Algebraic number fields and the LLL algorithm

TL;DR

This paper provides a detailed analysis of the computational complexity of algebraic operations and algorithms, including LLL, in algebraic number fields with exact arithmetic, extending classical algorithms to these fields.

Contribution

It extends the analysis of the LLL algorithm and Gaussian elimination to algebraic number fields, providing polynomial bounds on their exact computational costs.

Findings

Extended LLL algorithm to algebraic number fields

Derived polynomial upper bounds for algorithm running times

Analyzed real-specific operations in algebraic number fields

Abstract

In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let $K$ be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in $K$ in terms of the size of the input and the parameters of $K$. We include some earlier results about these, but we go further than them, e.g. we also analyze some $\mathbb{R}$-specific operations in $K$ like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from $\mathbb{Z}^n$ to $K^n$, and give a polynomial upper bound on the running time when the computations in $K$ are performed exactly (as opposed to floating-point approximations).