We present a variation of the modular algorithm for computing the Hermite Normal Form of an O  K  -module presented by Cohen [4], where O  K   is the ring of integers of a number field  K . An approach presented in [4] based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of [4] to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field  K .
['Jean-François Biasse', 'Claus Fieker']
A polynomial time algorithm for computing the HNF of a module over the integers of a number field