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

  • * Corresponding author: Jean-François Biasse

    * Corresponding author: Jean-François Biasse 

The first author is supported by NSF grant 1846166

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  • We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time $ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $ where $ -d $ is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of $ e^{\left(\sqrt{2}+o(1)\right)\sqrt{\log d\log\log d}} $ 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.

    Mathematics Subject Classification: Primary: 11Y16, 20C05, 11R32; Secondary: 11R29, 11R04, 11Y40, 11R18, 11R27.


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