Asymptotically Fast Computation of Hermite Normal Forms of Integer

This paper presents a new algorithm for computing the Hermite normal form H of an A 2 Z Z nm of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O~(m ?1 nM(m log jjAjj)) bit operations to produce both H and a candidate for U. Here, jjAjj = maxij jAijj, M(t) bit operations are suucient to multiply two dte-bit integers, and 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~(m 2 nM(mlog jjAjj)) bit operations to produce H, but does not produce a candidate for U. Previous methods require on the order of O~(n 3 M(m log jjAjj)) bit operations to produce a candidate for U | our algorithm improves on this signiicantly in both a theoretical and practical sense.