  In this article we give the details of an effective point counting algorithm
for genus two curves over finite fields of characteristic three. The algorithm
has an application in the context of curve based cryptography. One
distinguished property of the algorithm is that its complexity depends
quasi-quadratically on the degree of the finite base field. Our algorithm is a
modified version of an earlier method that was developed in joint work with
Lubicz. We explain how one can alter the original algorithm, on the basis of
new theory, such that it can be used to efficiently count points on genus two
curves over large finite fields. Examples of cryptographic size have been
computed using an experimental Magma implementation of the algorithm which has
been programmed by the author. Our computational results show that the
quasi-quadratic algorithm of Lubicz and the author, with some improvements, is
practical and relevant for cryptography.
