On the complexity of class group computations for large degree number fields

TL;DR

This paper analyzes the complexity of class group computations for large degree number fields without small defining polynomials, simplifying existing algorithms and optimizing parameters for better runtime performance.

Contribution

It simplifies and improves the complexity analysis of class group algorithms for large degree fields, and introduces a classification framework for these fields.

Findings

Simplified the algorithm based on Biasse and Fieker's work.

Identified optimal parameters to minimize runtime.

Provided a classification scheme for number fields by extension degree.

Abstract

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity analysis and identify the optimal parameters to reduce the runtime. We make use of the classes $\mathcal D$ defined in [GJ16] for classifying the fields according to the size of the extension degree and prove that they enable to describe all the number fields.