The Subexponential Method
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.
KeywordsPrime Ideal Factor Base Class Group Class Number Relation Matrix
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