Skip to main content
  • 224 Accesses

Synonyms

Lattice basis reduction

Related Concepts

Closest Vector Problem; Lattice; Lattice-Based Cryptography; Shortest Vector Problem

Definition

Among all the bases of a lattice, some are more useful than others. The goal of lattice reduction (also known as lattice basis reduction) is to find interesting bases, such as bases consisting of vectors which are relatively short and almost orthogonal. From a mathematical point of view, one is interested in proving the existence of at least one basis (in an arbitrary lattice) satisfying strong properties. From a computational point of view, one is rather interested in computing such bases in a reasonable time, given an arbitrary basis. In practice, one often has to settle for a trade-off between the quality of the basis and the running time.

Background

Lattice reduction goes back to the reduction theory of quadratic forms, initiated by Lagrange [11], Gauss [6], and Hermite [10]. Indeed, there is a natural relationship between lattices and...

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

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Recommended Reading

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Coppersmith D (1997) Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J Cryptol 10(4):233–260

    Article  MATH  MathSciNet  Google Scholar 

  3. Dirichlet JPGL (1850) Über die Reduction der positiven quadratischen Formen in drei unbestimmten ganzen Zahlen. J Reine Angew Math 40:209–227

    Article  MATH  Google Scholar 

  4. Gama N, Nguyen PQ (2008) Predicting lattice reduction. In: Proceedings of EUROCRYPT’08, Istanbul. LNCS, vol 4965. Springer, Berlin

    Google Scholar 

  5. Gama N, Nguyen PQ (2008) Finding short lattice vectors within Mordell’s inequality. In STOC’08: Proceedings of the 40th annual ACM symposium on theory of computing, Victoria. ACM, New York

    Google Scholar 

  6. Gauss CF (1801) Disquisitiones Arithmeticæ. Apud G. Fleischer, Leipzig

    Google Scholar 

  7. Gauss CF (1840) Recension der “Untersuchungen über die Eigenschaften der positiven tern ären quadratischen Formen von Ludwig August Seeber.” Göttingische Gelehrte Anzeigen, July 9, 1065ff, 1831. Repr J Reine Angew Math 20:312–320. http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN23599524X&DMDID=dmdlog22

  8. Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  9. Gruber M, Lekkerkerker CG (1987) Geometry of numbers. North-Holland, Groningen

    MATH  Google Scholar 

  10. Hermite C (1850) Extraits de lettres de M. Hermite à M. Jacobi sur différents objets de la théorie des nombres. J Reine Angew Math 40:279–290

    Google Scholar 

  11. Lagrange JL (1773) Recherches d’arithmétique. Nouv Mém Acad Roy Soc Belles Lett (Berlin):265–312

    Google Scholar 

  12. Lenstra AK, Lenstra Jr HW, Lovász L (1982) Factoring polynomials with rational coefficients. Math Ann 261: 513–534

    Article  Google Scholar 

  13. May A (2009) Using LLL-reduction for solving RSA and factorization problems: a survey. In: Nguyen PQ, Vallée B (eds) The LLL algorithm: survey and applications. Information security and cryptography. Springer, Heidelberg

    Google Scholar 

  14. Micciancio D, Goldwasser S (2002) Complexity of lattice problems: a cryptographic perspective. The Kluwer international series in engineering and computer science, vol 671. Kluwer, Boston

    Book  MATH  Google Scholar 

  15. Minkowski H (1896) Geometrie der Zahlen. Teubner, Leipzig

    Google Scholar 

  16. Nguyen PQ (2009) Hermite’s constant and lattice algorithms. In: Nguyen PQ, Vallée B (eds) The LLL algorithm: survey and applications. Information security and cryptography. Springer, Heidelberg

    Google Scholar 

  17. Nguyen PQ (2009) Public-key cryptanalysis. In: Recent trends in cryptography. Contemporary mathematics series, vol 477. AMS–RME, Providence

    Google Scholar 

  18. Nguyen PQ, Vallée B (2010) The LLL algorithm: survey and applications. Information security and cryptography. Springer, Heidelberg

    Book  MATH  Google Scholar 

  19. Nguyen PQ, Stern J (2001) The two faces of lattices in cryptology. In: Cryptography and lattices – proceedings of CALC ’01, Providence. LNCS, vol 2146. Springer, Berlin, pp 146–180

    Chapter  Google Scholar 

  20. Schnorr CP (1987) A hierarchy of polynomial lattice basis reduction algorithms. Theor Comput Sci 53:201–224

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this entry

Cite this entry

Nguyen, P. (2011). Lattice Reduction. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_457

Download citation

Publish with us

Policies and ethics