Upper bounding the quantum space complexity for computing class group and principal ideal problem

TL;DR

This paper establishes an upper bound on the quantum space complexity for algorithms solving class group and principal ideal problems, advancing understanding of quantum resource requirements in algebraic number theory.

Contribution

It provides the first explicit upper bound on quantum space complexity for these problems, based on recent algorithmic frameworks and reductions.

Findings

Derived an explicit quantum space complexity upper bound.

Applied recent algorithmic frameworks to algebraic number theory problems.

Connected quantum complexity bounds with classical reduction techniques.

Abstract

In this paper, we calculate the upper bound on quantum space complexity of the quantum algorithms proposed by Biasse and Song (SODA'16) for solving class group computation and the principal ideal problem using the reductions to $S$-unit group computation. We follow the approach of Barbulescu and Poulalion (AFRICACRYPT'23) and the framework given by de Boer, Ducas, and Fehr (EUROCRYPT'20) and Eisentr\"{a}ger, Hallgren, Kitaev, and Song (STOC'14).