[Paper Review] Quantum Algorithms for many-to-one Functions to Solve the Regulator and the Principal Ideal Problem

Arthur Schmidt|ArXiv.org|Dec 24, 2009

Quantum Computing Algorithms and Architecture9 references3 citations

TL;DR

This paper presents improved quantum algorithms for solving the regulator and principal ideal problems in real-quadratic number fields by leveraging many-to-one periodic functions in Shor's framework. The key contribution is a reduction of at least $2\log\Delta$ qubits compared to Hallgren's original algorithms, achieved by using irrational-period functions and reduced quadratic forms instead of ideals, with constant success probability despite non-injective periodicity.

ABSTRACT

We propose new quantum algorithms to solve the regulator and the principal ideal problem in a real-quadratic number field. We improve the algorithms proposed by Hallgren by using two different techniques. The first improvement is the usage of a period function which is not one-to-one on its period. We show that even in this case Shor's algorithm computes the period with constant probability. The second improvement is the usage of reduced forms (a, b, c) of discriminant D with a>0 instead of reduced ideals of the same discriminant. These improvements reduce the number of required qubits by at least 2 log D.

Motivation & Objective

• To develop more efficient quantum algorithms for the regulator and principal ideal problems in real-quadratic number fields.
• To reduce the qubit requirements of existing quantum algorithms by redefining the function space and period structure.
• To demonstrate that Shor’s algorithm can succeed with constant probability even when the function is many-to-one on its fundamental period.
• To provide a more efficient tool for assessing the quantum resistance of cryptosystems based on the principal ideal problem, such as the Buchmann-Williams cryptosystem.
• To lay the foundation for extending these techniques to class group computation and higher-degree number fields.

Proposed method

• The algorithm uses a period function that is not one-to-one on its fundamental period, yet Shor’s algorithm still computes the correct period with constant success probability.
• Instead of using reduced ideals, the method employs reduced quadratic forms $(a,b,c)$ of discriminant $\Delta$ with $a > 0$, which simplifies the representation and reduces qubit overhead.
• The quantum subroutine computes dual lattice vectors via the quantum Fourier transform, followed by classical post-processing using continued fractions or GCD-based methods.
• The algorithm estimates the regulator or ideal class distance using sampled dual vectors and applies classical refinement to improve approximation accuracy.
• A Monte Carlo approach is used: the algorithm may fail with bounded probability, but when it succeeds, the output is correct and verifiable.
• The method avoids the need for prior knowledge of logarithmic approximations by using functions with irrational periods that are always periodic but many-to-one.

Experimental results

Research questions

• RQ1 Can Shor’s algorithm compute the period of a function that is many-to-one on its fundamental period with constant success probability?
• RQ2 Can the use of reduced quadratic forms instead of ideals reduce the qubit count in quantum algorithms for number field problems?
• RQ3 Can the regulator and principal ideal problems be solved in quantum polynomial time without relying on the generalized Riemann hypothesis?
• RQ4 Can the framework be extended to compute the class group of real-quadratic fields or adapted to higher-degree number fields?
• RQ5 What is the minimal qubit overhead required for solving the regulator and PIP using periodic functions with non-injective structure?

Key findings

• The algorithm achieves a reduction of at least $2\log\Delta$ qubits compared to Hallgren’s original quantum algorithms for the regulator and principal ideal problems.
• Shor’s algorithm successfully computes the period of a many-to-one function with constant probability, even when the function is periodic over an irrational lattice.
• The success probability of the algorithm is at least $2^{-37}$, and it is a Monte Carlo algorithm that can be classically verified when it returns a solution.
• The use of reduced quadratic forms $(a,b,c)$ instead of ideals simplifies the function representation and reduces the required precision for quantum arithmetic.
• The algorithm runs in quantum polynomial time $O(\operatorname{polylog}(\log\Delta))$ and is independent of the generalized Riemann hypothesis.
• The method enables a more efficient analysis of quantum resistance for the Buchmann-Williams cryptosystem, as it directly tests principality of a given form.