The irreducible vectors of a lattice:

Abstract

The main idea behind lattice sieving algorithms is to reduce a sufficiently large number of lattice vectors with each other so that a set of short enough vectors is obtained. It is therefore natural to study vectors which cannot be reduced. In this work we give a concrete definition of an irreducible vector and study the properties of the set of all such vectors. We show that the set of irreducible vectors is a subset of the set of Voronoi relevant vectors and study its properties. For extremal lattices this set may contain as many as $$2^n$$ 2 n vectors, which leads us to define the notion of a complete system of irreducible vectors, whose size can be upper-bounded by the kissing number. One of our main results shows that modified heuristic sieving algorithms heuristically approximate such a set (modulo sign). We provide experiments in low dimensions which support this theory. Finally we give some applications of this set in the study of lattice problems such as SVP, SIVP and CVPP. The introduced notions, as well as various results derived along the way, may provide further insights into lattice algorithms and motivate new research into understanding these algorithms better.