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Designs, Codes and Cryptography
Erschienen in: Designs, Codes and Cryptography 5/2023

06.01.2023

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

verfasst von: Zhao Hu, Nian Li, Linjie Xu, Xiangyong Zeng, Xiaohu Tang

Erschienen in: Designs, Codes and Cryptography | Ausgabe 5/2023

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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.
Vorheriger Artikel Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes
Nächster Artikel An improved method for predicting truncated multiple recursive generators with unknown parameters
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Metadaten
Titel
The differential spectrum and boomerang spectrum of a class of locally-APN functions
verfasst von
Zhao Hu
Nian Li
Linjie Xu
Xiangyong Zeng
Xiaohu Tang
Publikationsdatum
06.01.2023
Verlag
Springer US
Erschienen in
Designs, Codes and Cryptography / Ausgabe 5/2023
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-022-01161-w

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