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Some classes of permutation binomials and trinomials of index \(q-1\) over \({\mathbb {F}_{q^n}}\)

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Abstract

In this paper, using the classification of degree 7 permutations over \(\mathbb {F}_q\), we classify certain sparse PPs of the form \(P(x)=x^rf(x^{\frac{q^n-1}{q-1}})\) of \(\mathbb {F}_{q^n}\) for \(n=2\) and 3. In particular, we give necessary and sufficient conditions for the polynomial \(f_{a,b}(x):=x(x^{2(q^2+q+1)}+ax^{q^2+q+1}+b)\) in \(\mathbb {F}_{q^3}[x]\) to be a permutation polynomial over \(\mathbb {F}_{q^3}\), where \(q >409\) is a prime power.

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Funding

Qiang Wang is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2017-06410). Luciane Quoos thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro - FAPERJ, and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-CAPES for the partial support of this research.

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Gupta, R., Quoos, L. & Wang, Q. Some classes of permutation binomials and trinomials of index \(q-1\) over \({\mathbb {F}_{q^n}}\). Cryptogr. Commun. 16, 387–402 (2024). https://doi.org/10.1007/s12095-023-00674-y

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