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

Abstract

<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 id="M1">\begin{document}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ -d $\end{document}</tex-math></inline-formula> is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of <inline-formula><tex-math id="M3">\begin{document}$ 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>

An algorithm for solving the principal ideal problem with subfields

“…where ∆ K is the discriminant of the field assuming the Generalized Riemann hypothesis (GRH) [7]. Buchmann [17] generalized this result to the case of infinite classes of number fields with fixed degree.…”

“…A second layer of parallelization was employed by computing individual roots on independent cores. After picking a principal ideal a = (α) of O K , the main steps of our computations are the following: (1) Finding the initial norm relation to determine the subfields K i , (2) Finding the norm relation in each of the subfields K i , (3) computing the subfields K i,j , (4) Computing the unit groups of the K i,j , (5) Computing the relative norms N K/Ki,j (a), (6) Computing generators of the ideals N K/Ki,j (a), (7) Identifying d-powers (without root computation), (8) Compact representation, and (9) Root computation. We also implemented the reduction from the SPIP to the PIP of [20,21] and we were able to retrieve a short generator of our challenge ideals in K (1) and K (2) (which is a solution to γ-SVP for a γ ∈ e Õ( √ n) in the input principal ideals).…”