The differential spectrum of a class of power functions over finite fields

Abstract

<p style='text-indent:20px;'>Functions with good differential-uniformity properties have important applications in coding theory and sequence design in addition to the applications in cryptography. The differential spectrum of a cryptographic function is useful for estimating its resistance to some variants of differential cryptanalysis. The objective of this paper is to determine the differential spectrum of the power function <inline-formula><tex-math id="M1">\begin{document}$ x^{p^{2k}-p^k+1} $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M2">\begin{document}$ \mathbb F_{p^n} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula> is an odd prime, <inline-formula><tex-math id="M4">\begin{document}$ n, k, e $\end{document}</tex-math></inline-formula> are integers with <inline-formula><tex-math id="M5">\begin{document}$ \gcd(n,k) = e $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \frac{n}{e} $\end{document}</tex-math></inline-formula> being odd. In particular, when <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula> is odd and <inline-formula><tex-math id="M8">\begin{document}$ e = 1 $\end{document}</tex-math></inline-formula>, our result includes a recent one (IEEE Trans. Inform. Theory 65(10): 6819-6826) as a special case.

Topics & Concepts