Abstract: Let K denote a number field, and G
a finite abelian group. The ring of algebraic
integers in K is denoted in this
paper by $\cal{O}_K$, and $\cal{A}$
denotes any $\cal{O}_K$-order
in K[G]. The paper describes an algorithm that
explicitly computes the Picard group Pic($\cal{A}$), and solves the corresponding (refined)
discrete logarithm problem. A tamely ramified extension L/K
of prime degree l of an imaginary
quadratic number field K is used as
an example; the class of $\cal{O}_L$ in Pic($\cal{O}_K[G]$) can be numerically determined.
