A proof of the conjectured run time of the Hafner-McCurley class group algorithm

Abstract

We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time \begin{document}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document} where \begin{document}$ -d $\end{document} is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of \begin{document}$ e^{\left(\sqrt{2}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document} was proven, and better bounds were conjectured. To achieve a proven result, we rely on a mild modification of the original algorithm, and on recent results on the properties of the Cayley graph of the ideal class group.