Combinatorica

, Volume 10, Issue 4, pp 333–348

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

  • J. C. Lagarias
  • H. W. LenstraJr.
  • C. P. Schnorr
Article
  • 288 Downloads

Abstract

Letλi(L), λi(L*) denote the successive minima of a latticeL and its reciprocal latticeL*, and let [b1,..., bn] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that
and
, where
andγj denotes Hermite's constant. As a consequence the inequalities
are obtained forn≥7. Given a basisB of a latticeL in ℝm of rankn andx∃ℝm, we define polynomial time computable quantitiesλ(B) andΜ(x,B) that are lower bounds for λ1(L) andΜ(x,L), whereΜ(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL*, then λ1(L)≤γn*λ(B) and
.

AMS subject classification (1980)

11 H 06 11 H 50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. Babai: On Lovász' lattice reduction and the nearest lattice point problem,Combinatorica,6 (1986), 1–13.Google Scholar
  2. [2]
    J. W. S. Cassels:An introduction to the geometry of numbers, Springer-Verlag, Berlin,1971.Google Scholar
  3. [3]
    J. H. Conway, andN. J. A. Sloane:Sphere packings, lattices and groups, Springer-Verlag, New York,1988.Google Scholar
  4. [4]
    M. Grötschel, L. Lovász, andA. Schrijver:Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin,1988.Google Scholar
  5. [5]
    P. M. Gruber, andC. G. Lekkerkerker:Geometry of numbers, North-Holland, Amsterdam,1987.Google Scholar
  6. [6]
    C. Hermite: Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objets de la théorie des nombres, Deuxième lettre,J. Reine Angew. Math. 40 (1850), 279–290.Google Scholar
  7. [7]
    F. John: Extremum problems with inequalities as subsidiary conditions, K. O. Friedrichs, O. E. Neugebauer, J. J. Stoker (eds),Studies and essays presented to R. Courant on his 60th birthday, 187–204, Interscience Publishers, New York,1948.Google Scholar
  8. [8]
    R. Kannan: Minkowski's convex body theorem and integer programming,Math. Oper. Res. 12 (1987), 415–440.Google Scholar
  9. [9]
    A. Korkine, andG. Zolotareff: Sur les formes quadratiques,Math. Ann. 6 (1873), 366–389.CrossRefMathSciNetGoogle Scholar
  10. [10]
    J. L.Lagrange: Recherches d'arithmétique,Nouv. Mém. Acad. Berlin (1773), 265–312; ∄uvres, vol. VIII, 693–753.Google Scholar
  11. [11]
    A. K. Lenstra, H. W. Lenstra, Jr., andL. Lovász: Factoring polynomials with rational coefficients,Math. Ann. 261 (1982), 515–534.CrossRefGoogle Scholar
  12. [12]
    H. W. Lenstra, Jr.: Integer programming with a fixed number of variables,Math. Oper. Res. 8 (1983), 538–548.Google Scholar
  13. [13]
    L. Lovász: An algorithmic theory of numbers, graphs and convexity,CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania,1986.Google Scholar
  14. [14]
    K. Mahler: A theorem on inhomogeneous diophantine inequalities,Nederl. Akad. Wetensch., Proc. 41 (1938), 634–637.Google Scholar
  15. [15]
    K.Mahler:The geometry of numbers, duplicated lectures, Boulder, Colorado,1950.Google Scholar
  16. [16]
    J. Milnor, andD. Husemoller:Symmetric bilinear forms, Springer-Verlag, Berlin,1973.Google Scholar
  17. [17]
    N. V. Novikova: Korkin-Zolotarev reduction domains of positive quadratic forms inn≤8 variables and a reduction algorithm for these domains,Dokl. Akad. Nauk SSSR 270 (1983), 48–51; English translation:Soviet Math. Dokl. 27 (1983), 557–560.Google Scholar
  18. [18]
    C. A. Rogers:Packing and covering, Cambridge University Press, Cambridge,1964.Google Scholar
  19. [19]
    S. S. Ryshkov: Geometry of positive quadratic forms (Russian),Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974),1, 501–506,Canad. Math. Congress, Montreal, Que., 1975.Google Scholar
  20. [20]
    S. S. Ryshkov, andE. P. Baranovskii: Classical methods in the theory of lattice packings,Uspekhi Mat. Nauk 34, 4 (208) (1979), 3–63; English translation:Russian Math. Surveys 34 (4) (1979), 1–68.Google Scholar
  21. [21]
    C. P. Schnorr: A hierarchy of polynomial time lattice basis reduction algorithms,Theoret. Comput. Sci. 53 (1987), 201–224.CrossRefGoogle Scholar
  22. [22]
    B. L. van der Waerden: Die Reduktionstheorie der positiven quadratischen Formen,Acta Math. 96 (1956), 265–309.Google Scholar
  23. [23]
    B. L. van der Waerden: H. Gross (eds),Studien zur Theorie der quadratischen Formen, BirkhÄuser-Verlag, Basel,1968.Google Scholar
  24. [24]
    P. van Emde Boas: AnotherNP-complete partition problem and the complexity of computing short vectors in a lattice,Report 81-04, Department of Mathematics, University of Amsterdam, Amsterdam,1981.Google Scholar

Copyright information

© Akadémiai Kiadó 1990

Authors and Affiliations

  • J. C. Lagarias
    • 1
  • H. W. LenstraJr.
    • 2
  • C. P. Schnorr
    • 3
  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.UniversitÄt FrankfurtFrankfurtF. R. Germany

Personalised recommendations