<p style='text-indent:20px;'>We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time <inline-formula><tex-math>\begin{document}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math>\begin{document}$ -d $\end{document}</tex-math></inline-formula> is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of <inline-formula><tex-math>\begin{document}$ e^{\left(\sqrt{2}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math></inline-formula> 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.</p>

中文翻译：

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Hafner-McCurley 类群算法的推测运行时间证明

<p style='text-indent:20px;'>我们在黎曼假设的推广下提出了一个证明，即 Hafner 和 McCurley 的类群算法在预期时间内运行 <inline-formula><tex-math>\begin{文档}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math>< /inline-formula> 其中 <inline-formula><tex-math>\begin{document}$ -d $\end{document}</tex-math></inline-formula> 是输入虚二次的判别式命令。在原始论文中，<inline-formula><tex-math>\begin{document}$ e^{\left(\sqrt{2}+o(1)\right)\sqrt{\log 的预期运行时间d\log\log d}} $\end{document}</tex-math></inline-formula> 被证明，更好的界限被推测出来。为取得经证实的结果，