Lattice-based cryptography derived from cylotomic rings, base their security on two hard mathematical problems, finding the shortest and the closest vector in a lattice. Even though there are many factors to consider when analyzing attacks against cryptosystems, this thesis focus on the mathematical theory. We study the properties of cyclotomic fields, S-units and how they relate to lattices. We present a detailed step by step approach on how to preform an S-unit attack and highlight the most important aspects to it. In addition, we develop new and comprehensive examples that demonstrates the different sides to such attacks.