Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over $\mathbb{K}$ in the jacobian of the curve?
Since for every reduced divisor we have the corresponding Mumford representation (U(x),V(x)), I would say that one only needs to look at the addition using Cantor's algorithm. Is there an algorithmic approach to computing it rapidly?
(A point to a reference material would also be greatly appreciated).