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- W3210319725 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 $CC$, 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 $CC$ 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 $FF_q$ 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.
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- W3210319725 date "2021-10-31" @default.
- W3210319725 modified "2023-09-25" @default.
- W3210319725 title "Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems" @default.
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- W3210319725 doi "https://doi.org/10.48550/arxiv.2111.00405" @default.
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