Asymptotically Fast Computation of the Hermite Normal Form of an Integer Matrix

This paper presents a new algorithm for computing the Hermite normal form H of an A ∈ Z n×m of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O (̃mnM(m log ||A||)) bit operations to produce both H and a candidate for U . Here, ||A|| = maxij |Aij |, M(t) bit operations are sufficient to multiply two dte-bit integers, and θ is the exponent for matrix multiplication over rings: two m×m matrices over a ring R can be multiplied in O(m) ring operations from R. The previously fastest algorithm of Hafner & McCurley requires O (̃mnM(m log ||A||)) bit operations to produce H, but does not produce a candidate for U . Previous methods require on the order of O (̃nM(m log ||A||)) bit operations to produce a candidate for U — our algorithm improves on this significantly in both a theoretical and practical sense.