Abstract
In this chapter we describe probabilistic algorithms for computing class numbers and regulators. We also show how they can be extended to solve the equivalence problem. Those algorithms are much faster than the deterministic algorithms presented in Chapter 9. They use an approach dubbed index calculus which originated in work by Kraichik [Kra22] and seemingly independent work by Western and Miller [WM68]. The first proposals to apply this approach in the context of imaginary quadratic class groups came from Seysen [Sey87] and Hafner/McCurley [HM89]. In this chapter we let Δ be a fundamental discriminant. It is rather straightforward, however, to extend the algorithms of this chapter to the case of nonfundamental discriminants.
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© 2007 Springer
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Buchmann, J., Vollmer, U. (2007). Subexponential Algorithms. In: Binary Quadratic Forms. Algorithms and Computation in Mathematics, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46368-9_12
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DOI: https://doi.org/10.1007/978-3-540-46368-9_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-46367-2
Online ISBN: 978-3-540-46368-9
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