# Approximating integer lattices by lattices with cyclic factor groups

## 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∈ℤ^{n} : 〈x,v〉=0 mod d} where v is a vector in ℤ^{n} and d an integer. An integer lattice L ⊂ ℤ^{n} can be written in this form if and only if L has rank n and if the abelian group ℤ^{n}/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 ℤ^{n} such that the abelian group ℤ^{n}/kL is cyclic.

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