Skip to main content
SpringerLink
Log in

Error analysis of weak Poly-LWE instances

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Error distribution plays a central role in the security of encryption based on the Learning with Errors (LWE) problem and its variants. In this paper, we investigate the error distribution of weak Poly-LWE instances. For this purpose, we derive a closed-form formula to compute the mapped error distribution. With this algebraic approach to evaluate the error, we examine the recently proposed attacks on Poly-LWE and Ring-LWE and reassess their parameters in order to include more instances. Notably, our method can also be applied to non-Gaussian error. We conduct experiments to investigate the shape of the mapped error distribution and confirm that in many cases it is no longer Gaussian nor uniform; our experimental results from distinguishers also validate our theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bos, J.W., Lauter, K.E., Loftus, J., Naehrig, M.: Improved security for a ring-based fully homomorphic encryption scheme. In: IMA International Conference, pp. 45–64. Springer (2013)

  2. Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 575–584. ACM (2013)

  3. Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Advances in cryptology—CRYPTO 2011, volume 6841 of Lecture Notes in Computer Science, pp. 505–524. Springer, Heidelberg (2011)

  4. Castryck, W., Iliashenko, I., Vercauteren, F.: Provably weak instances of ring-LWE revisited. In: Advances in cryptology—EUROCRYPT 2016. Part I, volume 9665 of Lecture Notes in Computer Science, pp. 147–167. Springer, Berlin (2016)

  5. Chen, H., Lauter, K., Stange, K.E.: Attacks on search rlwe (2015)

  6. Chen, H., Lauter, K.E., Stange, K.E.: Vulnerable galois rlwe families and improved attacks. In: IACR Cryptology ePrint Archive, vol. 2016, p. 193 (2016)

  7. Eisenträger, K., Hallgren, S., Lauter, K.: Weak instances of PLWE. In: Selected areas in cryptography—SAC 2014, volume 8781 of Lecture Notes in Computer Science, pp. 183–194. Springer, Cham (2014)

  8. Elias, Y., Lauter, K.E., Ozman, E., Stange, K.E.: Provably weak instances of ring-LWE. In: Advances in cryptology—CRYPTO 2015. Part I, volume 9215 of Lecture Notes in Computer Science, pp. 63–92. Springer, Heidelberg (2015)

  9. Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. In: IACR Cryptology ePrint Archive 2012, p. 144 (2012)

  10. Knuth, D.E., et al.: The art of computer programming. Sorting and searching 3, 426–458 (1999)

    Google Scholar 

  11. Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Advances in cryptology—EUROCRYPT 2010, volume 6110 of Lecture Notes in Computer Science, pp. 1–23. Springer, Berlin (2010)

  12. Peikert, C.: How (not) to instantiate ring-LWE. In: Security and Cryptography for Networks, volume 9841 of Lecture Notes in Computer Science, pp. 411–430. Springer, Cham (2016)

  13. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 84–93. ACM, New York (2005)

  14. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), Art. 34, 40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Yao Chen & Guang Gong

  2. Clemson University, Clemson, SC, 29634, USA

    Benjamin M. Case & Shuhong Gao

Authors
  1. Yao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benjamin M. Case
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuhong Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guang Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yao Chen.

Additional information

This article is part of the Topical Collection on Special Issue: Mathematical Methods for Cryptography

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Case, B.M., Gao, S. et al. Error analysis of weak Poly-LWE instances. Cryptogr. Commun. 11, 411–426 (2019). https://doi.org/10.1007/s12095-018-0301-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-018-0301-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation