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

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(n3(log n + log ||A||)2(log n)2) bit operations, where ||A|| = max ij|Aij| 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 (n3log ||A||)1 + o(1) bit operations, where the exponent `` + o(1)′′ captures additional factors \(c_1 (\log n)^{c_2} (\rm {loglog} ||A||)^{c_3} \) for positive real constants c1, c2, c3.