Combinatorica

, Volume 10, Issue 4, pp 333–348

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

Authors

  • J. C. Lagarias
    • AT&T Bell Laboratories
  • H. W. LenstraJr.
    • Department of MathematicsUniversity of California
  • C. P. Schnorr
    • UniversitÄt Frankfurt
Article

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

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
https://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f1.jpg
and
https://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f2.jpg
, where
https://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f3.jpg
andγj denotes Hermite's constant. As a consequence the inequalities
https://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f4.jpg
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
https://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f5.jpg
.

AMS subject classification (1980)

11 H 0611 H 50

Copyright information

© Akadémiai Kiadó 1990