Advertisement

The Reductions for the Approximating Covering Radius Problem

  • Wenwen Wang
  • Kewei Lv
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10631)

Abstract

We establish the direct connection between \(\text {CRP}\) (Covering Radius Problem) and other lattice problems. We first prove that there is a polynomial-time rank-preserving reduction from approximating \(\text {CRP}\) to \(\text {BDD}^{\rho }\) (Covering Bounded Distance Decoding Problem). Furthermore, we show that there are polynomial-time reductions from \(\text {BDD}^{\rho }\) to approximating \(\text {CVP}\) (Closest Vector Problem) and \(\text {SIVP}\) (Shortest Independent Vector Problem), respectively. Hence, \(\text {CRP}\) reduces to \(\text {CVP}\) and \(\text {SIVP}\) under deterministic polynomial-time reductions.

Keywords

Lattice Polynomial time reductions Covering Radius Problem Covering Bounded Distance Decoding Problem 

Notes

Acknowledgements

This work was supported by National Natural Key R&D Program of China (Grant No. 2017YFB0802502), the Science and Technology Plan Projects of University of Jinan (Grant No. XKY1714), the Doctoral Initial Foundation of the University of Jinan (Grant No. XBS160100335), the Social Science Program of the University of Jinan (Grant No. 17YB01).

References

  1. 1.
    Aharonov, D., Regev, O.: Lattice problems in NP intersect coNP. J. ACM 52, 749–765 (2005). Preliminary version in FOCS 2004MathSciNetCrossRefGoogle Scholar
  2. 2.
    Babai, L.: On Lovasz lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Banaszczyk, W.: New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296, 625–635 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blömer, J., Naewe, S.: Sampling methods for shortest vectors, closest vectors and successive minima. Theor. Comput. Sci. 410, 1648–1665 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blöer, J., Seifert, J.P.: On the complexity of computing short linearly independent vectors and short bases in a lattice. In: 31th Annual ACM Symposium on Theory of Computing, pp. 711–720. ACM (1999)Google Scholar
  6. 6.
    Dubey, C., Holenstein, T.: Approximating the closest vector problem using an approximate shortest vector oracle. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 184–193. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22935-0_16CrossRefGoogle Scholar
  7. 7.
    Goldreich, O., Goldwasser, S.: On the limits of nonapproximability of lattice problems. J. Comput. Syst. Sci. 60(3), 540–563 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guruswami, V., Micciancio, D., Regev, O.: The complexity of the covering radius problem on lattices and codes. Comput. Complex. 14(2), 90–121 (2005). Preliminary version in CCC 2004CrossRefGoogle Scholar
  9. 9.
    Haviv, I.: The remote set problem on lattice. Comput. Complex. 24, 103–131 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Haviv, I., Regev, O.: Hardness of the covering radius problem on lattices. Chicago J. Theor. Comput. Sci. 04, 1–12 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lenstra, A., Lenstra, H., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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).  https://doi.org/10.1007/978-3-642-03356-8_34CrossRefGoogle Scholar
  13. 13.
    Micciancio, D.: Efficient reductions among lattice problems. In: 19th SODA Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 84–98 (2008)Google Scholar
  14. 14.
    Micciancio, D.: Almost perfect lattices, the covering radius problem, and applications to Ajtai’s connection factor. Electron. Colloq. Comput. Complex. 66, 1–39 (2003)Google Scholar
  15. 15.
    Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computation. SIAM J. Comput. 42(3), 1364–1391 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: A Cryptographic Perspective. The Kluwer International Series in Engineering and Computer Science, vol. 671. Kluwer Academic Publishers, Boston (2002)CrossRefGoogle Scholar
  17. 17.
    Peikert, C.: Limits on the hardness of lattice problems in \(\ell _{p}\) norms. Comput. Complex. 17(2), 300–351 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Business SchoolUniversity of JinanJinanChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  3. 3.Data Assurance Communication Security Research CenterChinese Academy of SciencesBeijingChina

Personalised recommendations