On the SVP for low-dimensional circulant lattices

Abstract

Lattice is the main research subject in the geometry of numbers. SVP refers to finding a shortest nonzero lattice vector in a given lattice, which is thought to be a difficult optimization problem. For general lattice, the integer coefficients of a shortest nonzero vector under a lattice basis might be exponentially large, thus making the simple integer coefficient searching approach impractical. In this paper, we find that for low-dimensional circulant lattices(dimension \(n \in \{2,3,4,6\}\)), the integer coefficients of a shortest lattice vector under its circulant basis are actually in a small set \(S=\{-1,0,1\}\), which makes it easy to find the shortest vector in these cases. Moreover, we present the specific forms of the SVP solutions for low-dimensional circulant lattices.