp. 575–584 (2013)

    Google Scholar 

  8. Cadé, D., Pujol, X., Stehlé, D.: FPLLL (2013), http://perso.ens-lyon.fr/damien.stehle/fplll

  9. Chen, Y., Nguyen, P.Q.: BKZ 2.0: Better Lattice Security Estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  10. Gama, N., Nguyen, P.Q.: Predicting Lattice Reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  11. Gama, N., Nguyen, P.Q., Regev, O.: Lattice enumeration using extreme pruning. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 257–278. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  12. Kannan, R.: Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research 12(3), 415–440 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Lindner, R., Peikert, C.: Better key sizes (and attacks) for LWE-based encryption. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 319–339. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  14. Liu, M., Nguyen, P.Q.: Solving BDD by Enumeration: An Update. In: Dawson, E. (ed.) CT-RSA 2013. LNCS, vol. 7779, pp. 293–309. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  15. Lyubashevsky, V., Micciancio, D.: On Bounded Distance Decoding, Unique Shortest Vectors, and the Minimum Distance Problem. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 577–594. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  16. Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  17. Luzzi, L., Stehlé, D., Ling, C.: Decoding by Embedding: Correct Decoding Radius and DMT Optimality. IEEE Transactions on Information Theory 59(5), 2960–2973 (2013)

    CrossRef  Google Scholar 

  18. Micciancio, D., Mol, P.: Pseudorandom Knapsacks and the Sample Complexity of LWE Search-to-Decision Reductions. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 465–484. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  19. Micciancio, D., Peikert, C.: Hardness of SIS and LWE with Small Parameters. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 21–39. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  20. Micciancio, D., Regev, O.: Lattice-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post Quantum Cryptography, pp. 147–191. Springer (2009)

    Google Scholar 

  21. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC 2005, pp. 84–93. ACM (2005)

    Google Scholar 

  22. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM 56(6), article 34 (2009)

    Google Scholar 

  23. Schnorr, C.P.: Lattice reduction by random sampling and birthday methods. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 145–156. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Auckland, New Zealand

    Shi Bai & Steven D. Galbraith

Authors
  1. Shi Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven D. Galbraith
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centre for Computer and Information Security Research, School of Computer Science and Software Engineering, University of Wollongong, Northfields Avenue, 2522, Wollongong, NSW, Australia

    Willy Susilo & Yi Mu & 

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Bai, S., Galbraith, S.D. (2014). Lattice Decoding Attacks on Binary LWE. In: Susilo, W., Mu, Y. (eds) Information Security and Privacy. ACISP 2014. Lecture Notes in Computer Science, vol 8544. Springer, Cham. https://doi.org/10.1007/978-3-319-08344-5_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08344-5_21

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08343-8

  • Online ISBN: 978-3-319-08344-5

  • eBook Packages: Computer ScienceComputer Science (R0)

search

Navigation