Reducing the complexity for class group computations using small defining polynomials

TL;DR

This paper introduces an efficient algorithm for computing class groups of number fields with small defining polynomials, potentially reducing complexity to subexponential levels by testing smoothness of principal ideals.

Contribution

It presents a novel relation collection algorithm that leverages small defining polynomials to improve class group computation efficiency.

Findings

Complexity potentially reduced to $L_{| riangle_K|}(1/3)$

Efficient relation collection via smoothness testing of principal ideals

Applicable to number fields with small defining polynomials

Abstract

In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals generated by small algebraic integers. This strategy leads to an algorithm for computing the class group whose complexity is possibly as low as $L_{|\Delta_{\mathbf K}|}\left(\frac{1}{3}\right)$.