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A quantum query algorithm for computing the degree of a perfect nonlinear Boolean function

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Abstract

The degree of a Boolean function is a basic primitive that has applications in coding theory and cryptography. This paper considers a problem of computing the degree of a perfect nonlinear Boolean function in a quantum system. The details are as follows: Given a promise that the function f is either linear or perfect nonlinear in \(F^{n}_{d}\), we propose a quantum algorithm 1 to distinguish which case it is with a high probability, where d is an even number. Furtherly, for computing the degree of a perfect nonlinear Boolean function f, we present a quantum Algorithm 2 to solve it by calling quantum Algorithm 1 when \(d=2\). The quantum query complexity of the proposed quantum Algorithm 2 is O(s), and the space complexity (the number of quantum logic gate) is \(O(2^{s})\), where \(s+1=\text {deg}(f)\). The analysis shows that the quantum Algorithm 2 proposed in this paper is more efficient than any classical algorithm for solving this problem.

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Acknowledgements

The authors are supported by the Major State Basic Research Development Program of China No. 2014CB340600, Natural Science Foundation of HeBei Province No. F2017201199, Science and Technology Research Project of Hebei Higher Education No. QN2017020.

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Authors and Affiliations

  1. School of Cyber Security and Computers, Hebei University, Baoding, 071002, People’s Republic of China

    WanQing Wu

  2. Computer School of Wuhan University, Wuhan, 430072, People’s Republic of China

    HuanGuo Zhang

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  1. WanQing Wu
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  2. HuanGuo Zhang
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Correspondence to WanQing Wu.

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Wu, W., Zhang, H. A quantum query algorithm for computing the degree of a perfect nonlinear Boolean function. Quantum Inf Process 18, 62 (2019). https://doi.org/10.1007/s11128-019-2175-z

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