Comparing Real and Imaginary Arithmetics for Divisor Class Groups of Hyperelliptic Curves

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 Ruck 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.