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A rigorous subexponential algorithm for computation of class groups

Authors: James L. Hafner and Kevin S. McCurley
Journal: J. Amer. Math. Soc. 2 (1989), 837-850
MSC: Primary 11Y40; Secondary 11R29
MathSciNet review: 1002631
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Abstract: 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)$.

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