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

Jintai Ding1, Vlad Gheorghiu2, András Gilyén3, Sean Hallgren4, and Jianqiang Li4

1University of Cincinnati, OH, USA
2Institute for Quantum Computing / Dept. of Combinatorics & Optimization, University of Waterloo, ON, Canada
3Institute for Quantum Information and Matter, Caltech, Pasadena CA, USA
4Department of Computer Science and Engineering, Pennsylvania State University, PA, USA

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Abstract

Recently Chen and Gao [15] 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 \citeChenGao2018. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem [2].

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