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Approximating integer lattices by lattices with cyclic factor groups

  • A. Paz
  • C. P. Schnorr
Algorithms And Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)

Abstract

We reduce in polynomial time various computational problems concerning integer lattices to the case that the lattice L is defined by a single modular (linear, homogeneous) equation, L = {x∈ℤn : 〈x,v〉=0 mod d} where v is a vector in ℤn and d an integer. An integer lattice L ⊂ ℤn can be written in this form if and only if L has rank n and if the abelian group ℤn/L is cyclic. The shortest vector problem, the problem to compute the successive minima of a lattice and the problem to reduce (in the sense of Minkowski or in the sense of Korkine, Zolotareff) a lattice basis is transformed in polynomial time to lattices of the above special form. Our method shows that every integer lattice can be approximated efficiently by rational lattices L ⊂ 1/k ℤn such that the abelian group ℤn/kL is cyclic.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • A. Paz
    • 1
  • C. P. Schnorr
    • 2
  1. 1.Computer Science Department TechnionHaifaIsrael
  2. 2.Fachbereich Mathematik/InformatikUniversität FrankfurtFrankfurtGermany

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