A fast, rigorous technique for computing the regulator of a real quadratic field

Abstract

Abstract. We present a new algorithm for computing the regulator of a real quadratic field Q( √ D), based on an algorithm for unconditionally verifying the correctness of the regulator produced by a subexponential algorithm, that runs in expected time O(D 1/6+ ) under the Generalized Riemann Hypothesis. The correctness of our algorithm relies on no unproven hypotheses and is currently the fastest known unconditional algorithm for computing the regulator. A number of implementation issues and performance enhancements are discussed, and we present the results of computations demonstrating the efficiency of the new algorithm.

Quadratic class numbers and character sums

“…The error term (14) shows that for d of practical size these terms in general will be negligible, although that must still be checked; in our case we get…”

“…The latter may be computed deterministically to high precision (within an absolute error of d −1 , say) in time O ε (d 1/4+ε ) [2], and heuristically in time O ε (d 1/6+ε ) [14].…”

Rigorous Computation of Fundamental Units in Algebraic Number Fields

“…These questions were answered affirmatively for the case of real quadratic fields in [dHJW07]. The unit group of a real quadratic field of discriminant ∆ has rank one, generated by a single fundamental unit ε ∆ > 1.…”

“…The corresponding lattice of logarithms is generated by a single real number, the regulator R ∆ = log ε ∆ . In [dHJW07], it is proved that an unconditionally correct approximation of R ∆ can be computed in time O(S 1/3 ∆ ǫ ) given an integer multiple S of R ∆ . Furthermore, if it is assumed that S is the output of the index-calculus algorithm, then, assuming the GRH, S is the regulator and hence of size O(∆ 1/2+ǫ ).…”

Ρητές Καμπύλες Επί Αλγεβρικών Σωμάτων Αριθμών Και Διοφαντικές Εξισώσεις

“…Πρόσφατα προτάθηκε ένας αλγόριθµος υπολογισµού του κανονικοποιητή που υλοποιείται σε χρόνο ( 1/6+ ), υποθέτοντας αρχικά την ισχύ της Γενικευµένης Υπόθεσης του Riemann [18]. Με την ίδια υπόθεση µπορούµε να ϐρούµε κάποιους υποεκθετικούς στοχαστικούς αλγορίθµους στα [1,52].…”