Computing the multiplicative group of residue class rings

Let k be a global field with maximal order o(k) and let m(o) be an ideal of o(k). We present algorithms for the computation of the multiplicative group (o(k)/m(o))* of the residue class ring o(k)/m(o) and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group Cl-k(m) modulo m = m(o)m(infinity), where m(infinity) denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.