Generating Random Factored Ideals in Number Fields

TL;DR

This paper introduces a randomized polynomial-time algorithm for generating random ideals in number fields and function fields, with applications to sampling ideals uniformly by norm.

Contribution

The paper presents the first efficient randomized algorithms for generating and factoring ideals in number and function fields according to their norm distribution.

Findings

Algorithm runs in polynomial time

Able to generate ideals uniformly at random

Includes a method for factoring the generated integers

Abstract

We present a randomized polynomial-time algorithm to generate a random integer according to the distribution of norms of ideals at most N in any given number field, along with the factorization of the integer. Using this algorithm, we can produce a random ideal in the ring of algebraic integers uniformly at random among ideals with norm up to N, in polynomial time. We also present a variant of this algorithm for generating ideals in function fields.