Quantum Advantage via Solving Multivariate Polynomials (2509.07276v1)

Abstract: In this work, we propose a new way to (non-interactively, verifiably) demonstrate quantum advantage by solving the average-case $\mathsf{NP}$ search problem of finding a solution to a system of (underdetermined) constant degree multivariate equations over the finite field $\mathbb{F}2$ drawn from a specified distribution. In particular, for any $d \geq 2$, we design a distribution of degree up to $d$ polynomials ${p_i(x_1,\ldots,x_n)}{i\in [m]}$ for $m<n$ over $\mathbb{F}_2$ for which we show that there is a expected polynomial-time quantum algorithm that provably simultaneously solves $\{p_i(x_1,\ldots,x_n)=y_i\}_{i\in [m]}$ for a random vector $(y_1,\ldots,y_m)$. On the other hand, while solutions exist with high probability, we conjecture that for constant $d > 2$, it is classically hard to find one based on a thorough review of existing classical cryptanalysis. Our work thus posits that degree three functions are enough to instantiate the random oracle to obtain non-relativized quantum advantage. Our approach begins with the breakthrough Yamakawa-Zhandry (FOCS 2022) quantum algorithmic framework. In our work, we demonstrate that this quantum algorithmic framework extends to the setting of multivariate polynomial systems. Our key technical contribution is a new analysis on the Fourier spectra of distributions induced by a general family of distributions over $\mathbb{F}_2$ multivariate polynomials -- those that satisfy $2$-wise independence and shift-invariance. This family of distributions includes the distribution of uniform random degree at most $d$ polynomials for any constant $d \geq 2$. Our analysis opens up potentially new directions for quantum cryptanalysis of other multivariate systems.

• The paper introduces a novel quantum algorithm demonstrating quantum advantage by efficiently solving underdetermined multivariate polynomial equations over F2.
• The paper employs advanced Fourier spectra analysis and dual Reed-Solomon based error correction to process the polynomial system effectively.
• The paper argues that degree-three or higher polynomial systems are classically intractable, thereby reshaping approaches in quantum cryptanalysis.

Quantum Advantage via Solving Multivariate Polynomials

Introduction

The paper introduces a novel quantum algorithm aimed at demonstrating quantum advantage by addressing an average-case NP search problem involving multivariate polynomial equations over a finite field. Specifically, it presents a scheme where the problem, solved efficiently by quantum algorithms, is conjecturally intractable for classical computation when using polynomials of degree three or higher. Building on the Yamakawa-Zhandry framework, the authors extend the approach to accommodate multivariate polynomial systems. The central contribution lies in analyzing the Fourier spectra of induced distributions by multivariate polynomials that exhibit specific independence properties and invariance, offering insights into quantum cryptanalysis.

Problem Formulation

The paper focuses on solving systems of constant-degree multivariate polynomial equations over the finite field $F_2$. The structured problem involves two main components: a set of degree-three polynomials and linear constraints derived from a Reed-Solomon code, where the algebraic system is heavily underdetermined. The core challenge is to find solutions over $F_2$ such that both the polynomial equations and the linear constraints are satisfied. While quantum algorithms provide polynomial-time solutions, classical algorithms are conjectured to struggle due to the structural and independence properties discussed within the paper.

Quantum Algorithm and Technical Contributions

The quantum algorithm extends the work of Yamakawa and Zhandry while adapting it to structured algebraic systems. It constructs quantum states that superimpose solutions to polynomial constraints and code words, with unique reliance on the Quantum Fourier Transform (QFT) and error correction based on dual Reed-Solomon codes. Key innovations include:

• Fourier Spectra Analysis: The work introduces a novel analysis of the Fourier spectra for distributions over multivariate polynomials, enabling efficient quantum processing.
• Error Distribution and Decoding: The authors demonstrate that the error distribution from the polynomial system is amenable to quantum error correction techniques, facilitating unique decoding necessary for quantum advantage.
• Independence and Shift Invariance: By leveraging 2-wise independence and shift invariance, the algorithm ensures consistency and amplifies the distinction between quantum and classical complexities.

Conjectured Classical Hardness and Cryptanalytic Impact

The paper argues for the classical hardness based on the underdetermined nature and degree structure of the polynomials. While degree-two systems can be reduced using specialization approaches, degree-three or higher resist classical algorithms' typical strategies, including exhaustive search or algebraic techniques using Gröbner bases. Therefore, the authors suggest that these results challenge the prevailing belief in the uniform difficulty of such systems across both quantum and classical paradigms.

Implementation Considerations

Implementing the described quantum algorithm requires considerations such as:

• Quantum Resources: The deployment involves QFT operations and criteria for error correction, demanding a quantum system with sufficient qubit coherence and gate fidelity.
• Complexity and Scalability: The algorithm supports polynomial scalability concerning quantum resources, though classical systems exhibit exponential growth in complexity.
• Structural Dependence: The success hinges on exploiting the structured nature of chosen cryptographic codes and ensuring compatibility with existing quantum hardware capabilities.

Conclusion and Future Directions

This work contributes a significant practical approach to realizing quantum advantage through multivariate polynomial systems. It stands at the intersection of theoretical advances in quantum cryptography and applied quantum computing, paving the way for future explorations into quantum-solving strategies for structured problems. Potential developments may encompass diverse polynomial systems and applications across cryptographic domains, fostering further research into harnessing quantum technologies for complex algebraic challenges. These insights may also inspire classical approximation methods, enhancing overall problem-solving methodologies in cryptanalysis.