These lecture notes were translated from the Finnish lecture notes for the TUT course "Matemaattinen kryptologia". The laborious bulk translation was taken care of by the students Jussi Kangas (visiting from the University of Tampere) and Paul Coughlan (visiting from the University of Dublin, Trinity College). I want to thank the translation team for their effort.
The notes form the base text for the course "MAT-52606 Mathematical Cryptology". They contain the central mathematical background needed for understanding modern data encryption methods, and introduce applications in cryptography and various protocols.