Computing bases in Hermite normal form of lattices of integer relations

2026-05-08

AI Summary

This work addresses the efficient computation of a Hermite normal form (HNF) basis for lattices defined by integer relations. Specifically, for the lattice consisting of all integer row vectors \( p \) satisfying \( pF \in \mathcal{L}(M) \), the paper proposes a Las Vegas randomized algorithm. The method generalizes HNF computation to arbitrary integer-relation lattices by integrating randomization techniques, lattice theory, and integer matrix arithmetic, producing a correct basis with failure probability at most \( 1/2 \). Notably, when \( M \) is square and \( F \) is the identity matrix, the algorithm achieves a bit complexity nearly matching the lower bound dictated by matrix multiplication, thereby significantly improving computational efficiency.

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 $M$.

Problem

• Hermite normal form
• integer lattices
• integer relations
• lattice basis
• matrix computation

Innovation

• Hermite normal form
• integer lattices
• Las Vegas algorithm
• matrix multiplication complexity
• integer relations