Rigorous Computation of Fundamental Units in Algebraic Number Fields

TL;DR

This paper introduces an efficient algorithm for computing the unit group of algebraic number fields, improving upon previous methods and capable of certifying correctness under certain hypotheses.

Contribution

The authors develop a new algorithm that unconditionally computes the unit group with fewer bit operations, and can certify correctness under the Generalized Riemann Hypothesis.

Findings

Algorithm operates in expected time $O( ext{discriminant}^{1/4 - 1/(8n+4) + ext{epsilon}})$

Unconditionally computes the unit group with fewer bit operations than baby-step giant-step

Can certify correctness under the Generalized Riemann Hypothesis

Abstract

We present an algorithm that unconditionally computes a representation of the unit group of a number field of discriminant $\Delta_K$, given a full-rank subgroup as input, in asymptotically fewer bit operations than the baby-step giant-step algorithm. If the input is assumed to represent the full unit group, for example, under the assumption of the Generalized Riemann Hypothesis, then our algorithm can unconditionally certify its correctness in expected time $O(\Delta_K^{n/(4n + 2) + \epsilon}) = O(\Delta_K^{1/4 - 1/(8n+4) + \epsilon})$ where $n$ is the unit rank.