We compare optimized arithmetics with ideals in real resp. imaginary quadratic function fields for divisor class groups of
hyperelliptic curves. Our analysis shows that the new real quadratic arithmetic presented by Rück and the first author in
[6] and an appropriate modification of the algorithm of Cantor both require a number of operations which is O(g
2
) in the field of constants, where g is the genus of a hyperelliptic curve.