Practical improvements to class group and regulator computation of real quadratic fields

TL;DR

This paper introduces practical enhancements to the index-calculus algorithm, significantly speeding up the computation of class groups and regulators for real quadratic fields with large discriminants.

Contribution

It applies novel strategies like double large prime, structured Gaussian elimination, and Bernstein's batch smoothness to improve existing algorithms.

Findings

Achieved substantial speed-up in computations.

Successfully computed class group and regulator for a field with 110-digit discriminant.

Demonstrated practical applicability of the improved algorithm.

Abstract

We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein's batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number field with a 110-decimal digit discriminant.