On Lovász' lattice reduction and the nearest lattice point problem

Shortened version
  • László Babai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)


Answering a question of Vera Sós, we show how Lovász' lattice reduction can be used to find a point of a given lattice, nearest within a factor of cd (c = const.) to a given point in Rd. We prove that each of two straightforward fast heuristic procedures achieves this goal when applied to a lattice given by a Lovász-reduced basis. The verification of one of them requires proving a geometric feature of Lovász-reduced bases: a c 1 d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, where \(c_1 = \sqrt 2 /3\).

As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor of Cd.

In another application, we improve the Grötschel-Lovász-Schrijver version of H. W. Lenstra's integer linear programming algorithm.

The algorithms, when applied to rational input vectors, run in polynomial time.

For lack of space, most proofs are omitted. A full version will appear in Combinatorica.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Adleman, On breaking the iterated Merkle-Hellman public key cryptosystem, Proc. 15th ACM Symp. on Theory of Computing, Boston 1983, 402–412.Google Scholar
  2. 2.
    J. W. S. Cassels, An introduction to the geometry of numbers, Springer, New York 1971.Google Scholar
  3. 3.
    P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Rep. MI/UVA 81-04, Amsterdam 1981.Google Scholar
  4. 4.
    M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 186–197.Google Scholar
  5. 5.
    M. Grötschel, L. Lovász and A. Schrijver, Geometric methods in combinatorial optimization, in: Progress in Combinatorial Optimization (W. R. Pulleyblank, ed.), Proc. Silver Jubilee Conf. on Comb., Univ. Waterloo, Vol. 1, 1982, Acad. Press, N.Y. 1984, 167–183.Google Scholar
  6. 6.
    R. Kannan, A. K. Lenstra and L. Lovász, Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers, in: Proc. 16th Ann. ACM Symp. on Theory of Computing, Washington, D.C. 1984, 191–200.Google Scholar
  7. 7.
    J. Lagarias and A. M. Odlyzko, Solving low density subset sum problems, in: Proc. 24th IEEE Symp. on Foundations of Comp. Sci., 1983, 1–10.Google Scholar
  8. 8.
    A. K. Lenstra, Lattices and factorization of polynomials, Report IW 190/81, Mathematisch Centrum, Amsterdam 1981.Google Scholar
  9. 9.
    A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.Google Scholar
  10. 10.
    H. W. Lenstra, Jr., Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), 538–548.Google Scholar
  11. 11.
    L. Lovász, private communications, 1981–1982.Google Scholar
  12. 12.
    A. M. Odlyzko and H. te Riele, Disproof of the Mertens conjecture, to appear.Google Scholar
  13. 13.
    A. Shamir, A polynomial time algorithm for breaking the Merkle-Hellman cryptosystem, Proc. 23rd IEEE Symp. on Foundations of Comp. Sci., Chicago, Illinois (1982), 145–152.Google Scholar
  14. 14.
    Vera T. Sós, On the theory of diophantine approximation II, Acta Math. Acad. Sci. Hung. (1958), 229–241.Google Scholar
  15. 15.
    Vera T. Sós, Irregularities of partitions: Ramsey theory, uniform distribution, in: Surveys in Combinatorics, Proc. 9th British Combinatorial Conference, 1983 (E. Keith Lloyd, ed.) London Math. Soc. Lect. Notes 82, Cambridge Univ. Press 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • László Babai
    • 1
  1. 1.Department of AlgebraEötvös UniversityBudapestHungary

Personalised recommendations