A rigorous subexponential algorithm for computation of class groups

Let $C( - d)$ denote the Gauss Class Group of quadratic forms of a negative discriminant $- d$ (or equivalently, the class group of the imaginary quadratic field $Q(\sqrt { - d} )$). We give a rigorous proof that there exists a Las Vegas algorithm that will compute the structure of $C( - d)$ with an expected running time of $L{(d)^{\sqrt 2 + o(1)}}$ bit operations, where $L(d) = {\text {exp}}(\sqrt {\log d\;\log \log d} )$. Thus, of course, also includes the computation of the class number $h( - d)$, the cardinality of $C( - d)$.