Skip to main content
SpringerLink
Log in
Book cover

Modern Discrete Mathematics and Analysis pp 81–95Cite as

Solutions of Hard Knapsack Problems Using Extreme Pruning

  • Chapter
  • First Online:

  • 1537 Accesses

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 131))

Abstract

In the present study we provide a review for the state-of-the-art attacks to the knapsack problem. We implemented the Schnorr-Shevchenko lattice attack, and we applied the new reduction strategy, BKZ 2.0. Finally, we compared the two implementations.

This is a preview of subscription content, 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   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
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

Learn about institutional subscriptions

References

  1. Ajtai, M.: The shortest vector problem in L 2 is NP-hard for randomized reduction. In: Proceedings of the 30th ACM Symposium on Theory of Computing (1998)

    Google Scholar 

  2. Becker, A., Coron, J.-S., Joux, A.: Improved generic algorithm for hard knapsacks. In: Eurocrypt 2011. Lecture Notes in Computer Science, vol. 6632 (2011)

    Chapter  Google Scholar 

  3. Chen, Y., Nguyen, P.Q.: BKZ 2.0: better lattice security estimates. In: Asiacrypt. Lecture Notes in Computer Science, vol. 7073. Springer, Berlin (2011)

    Google Scholar 

  4. Coster, M.J., Joux, A., LaMacchia, B.A., Odlyzko, A.M., Schnorr, C.-P., Stern, J.: Improved low-density subset sum algorithms. Comput. Complex. 2, 111–128 (1992)

    Article  MathSciNet  Google Scholar 

  5. Freize, A.M.: On the Lagarias-Odlyzko algorithm for the subset sum problem. SIAM J. Comput. 15(2), 536–539 (1986)

    Article  MathSciNet  Google Scholar 

  6. Galbraith, S.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  7. Gama, N., Nguyen, P.Q.: Predicting Lattice Reduction. Lecture Notes in Computer Science, vol. 4965 (2008)

    Google Scholar 

  8. Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Eurocrypt 2008. Advances in Cryptology. Springer, Berlin (2008)

    Google Scholar 

  9. Gama, N., Nguyen, P.Q., Regev, O.: Lattice enumeration using extreme pruning. In: Eurocrypt 2010. Advances in Cryptology. Springer, Berlin (2010)

    Chapter  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)

    Google Scholar 

  11. Howgrave-Graham, N., Joux, A.: New generic algorithms for hard knapsacks. In: Eurocrypt 2010. Lecture Notes in Computer Science. vol. 6110 (2010)

    Chapter  Google Scholar 

  12. Joux, A.: Algorithmic Cryptanalysis. Chapman & Hall/CRC Cryptography and Network Security. CRC Press, Boca Raton (2009)

    Book  Google Scholar 

  13. Lagarias, J., Odlyzko, A.: Solving low-density attacks to low-weight knapsacks. In: Advance in Cryptology-ASIACRYPT 2005. Lecture Notes in Computer Science, vol. 3788 (2005)

    Google Scholar 

  14. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)

    Google Scholar 

  15. Micciancio, D., Goldwasser, S.: Complexity of Lattices : A Cryptographic Perspective. Springer, Berlin (2003)

    Google Scholar 

  16. Nguyen, P.Q., Stern, J.: Adapting density attacks to low-weight knapsacks. In: Advances in Cryptology - ASIACRYPT 2005. Lecture Notes in Computer Science, vol. 3788 (2005)

    Google Scholar 

  17. Nguyen, P.Q., Stehl\(\acute {\text{e}}\), D.: An LLL algorithm with quadratic complexity. SIAM J. Comput. 39(3), 874–903 (2009)

    Article  MathSciNet  Google Scholar 

  18. Okamoto, T., Tanaka, K., Uchiyama, S.: Quantum public-key cryptosystems. In: On Proceedings of Crypto 2000 (2000)

    Google Scholar 

  19. Radziszowski, S., Kreher, D.: Solving subset sum problems with the L 3 algorithm. J. Comb. Math. Comb. Comput. 3, 49–63 (1988)

    Google Scholar 

  20. Schnorr, C.-P.: A more efficient algorithm for lattice basis reduction algorithms. J. Algorithms 9, 47–62 (1987)

    Article  MathSciNet  Google Scholar 

  21. Schnorr, C.-P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66(1–3), 181–199 (1994)

    Article  MathSciNet  Google Scholar 

  22. Schnorr, C.-P., Shevchenko, T.: Solving Subset Sum Problems of Density close to 1 by “randomized” BKZ-reduction. Cryptology. ePrint Archive: Report 2012/620

    Google Scholar 

  23. Shoup, V.: NTL: A library for doing number theory. http://www.shoup.net/ntl/

  24. Stein, W.A., et al. Sage Mathematics Software. The Sage Development Team (2012). http://www.sagemath.org

  25. The FPLLL development team, fplll, a lattice reduction library (2016). Available at https://github.com/fplll/fplll

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    E. Daravigkas & K. A. Draziotis

  2. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    A. Papadopoulou

Authors
  1. E. Daravigkas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. A. Draziotis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Papadopoulou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. A. Draziotis .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Hellenic Military Academy, Vari, Greece

    Nicholas J. Daras

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Daravigkas, E., Draziotis, K.A., Papadopoulou, A. (2018). Solutions of Hard Knapsack Problems Using Extreme Pruning. In: Daras, N., Rassias, T. (eds) Modern Discrete Mathematics and Analysis . Springer Optimization and Its Applications, vol 131. Springer, Cham. https://doi.org/10.1007/978-3-319-74325-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation