  In this paper we describe a generalisation and adaptation of Kedlaya's
algorithm for computing the zeta function of a hyperelliptic curve over a
finite field of odd characteristic that the author used for the implementation
of the algorithm in the Magma library. We generalise the algorithm to the case
of an even degree model. We also analyse the adaptation of working with the
$x^idx/y^3$ rather than the $x^idx/y$ differential basis. This basis has the
computational advantage of always leading to an integral transformation matrix
whereas the latter fails to in small genus cases. There are some theoretical
subtleties that arise in the even degree case where the two differential bases
actually lead to different redundant eigenvalues that must be discarded.
