Skip to main content
SpringerLink
Log in

Inapproximability Results for Computational Problems on Lattices

  • Chapter
  • First Online:
The LLL Algorithm

Part of the book series: Information Security and Cryptography ((ISC))

Abstract

In this article, we present a survey of known inapproximability results for computational problems on lattices, viz. the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), the Closest Vector Problem with Preprocessing (CVPP), the Covering Radius Problem (CRP), the Shortest Independent Vectors Problem (SIVP), and the Shortest Basis Problem (SBP).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Micciancio, S. Goldwasser. Complexity of lattice problems, A cryptographic perspective. Kluwer Academic Publishers, 2002

    Google Scholar 

  2. R. Kumar, D. Sivakumar. Complexity of SVP – A reader’s digest. SIGACT News, 32(3), Complexity Theory Column (ed. L. Hemaspaandra), 2001, pp 40–52

    Google Scholar 

  3. O. Regev. On the Complexity of Lattice Problems with polynomial Approximation Factors. In Proc. of the LLL+25 Conference, Caen, France, June 29-July 1, 2007

    Google Scholar 

  4. C.F. Gauss. Disquisitiones arithmeticae. (leipzig 1801), art. 171. Yale University. Press, 1966. English translation by A.A. Clarke

    Google Scholar 

  5. H. Minkowski. Geometrie der zahlen. Leizpig, Tuebner, 1910

    MATH  Google Scholar 

  6. A.K. Lenstra, H.W. Lenstra, L. Lovász. Factoring polynomials with rational coefficients. Mathematische Ann., 261, 1982, pp 513–534

    Google Scholar 

  7. J.C. Lagarias, A.M. Odlyzko. Solving low-density subset sum problems. Journal of the ACM, 32(1), 1985, pp 229–246

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Landau, G.L. Miller. Solvability of radicals is in polynomial time. Journal of Computer and Systems Sciences, 30(2), 1985, pp 179–208

    Article  MATH  MathSciNet  Google Scholar 

  9. H.W. Lenstra. Integer programming with a fixed number of variables. Tech. Report 81–03, Univ. of Amsterdam, Amstredam, 1981

    Google Scholar 

  10. R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of the 15th Annual ACM Symposium on Theory of Computing, 1983, pp 193–206

    Google Scholar 

  11. C.P. Schnorr. A hierarchy of polynomial-time basis reduction algorithms. In Proc. of Conference on Algorithms, P\(\acute{\mathrm{e}}\)ecs (Hungary), 1985, pp 375–386

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Ajtai, R. Kumar, D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In Proc. of the 33rd Annual ACM Symposium on the Theory of Computing, 2001, pp 601–610

    Google Scholar 

  14. P. van Emde Boas. Another NP-complete problem and the complexity of computing short vectors in a lattice. Tech. Report 81-04, Mathematische Instiut, University of Amsterdam, 1981

    Google Scholar 

  15. M. Ajtai. The shortest vector problem in L 2 is NP-hard for randomized reductions. In Proc. of the 30th Annual ACM Symposium on the Theory of Computing, 1998, pp 10–19

    Google Scholar 

  16. J.Y. Cai, A. Nerurkar. Approximating the SVP to within a factor \((1 + 1/{\mathrm{dim}}^{\varepsilon })\) is NP-hard under randomized reductions. In Proc. of the 13th Annual IEEE Conference on Computational Complexity, 1998, pp 151–158

    Google Scholar 

  17. D. Micciancio. The shortest vector problem is NP-hard to approximate to within some constant. In Proc. of the 39th IEEE Symposium on Foundations of Computer Science, 1998

    Google Scholar 

  18. S. Khot. Hardness of approximating the shortest vector problem in lattices. Journal of the ACM, 52(5), 2005, pp 789–808

    Article  MathSciNet  Google Scholar 

  19. I. Haviv, O. Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. To appear in Proc. of the 39th Annual ACM Symposium on the Theory of Computing, 2007

    Google Scholar 

  20. M. Ajtai. Generating hard instances of lattice problems. In Proc. of the 28th Annual ACM Symposium on the Theory of Computing, 1996, pp 99–108

    Google Scholar 

  21. M. Ajtai, C. Dwork. A public-key cryptosystem with worst-case/average-case equivalence. In Proc. of the 29th Annual ACM Symposium on the Theory of Computing, 1997, pp 284–293

    Google Scholar 

  22. J.Y. Cai, A. Nerurkar. An improved worst-case to average-case connection for lattice problems. In 38th IEEE Symposium on Foundations of Computer Science, 1997

