A cubic algorithm for computing the Hermite normal form of a nonsingular integer matrix

TL;DR

This paper presents a randomized cubic-time algorithm for computing the Hermite normal form of nonsingular integer matrices, improving efficiency with variants using advanced multiplication techniques.

Contribution

It introduces a new Las Vegas randomized algorithm with cubic complexity and a variant employing pseudo-linear multiplication for faster computation.

Findings

Algorithm runs in $O(n^3 ( ext{log } n + ext{log } ||A||)^2 ( ext{log } n)^2)$ bit operations.

A variant achieves near-linear time complexity $(n^3 ext{log } ||A||)^{1+o(1)}$ with advanced multiplication.

The approach is practical for large integer matrices due to improved theoretical bounds.

Abstract

A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix $A$ of dimension $n$. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by $O(n^3 (\log n + \log ||A||)^2(\log n)^2)$ bit operations, where $||A||= \max_{ij} |A_{ij}|$ denotes the largest entry of $A$ in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time $(n^3 \log ||A||)^{1+o(1)}$ bit operations, where the exponent $"+o(1)"$ captures additional factors $c_1 (\log n)^{c_2} (\log \log ||A||)^{c_3}$ for positive real constants $c_1,c_2,c_3$.