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21.10.2023 | Original Paper

Further results on the \((-1)\)-differential uniformity of some functions over finite fields with odd characteristic

verfasst von: Qian Liu, Ximeng Liu, Meixiang Chen, Jian Zou, Zhiwei Huang

Erschienen in: Applicable Algebra in Engineering, Communication and Computing

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Abstract

Functions with low differential uniformity have wide applications in cryptography. In this paper, by using the quadratic character of \({\mathbb {F}}_{p^n}^*\), we further investigate the \((-1)\)-differential uniformity of these functions in odd characteristic: (1) \(f_1(x)=x^d\), where \(d=-\frac{p^n-1}{2}+p^k+1\), n and k are two positive integers satisfying \(\frac{n}{\gcd (n,k)}\) is odd; (2) \(f_2(x)=(x^{p^m}-x)^{\frac{p^n-1}{2}+1}+x+x^{p^m}\), where \(n=3m\); (3) \(f_3(x)=(x^{3^m}-x)^{\frac{3^n-1}{2}+1}+(x^{3^m}-x)^{\frac{3^n-1}{2}+3^m}+x\), where \(n=3m\). The results show that the upper bounds on the \((-1)\)-differential uniformity of the power function \(f_1(x)\) are derived. Furthermore, we determine the \((-1)\)-differential uniformity of two classes of permutation polynomials \(f_2(x)\) and \(f_3(x)\) over \({\mathbb {F}}_{p^n}\) and \({\mathbb {F}}_{3^n}\), respectively. Specifically, a class of permutation polynomial \(f_3(x)\) that is of P\(_{-1}\)N or AP\(_{-1}\)N function over \({\mathbb {F}}_{3^n}\) is obtained.

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Metadaten
Titel
Further results on the -differential uniformity of some functions over finite fields with odd characteristic
verfasst von
Qian Liu
Ximeng Liu
Meixiang Chen
Jian Zou
Zhiwei Huang
Publikationsdatum
21.10.2023
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-023-00632-4