A Las Vegas randomized algorithm computes the n by n Hermite normal form basis of the integer lattice of rows p where pF lies in the row lattice generated by M, with bit complexity matching matrix multiplication when M is square and F is the identity.
Saunders and Zhendong Wan
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Computing bases in Hermite normal form of lattices of integer relations
A Las Vegas randomized algorithm computes the n by n Hermite normal form basis of the integer lattice of rows p where pF lies in the row lattice generated by M, with bit complexity matching matrix multiplication when M is square and F is the identity.