Abstract
In this paper, we present quantum algorithms to solve the linear structures of Boolean functions. “Suppose Boolean function \(f\): \(\{0, 1\}^{n}\rightarrow \{0, 1\}\) is given as a black box. There exists an unknown n-bit string \(\alpha \) such that \(f(x)=f(x\oplus \alpha )\). We do not know the n-bit string \(\alpha \), excepting the Hamming weight \(W(\alpha )=m, 1\le m\le n\). Find the string \(\alpha \).” In case \(W(\alpha )=1\), we present an efficient quantum algorithm to solve this linear construction for the general \(n\). In case \(W(\alpha )>1\), we present an efficient quantum algorithm to solve it for most cases. So, we show that the problem can be ”solved nearly” in quantum polynomial times \(O(n^{2})\). From this view, the quantum algorithm is more efficient than any classical algorithm.
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Acknowledgments
We wish to thank reviewers’ comments for helpful readability about this paper. The authors are supported by the National Science Foundation of China (Nos. 61303212, 61202386), the State Key Program of National Natural Science of China (No. 613320194), and Major State Basic Research Development Program of China (No. 2014CB340600) and the Fundamental Research Funds for the Central Universities (2012211020213).
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Wu, W., Zhang, H., Wang, H. et al. Polynomial-time quantum algorithms for finding the linear structures of Boolean function. Quantum Inf Process 14, 1215–1226 (2015). https://doi.org/10.1007/s11128-015-0940-1
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DOI: https://doi.org/10.1007/s11128-015-0940-1