An algorithm of subexponential type computing the class group of quadratic orders over principal ideal domains

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We present an algorithm which computes the class group of a quadratic order over a principal ideal domain that fulfills some properties which are implicated by computational requirements. It is a generalization of the subexponential method of Hafner and McCurley which computes the class group of an order in an imaginary quadratic number field. We discuss the concept of reduction theory of binary quadratic forms over a Euclidean domain which makes our algorithm practical. Several examples of principal ideal domains are given for which we have applied our generic algorithm implemented in C++ using template techniques.