Computing bases in Hermite normal form of lattices of integer relations

TL;DR

The paper introduces a Las Vegas randomized algorithm for computing Hermite normal form bases of lattices of integer relations, with efficiency comparable to matrix multiplication.

Contribution

It presents a novel randomized algorithm for computing Hermite bases of integer relation lattices with probabilistic correctness guarantees.

Findings

Algorithm is Las Vegas type, with at most 1/2 failure probability.

When F=I, the algorithm computes the Hermite normal form of M efficiently.

Complexity is comparable to matrix multiplication for similar-sized matrices.

Abstract

Given a full column rank $M \in \Z^{\ell \times m}$ and an $F \in \Z^{n \times m}$ we present an algorithm to compute the $n \times n$ basis in Hermite form of the integer lattice comprised of all rows $p \in \Z^{1 \times n}$ such that $pF \in \Z^{1 \times m}$ is in the integer lattice generated by the rows of $M$. The algorithm is randomized of the Las Vegas type, that is, it can fail with probability at most $1/2$, but if fail is not returned it guarantees to produce the correct result. When $M$ is square and $F=I_m$, then the computed basis is the Hermite normal form of $M$, and the algorithm uses about the same number of bit operations as required to multiply together two matrices of the same dimension and size of entries as $M$.