  We describe an algorithm to count the number of rational points of an
hyperelliptic curve defined over a finite field of odd characteristic which is
based upon the computation of the action of the Frobenius morphism on a basis
of the Monsky-Washnitzer cohomology with compact support. This algorithm
follows the vein of a systematic exploration of potential applications of
cohomology theories to point counting.
  Our algorithm decomposes in two steps. A first step which consists in the
computation of a basis of the cohomology and then a second step to obtain a
representation of the Frobenius morphism. We achieve a $\tilde{O}(g^4 n^{3})$
worst case time complexity and $O(g^3 n^3)$ memory complexity where $g$ is the
genus of the curve and $n$ is the absolute degree of its base field. We give a
detailed complexity analysis of the algorithm as well as a proof of
correctness.
