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The Subexponential Method

Part of the CMS Books in Mathematics book series (CMSBM)

Up to this point, all the algorithms we have presented for computing the regulator R Δ of the real quadratic order O Δ , and, hence, for solving Pell’s equation, have exponential complexity in the size of the discriminant Δ. The most exciting recent development has certainly been the discovery of a Las Vegas algorithm1 by Buchmann for computing R Δ and h Δ whose expected running time is subexponential in log ¦Δ¦. This algorithm has enabled the computation of R Δ for discriminants Δ as large as 101 decimal digits, a dramatic improvement over what had been attainable previously. Unfortunately, as we will discuss in more detail below, this improvement comes at a price. Like Lenstra’s algorithm for computing R Δ described in Chapter 10, the complexity result is conditional on the generalized Riemann hypothesis (GRH) for Hecke L-functions and the extended Riemann hypothesis (ERH), but in the case of the subexponential algorithm, the correctness of the output is conditional as well. Nevertheless, the fact that R Δ can be computed in subexponential time, even assuming the GRH and ERH, remains an important breakthrough.

Keywords

Prime Ideal Factor Base Class Group Class Number Relation Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, LLC 2009

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