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

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+e ) 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.

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