Computing a Basis for an Integer Lattice

Abstract

The extended gcd problem takes as input two integers, and asks as output an integer linear combination of the integers that are equal to their gcd. The classical extended Euclidean algorithm and fast variants such as the half-gcd algorithm give efficient algorithmic solutions. In this thesis, we give a fast algorithm to solve the simplest — but not trivial — extension of the scalar extended gcd problem on two integers to the case of integer input matrices.

Given a full column rank (n + 1) × n integer matrix A, we present an algorithm that produces a square nonsingular integer matrix B such that the lattice generated by the rows of B — the set of all integer linear combinations of the rows of B — is equal to the lattice generated by the rows of A. The magnitude of entries in the basis B are guaranteed to be not much larger than those of the input matrix A. The cost of our algorithm to produce B is about the same as that required to multiply together two square integer matrices of dimension n and with the size of entries about that of the input matrix. This running time bound improves by about a factor of n on the fastest previously known algorithm.