Differential spectra of a class of power permutations with Niho exponents

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ m\geq3 $\end{document}</tex-math></inline-formula> be a positive integer and <inline-formula><tex-math id="M2">\begin{document}$ n = 2m $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M3">\begin{document}$ f(x) = x^{2^m+3} $\end{document}</tex-math></inline-formula> be a power permutation over <inline-formula><tex-math id="M4">\begin{document}$ {\mathrm {GF}}(2^n) $\end{document}</tex-math></inline-formula>, which is a monomial with a Niho exponent. In this paper, the differential spectrum of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> is investigated. It is shown that the differential spectrum of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ \mathbb S = \{\omega_0 = 2^{2m-1}+2^{2m-3}-1,\omega_2 = 2^{2m-2}+2^{m-1}, \omega_4 = 2^{2m-3}-2^{m-1},\omega_{2^m} = 1\} $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula> is even, and <inline-formula><tex-math id="M9">\begin{document}$ \mathbb S = \{\omega_0 = \frac{7\cdot2^{2m-2}+2^m}3, \omega_2 = 3\cdot2^{2m-3}-2^{m-2}-1, \omega_6 = \frac{2^{2m-3}-2^{m-2}}3, \omega_{2^m+2} = 1\} $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M10">\begin{document}$ m $\end{document}</tex-math></inline-formula> is odd.</p>

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.