We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.