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}.

翻译：最近，陈和高等人~\cite{ChenGao2017}提出了一种用于布尔多项式系统求解的新型量子算法，其动机源于对某些后量子密码系统的密码分析。该方法的核心思想是将量子线性系统（QLS）算法应用于由布尔多项式系统导出的复数域$\mathbb{C}$上的Macaulay线性系统。该算法的效率取决于Macaulay矩阵的条件数。本文给出了条件数关于布尔解汉明权重的强下界，并表明在许多（若非全部）情况下，基于Grover的穷举搜索算法优于该算法。随后，我们通过约化原始Macaulay线性系统，引入复数域$\mathbb{C}$上的布尔Macaulay线性系统，对陈和高算法进行了改进。当解的汉明权重与布尔变量数量呈对数关系时，这种改进算法可能显著优于暴力搜索算法。此外，我们利用Valiant-Vazirani仿射哈希方法给出了改进算法正确性的更简单、更初等的证明，并将结果推广到$\mathbb{F}_q$上的多项式系统，改进了陈、高和袁~\cite{ChenGao2018}的后续工作。我们还提出了一种通过推广量子优惠券收集问题~\cite{arunachalam2020QuantumCouponCollector}来提取布尔多项式系统解的新方法。