Tittel: An elementary look at hyperelliptic curves and their usefulness in cryptography

Veileder: Kristian Gjøsteen

Sammendrag: The purpose of this project was to obtain an elementary understanding of the mathematics behind cryptosystems based on hyperelliptic curves. The reason for this choice this topic was because of how beautiful I thought the elliptic curves and their group structures were. Here I first present the regular elliptic curves and their group structures, before moving on to hyperelliptic curves, and the building blocks for constructing the Picard group. I present a general algorithm for adding elements in the Picard group, and give a geometric interpretation of the algorithm. Lastly I look at the index calculus algorithm adapted to these groups, which shows that the discrete logarithm problem can be solved quite quickly for a large amount of these curves.