Skip to main content
SpringerLink
Log in

Solving Closest Vector Instances Using an Approximate Shortest Independent Vectors Oracle

  • Regular Paper
  • Published:
Journal of Computer Science and Technology Aims and scope Submit manuscript

Abstract

Given an n-dimensional lattice L and some target vector, this paper studies the algorithms for approximate closest vector problem (CVPγ) by using an approximate shortest independent vectors problem oracle (SIVPγ). More precisely, if the distance between the target vector and the lattice is no larger than c \( {\scriptscriptstyle \frac{c}{\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \) for arbitrary large but finite constant c > 0, we give randomized and deterministic polynomial time algorithms to find a closest vector, while previous reductions were only known for \( {\scriptscriptstyle \frac{c}{2\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \). Moreover, if the distance between the target vector and the lattice is larger than some quantity with respect to λ n (L), using SIVPγ oracle and Babai’s nearest plane algorithm, we can solve \( \mathrm{CVP}\upgamma \sqrt{n} \) in deterministic polynomial time. Specially, if the approximate factor γ ϵ (1, 2) in the SIVPγ oracle, we obtain a better reduction factor for CVP.

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

Similar content being viewed by others

References

  1. Goldreich O, Micciancio D, Safra S, Seifert J P. Approximating shortest lattice vectors is not harder than approxi mating closest lattice vectors. Information Processing Letters, 1999, 71(2): 55–61.

    Article  MathSciNet  MATH  Google Scholar 

  2. Dinur I, Kindler G, Raz R, Safra S. Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica, 2003, 23(2): 205–243.

    Article  MathSciNet  MATH  Google Scholar 

  3. Haviv I, Regev O. Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In Proc. the 39th Annual ACM Symp. Theory of Computing, June 2007, pp.469-477.

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Ajtai M, Kumar R, Sivakumar D. Sampling short lattice vectors and the closest lattice vector problem. In Proc. the 17th IEEE Annual Conf. Computational Complexity, May 2002, pp.41-45.

  6. Kannan R. Algorithmic geometry of numbers. Annual Review of Computer Science, 1987, 2: 231–267.

  7. Banaszczyk W. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 1993, 296(1): 625–635.

    Article  MathSciNet  MATH  Google Scholar 

  8. Lyubashevsky V, Micciancio D. On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In Lecture Notes in Computer Science 5677, Halevi S (ed.), Springer Berlin Heidelberg, 2009, pp.577-594.

    Google Scholar 

  9. Dubey C, Holenstein T. Approximating the closest vector problem using an approximate shortest vector oracle. In Lecture Notes in Computer Science 6845, Goldberg L A, Jansen K, Ravi R, Rolim J D P (eds.), Springer Berlin Heidelberg, 2011, pp.184-193.

  10. Ajtai M. Generating hard instances of lattice problems (extended abstract). In Proc. the 28th ACM Annual Symp. Theory of Computing, May 1996, pp.99-108.

  11. Regev O. On lattices, learning with errors, random linear codes, and cryptography. In Proc. the 37th Annual ACM Symp. Theory of Computing, May 2005, pp.84-93.

  12. Blömer J, Seifert J. On the complexity of computing short linearly independent vectors and short bases in a lattice. In Proc. the 31st Annual ACM Symp. Theory of Computing, May 1999, pp.711-720.

  13. Micciancio D. Efficient reductions among lattice problems. In Proc. the 19th Annual ACM-SIAM Symp. Discrete Algorithms, January 2008, pp.84-93.

  14. Babai L. On lovász lattice reduction and the nearest lattice point problem. Combinatorica, 1986, 6(1): 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  15. Micciancio D, Goldwasser S. Complexity of Lattice Problems: A Cryptographic Perspective. Kluwer Academic Publishers, 2002.

  16. Klein P. Finding the closest lattice vector when it’s unusually close. In Proc. the 11th Annual ACM-SIAM Symp. Discrete Algorithms, January 2000, pp.937-941.

  17. Cai J. A new transference theorem in the geometry of numbers and new bounds for Ajtai’s connection factor. Discrete Applied Mathematics, 2003, 126(1): 9–31.

    Article  MathSciNet  MATH  Google Scholar 

  18. Micciancio D. The geometry of lattice cryptography. In Lecture Notes in Computer Science 6858, Aldini A, Gorrieri R (eds.), Springer Berlin Heidelberg, 2011, pp.185-210.

    Google Scholar 

  19. Guruswami V, Micciancio D, Regev O. The complexity of the covering radius problem. Computational Complexity, 2005, 14(2): 90–121.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Cheng-Liang Tian & Dong-Dai Lin

  2. Institute for Advanced Study, Tsinghua University, Beijing, 100084, China

    Wei Wei

Authors
  1. Cheng-Liang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dong-Dai Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cheng-Liang Tian.

Additional information

This work is partially supported by the National Basic Research 973 Program of China under Grant No. 2011CB302400, the National Natural Science Foundation of China under Grant Nos. 61379139 and 61133013, and the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA06010701.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, CL., Wei, W. & Lin, DD. Solving Closest Vector Instances Using an Approximate Shortest Independent Vectors Oracle. J. Comput. Sci. Technol. 30, 1370–1377 (2015). https://doi.org/10.1007/s11390-015-1604-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11390-015-1604-4

Keywords

Search

Navigation