Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

TL;DR

This paper analyzes the limitations of the Macaulay matrix approach with the HHL quantum algorithm for solving multivariate polynomial systems, demonstrating lower bounds, comparing with classical methods, and proposing improvements.

Contribution

It provides a lower bound on the condition number of the Macaulay matrix, compares quantum and classical algorithms, and introduces an improved quantum algorithm for Boolean polynomial systems.

Findings

Grover-based search outperforms the quantum approach in many cases

The improved algorithm is effective when the solution's Hamming weight is logarithmic

A new solution extraction method via quantum coupon collector is proposed

Abstract

Recently Chen and Gao~\cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over $\mathbb{C}$, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over $\mathbb{C}$ by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables. Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over $\mathbb{F}_q$ improving on subsequent work by Chen, Gao and Yuan \cite{ChenGao2018}. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.