Abstract
In this paper, we study the cycle structure of a permutation polynomial of the form \(f(x)=x^{(q+1)s_1}(x+x^{q})^{s_2}+x^{s_3}\) over \({\mathbb {F}}_{q^2}\), where \((s_1,s_2,s_3)\in \{(q^2-2,3,q), (1,q^2-2,1), (1,q^2-1,2), (1,2,4)\}\) and q is even. By calculating the sum of all elements in each cycle of a fraction polynomial \(\frac{x}{x^3+x^2+1}\) or a linearized polynomial \(x^{2^e}+x^2+x\) with \(e\ge 0\), the cycle structure of f(x) over \({\mathbb {F}}_{q^2}\) in the first three cases, that is, \((s_1,s_2,s_3)=\{(q^2-2,3,q), (1,q^2-2,1)\) or \((1,q^2-1,2)\}\), is characterized. For the case \((s_1,s_2,s_3)=(1,2,4)\), we give the cycle structure of f(x) over \({\mathbb {F}}_{q^2}\) for \(q={2^{{2}^{k}}}\) with a positive integer k. For \(q=2^{{2}^{k}_p}\) with an odd prime p, it needs more techniques to determine the cycle structure of f(x). We only give its cycle length.
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The data used to support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We thank professor Xiang-Dong Hou for his helpful suggestion. The work was supported by the National Nature Science Foundation of China (NSFC) under Grant Nos. 62072161 and 12101207 and by the Application Foundation Frontier Project of Wuhan Science and Technology Bureau under Grant No. 2020010601012189.
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Zeng, D., Zeng, X., Li, L. et al. The cycle structure of a class of permutation polynomials. Des. Codes Cryptogr. 91, 1373–1400 (2023). https://doi.org/10.1007/s10623-022-01155-8
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DOI: https://doi.org/10.1007/s10623-022-01155-8