A Cubic Algorithm for Computing the Hermite Normal Form of a Nonsingular Integer Matrix

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 ) c2 (loglog|| A ||) c3 for positive real constants c 1 ,c 2 ,c 3 .