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∈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.