  We present a variation of the modular algorithm for computing the Hermite
Normal Form of an $\OK$-module presented by Cohen, where $\OK$ is the ring of
integers of a number field K. The modular strategy was conjectured to run in
polynomial time by Cohen, but so far, no such proof was available in the
literature. In this paper, we provide a new method to prevent the coefficient
explosion and we rigorously assess its complexity with respect to the size of
the input and the invariants of the field K.
