Sammendrag
In this thesis we will look at elliptic and hyperelliptic curves. There are three
abelian groups that are isomorphic to hyperelliptic curves. The Jacobian of
hyperelliptic curves, the ideal class group and the form class group, will all be
defined and given abelian group structure. We will give an algorithm for point
addition and point doubling done exclusively in the jacobian of the curve. We
will end the thesis with proving that there exists an isomorphism between the
form class group and the ideal class group.