The infrastructure of a global field and baby step-giant step algorithms

In Computational Number Theory, one is interested in the computation of invariants. One such invariant is the regulator of a number field or a global function field. The regulator can be obtained from the unit lattice, whose structure corresponds to the structure of the so called infrastructure. In this thesis, we generalize the infrastructure to the n-dimensional case; so far, the infrastructure was mainly investigated in the one-dimensional case. For that purpose, we generalize f -representations and use them to obtain a reduction map. Furthermore, we relate the infrastructure to the (Arakelov) divisor class group and describe the divisor class group using f -representations. This allows both to do explicit arithmetic in the divisor class group and to compute giant steps in the infrastructure. This extends work particularly by J. Buchmann and R. Schoof in the number field case and S. Paulus and H.-G. Rück in the function field case. We also discuss an implementation of computation of boxes in the function field case, and explain how it can be used to compute giant steps and baby steps. Moreover, we describe existing algorithms for computation of the unit lattice and, hence, the regulator. We present two approaches for the function field case, one using algorithms designed for operating on finite abelian groups in the case that one infinite place has degree one, and one using a lifting strategy for reducing to the case of at least one infinite place of degree one. Finally, we extend J. Buchmann’s baby step-giant step algorithm for number fields to the global field case and combine it with an optimization by D. Terr for the classic baby step-giant step algorithm.