In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to compute it knowing the eigenvalues of the Frobenius action on $H^1(C)$, which could be recovered knowing $\sharp C(F_{q^l})$ for all $l$ between $1$ and the genus of $C$.

Some algorithms working in polynomial time are available, but for high values of the genus the exponent is high and the implementation is difficult. A nice survey is the paper of Gaudry and Harley "Counting Points on Hyperelliptic Curves over Finite Fields" Lecture Notes in Computer Science, 2000, Volume 1838/2000, 313332, which also contains some explicit computations on Jacobians in the case of genus $2$. 

