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This survey, mostly written in the years 2022-2023, is meant as an as short as possible description of the current state-of-the-art lattice attacks on lattice-based cryptosystems, without losing the essence of the matter.

The main topic of this PhD thesis is the Arakelov ray class group of a number field, an algebraic object that contains both the ideal class group structure and the unit group structure. The main result consists of the fact that certain specific random walks on the Arakelov ray class group result in a target point that is uniformly distributed on this group, under the assumption of an extended version of the Riemann Hypothesis. Almost all other results of this work are consequences of this fact. [(Errata)](publication/phdthesis/errata.pdf)