One of the greatest achievements in pure mathematics is Wiles' proof of Fermat's Last Theorem in 1994. One of the biggest challenges in applied mathematics is to provide tools for secure data storage and transmission.

It is very remarkable that for both problems one can use methods from a very active field of contemporary research called Arithmetic Geometry. In this area one combines methods from Algebraic Number Theory, Algebraic Geometry and Algebra to obtain results for diophantine problems. The key ingredient is Galois theory of number fields and their completions. Representations induced by actions of the Galois groups of these fields on geometric objects play a crucial role. In the lecture we shall explain why Fermat's Last Theorem can be solved in this way and why an important part of public key cryptography leads to questions about Galois representations. We shall discuss constructive and destructive aspects of this approach.