Solving the Decision Principal Ideal Problem with Pre-processing

TL;DR

This paper explores classical algorithms for the principal ideal problem in algebraic number theory, demonstrating efficient solutions under certain conditions after pre-processing related to class field computation.

Contribution

It establishes a connection between the decision principal ideal problem and class field computation, enabling efficient solutions with pre-processing when the class group is smooth.

Findings

Decision problem solvable efficiently with pre-computation

Pre-processing involves collecting information about the Hilbert class field

Efficiency depends on the smoothness of the class group

Abstract

The principal ideal problem constitutes a fundamental problem in algebraic number theory and has attracted significant attention due to its applications in ideal lattice based cryptosystems. Efficient quantum algorithm has been found to address this problem. The situation is different in the classical computational setting. In this work, we delve into the relationship between the principal ideal problem and the class field computation. We show that the decision version of the problem can be solved efficiently if the class group is smooth, after pre-computation has been completed to collect information about the Hilbert class field.