, 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

DOI: 10.1007/BF02128669

Cite this article as:
Lagarias, J.C., Lenstra, H.W. & Schnorr, C.P. Combinatorica (1990) 10: 333. doi:10.1007/BF02128669


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
, 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 0611 H 50

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