    Google Scholar 

  23. J.Y. Cai. Applications of a new transference theorem to Ajtai’s connection factor. In Proc. of the 14th Annual IEEE Conference on Computational Complexity, 1999

    Google Scholar 

  24. O. Regev. New lattice based cryptographic constructions. To appear in Proc. of the 35th Annual ACM Symposium on the Theory of Computing, 2003

    Google Scholar 

  25. O. Goldreich, D. Micciancio, S. Safra, J.P. Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 1999

    Google Scholar 

  26. S. Arora, L. Babai, J. Stern, E.Z. Sweedyk. The hardness of approximate optima in lattices, codes and systems of linear equations. Journal of Computer and Systems Sciences (54), 1997, pp 317–331

    Google Scholar 

  27. I. Dinur, G. Kindler, S. Safra. Approximating CVP to within almost-polynomial factors is NP-hard. In Proc. of the 39th IEEE Symposium on Foundations of Computer Science, 1998

    Google Scholar 

  28. D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Transactions on Information Theory, vol 47(3), 2001, pp 1212–1215

    Google Scholar 

  29. U. Feige and D. Micciancio. The inapproximability of lattice and coding problems with preprocessing. Computational Complexity, 2002, pp 44–52

    Google Scholar 

  30. O. Regev. Improved inapproximability of lattice and coding problems with preprocessing. IEEE Transactions on Information Theory, 50(9), 2004, pp 2031–2037

    Article  MathSciNet  Google Scholar 

  31. M. Alekhnovich, S. Khot, G. Kindler, N. Vishnoi. Hardness of approximating the closest vector problem with pre-processing. In Proc. of the 46th IEEE Symposium on Foundations of Computer Science, 2005

    Google Scholar 

  32. D. Aharonov, O. Regev. Lattice problems in NP ∩ coNP. Journal of the ACM, 52(5), 2005, pp 749–765

    Article  MathSciNet  Google Scholar 

  33. I. Haviv, O. Regev. Hardness of the covering radius problem on lattices. In Proc. of the 21st Annual IEEE Computational Complexity Conference, 2006

    Google Scholar 

  34. J. Blömer, J.P. Seifert. On the complexity of compuing short linearly independent vectors and short bases in a lattice. In Proc. of the 31st Annual ACM Symposium on the Theory of Computing, 1999, pp 711–720

    Google Scholar 

  35. O. Regev, R. Rosen. Lattice problems and norm embeddings. In Proc. of the 38th Annual ACM Symposium on the Theory of Computing, 2006

    Google Scholar 

  36. I. Dinur. Approximating SVP ∞ to within almost polynomial factors is NP-hard. Proc. of the 4th Italian Conference on Algorithms and Complexity, LNCS, vol 1767, Springer, 2000

    Google Scholar 

  37. W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, vol. 296, 1993, pp 625–635

    Article  MATH  MathSciNet  Google Scholar 

  38. O. Goldreich, S. Goldwasser. On the limits of non-approximability of lattice problems. In Proc. of the 30th Annual ACM Symposium on the Theory of Computing, 1998, pp 1–9

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. S. Arora and S. Safra. Probabilistic checking of proofs : A new characterization of NP. Journal of the ACM, 45(1), 1998, pp 70–122

    Article  MATH  MathSciNet  Google Scholar 

  41. S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3), 1998, pp 501–555

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Raz. A parallel repetition theorem. SIAM Journal of Computing, 27(3), 1998, pp 763–803

    Article  MATH  MathSciNet  Google Scholar 

  43. J. Milnor, D. Husemoller. Symmetric bilinear forms. Springer, Berlin, 1973

    MATH  Google Scholar 

  44. J.C. Lagarias, H.W. Lenstra, C.P. Schnorr. Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, vol 10, 1990, pp 333–348

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. New York University, New York, NY, 10012, USA

    Subhash Khot

Authors
  1. Subhash Khot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subhash Khot .

Editor information

Editors and Affiliations

  1. Dépt. Mathématiques et d'Informatique, Ecole Normale Supérieure, rue d'Ulm 45, Paris CX 05, 75230, France

    Phong Q. Nguyen

  2. Dept. Informatique, Université de Caen, Caen CX, 14032, France

    Brigitte Vallée

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Khot, S. (2009). Inapproximability Results for Computational Problems on Lattices. In: Nguyen, P., Vallée, B. (eds) The LLL Algorithm. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02295-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02295-1_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02294-4

  • Online ISBN: 978-3-642-02295-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation