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The differential spectrum and boomerang spectrum of a class of locally-APN functions

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Abstract

In this paper, we study the boomerang spectrum of the power mapping \(F(x)=x^{k(q-1)}\) over \({\mathbb {F}}_{q^2}\), where \(q=p^m\), p is a prime, m is a positive integer and \(\gcd (k,q+1)=1\). We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from \((p,k)=(2,1)\) to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if \(p=2\) and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example \(x^{45}\) over \({\mathbb F}_{2^8}\) in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2.

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Acknowledgements

This work was supported by the National Key Research and Development Program of China (No. 2021YFA1000600), the National Natural Science Foundation of China (Nos. 62072162, 12001176), the Natural Science Foundation of Hubei Province of China (No. 2021CFA079) and the Knowledge Innovation Program of Wuhan-Basic Research (No. 2022010801010319).

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Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Zhao Hu, Xiangyong Zeng & Xiaohu Tang

  2. Hubei Key Laboratory of Applied Mathematics, School of Cyber Science and Technology, Hubei University, Wuhan, 430062, China

    Nian Li

  3. Wuhan Marine Communication Research Institute, Wuhan, 430000, China

    Linjie Xu

  4. Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, 610031, China

    Xiaohu Tang

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  1. Zhao Hu
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Correspondence to Nian Li.

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Hu, Z., Li, N., Xu, L. et al. The differential spectrum and boomerang spectrum of a class of locally-APN functions. Des. Codes Cryptogr. 91, 1695–1711 (2023). https://doi.org/10.1007/s10623-022-01161-w

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