Nothing Special   »   [go: up one dir, main page]

Skip to main content

New Variants of Lattice Problems and Their NP-Hardness

  • Conference paper
Information Security Practice and Experience (ISPEC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8434))

  • 1838 Accesses

Abstract

We introduce some new variants of lattice problems: Quadrant-SVP, Quadrant-CVP and Quadrant-GapCVP’. All of them are NP-hard under deterministic reductions from subset sum problem. These new type of lattice problems have potential in construction of cryptosystems. Moreover, these new variant problems have reductions with standard SVP (shortest vector problem) and CVP (closest vector problem), this feature gives new way to study the complexity of SVP and CVP, especially for the proof of NP-hardness of SVP under deterministic reductions, which is an open problem up to now.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ajtai, M.: The shortest vector problem in l 2 is np-hard for randomized reductions. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 10–19. ACM (1998)

    Google Scholar 

  2. Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp. 284–293. ACM (1997)

    Google Scholar 

  3. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing, pp. 601–610. ACM (2001)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Blömer, J., Naewe, S.: Sampling methods for shortest vectors, closest vectors and successive minima. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 65–77. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Cai, J.-Y., Nerurkar, A.: Approximating the svp to within a factor (1+ 1/dim e) is np-hard under randomized reductions. Journal of Computer and System Sciences 59(2), 221–239 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gary, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of np-completeness (1979)

    Google Scholar 

  8. Gentry, C.: Fully homomorphic encryption using ideal lattices (2009)

    Google Scholar 

  9. Goldreich, O., Goldwasser, S.: On the limits of non-approximability of lattice problems. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 1–9. ACM (1998)

    Google Scholar 

  10. Goldreich, O., Goldwasser, S., Halevi, S.: Public-key cryptosystems from lattice reduction problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  11. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: A ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Micciancio, D.: The shortest vector in a lattice is hard to approximate to within some constant. SIAM Journal on Computing 30(6), 2008–2035 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Micciancio, D.: Inapproximability of the shortest vector problem: Toward a deterministic reduction. Theory of Computing 8(1), 487–512 (2012)

    Article  MathSciNet  Google Scholar 

  16. Micciancio, D., Goldwasser, S.: Complexity of lattice problems: a cryptographic perspective, vol. 671. Springer (2002)

    Google Scholar 

  17. Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 351–358. ACM (2010)

    Google Scholar 

  18. Minkowski, H.: Geometrie der zahlen. BG Teubner (1910)

    Google Scholar 

  19. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), 34:1–34:40 (2009)

    Google Scholar 

  20. Schnorr, C.-P.: A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical Computer Science 53(2), 201–224 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. van Emde-Boas, P.: Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Department, Univ. (1981)

    Google Scholar 

  22. Voronoï, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. deuxième mémoire. recherches sur les parallélloèdres primitifs. Journal für die reine und angewandte Mathematik 134, 198–287 (1908)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Li, W. (2014). New Variants of Lattice Problems and Their NP-Hardness. In: Huang, X., Zhou, J. (eds) Information Security Practice and Experience. ISPEC 2014. Lecture Notes in Computer Science, vol 8434. Springer, Cham. https://doi.org/10.1007/978-3-319-06320-1_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06320-1_37

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06319-5

  • Online ISBN: 978-3-319-06320-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics