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Subexponential Algorithms for Class Group and Unit Computations

Abstract We describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use of complex logarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice.