Finding a closest point in a lattice of Voronoi's first kind

TL;DR

This paper presents an efficient method to find the closest point in certain lattices of Voronoi's first kind using a convergent series and minimum cut algorithms, reducing computational complexity to polynomial time.

Contribution

It introduces a polynomial-time algorithm for finding closest points in Voronoi's first kind lattices with known obtuse superbasis, utilizing a convergent series and flow network min-cut computations.

Findings

Closest point can be computed in O(n^4) operations.

Series converges after at most n terms.

Each vector computed in O(n^3) operations.

Abstract

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network.