There are algorithms working in polynomial time, 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, 313-332,

which also contains explicit computations on Jacobians in the genus $2$ case.

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, 313-332,

which also contains some explicit computations on Jacobians in the case of genus $2$.

There are algorithms working in polynomial time, but for higher genus the exponent is high and 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, 313-332.

There are algorithms working in polynomial time, 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, 313-332,

which also contains explicit computations on Jacobians in the genus $2$ case.