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- W4385262785 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}." @default.
- W4385262785 created "2023-07-26" @default.
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- W4385262785 date "2023-07-26" @default.
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- W4385262785 title "Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems" @default.
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- W4385262785 doi "https://doi.org/10.22331/q-2023-07-26-1069" @default.
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