Solving the Pell Equation pp 307-352 | Cite as

# 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 algorithm^{1} 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## Preview

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