Advertisement

The History of the LLL-Algorithm

  • Ionica SmeetsEmail author
  • Arjen Lenstra
  • Hendrik Lenstra
  • László Lovász
  • Peter van Emde Boas
Chapter
Part of the Information Security and Cryptography book series (ISC)

Abstract

The 25th birthday of the LLL-algorithm was celebrated in Caen from 29th June to 1st July 2007. The three day conference kicked off with a historical session of four talks about the origins of the algorithm. The speakers were the three L’s and close bystander Peter van Emde Boas. These were the titles of their talks.
  • A tale of two papers – Peter van Emde Boas.

  • The early history of LLL – Hendrik Lenstra.

  • The ellipsoid method and basis reduction – László Lovász.

  • Polynomial factorization and lattices in the very early 1980s – Arjen Lenstra.

This chapter is based on those talks, conversations with these four historic characters, the notes that Peter van Emde Boas and Arjen Lenstra wrote for the preproceedings, and many artifacts from the phenomenal archive of Van Emde Boas.

Keywords

Polynomial Time Integer Programming Polynomial Algorithm Basis Reduction Ellipsoid Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2): 169–197, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L. G. Hačijan. A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR, 244(5): 1093–1096, 1979.MathSciNetGoogle Scholar
  3. 3.
    H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Report 81-03 (First version), University of Amsterdam, April 1981.Google Scholar
  4. 4.
    H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Report 81-03 (Second version), University of Amsterdam, November 1981.Google Scholar
  5. 5.
    H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4): 538–548, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    H. W. Lenstra, Jr. Euclidean number fields of large degree. Report 76-09, University of Amsterdam, May 1976.Google Scholar
  7. 7.
    P. Gács, and L. Lovász. Khachiyan’s algorithm for linear programming. Math. Programming Stud., 14: 61–68, 1981.zbMATHGoogle Scholar
  8. 8.
    A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4): 515–534, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    P. J. Weinberger and L. P. Rothschild. Factoring polynomials over algebraic number fields. ACM Trans. Math. Software, 2(4): 335–350, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Zassenhaus. On Hensel factorization. I. J. Number Theory, 1: 291–311, 1969.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ionica Smeets
    • 1
    Email author
  • Arjen Lenstra
  • Hendrik Lenstra
  • László Lovász
  • Peter van Emde Boas
  1. 1.Mathematisch InstituutUniversiteit LeidenLeidenthe Netherlands

Personalised recommendations