Differential Spectrum and Boomerang Spectrum of Some Power Mapping

TL;DR

This paper extends the analysis of differential and boomerang spectra for a class of power mappings over finite fields from a special case to a general case, using a new method involving rational points on curves.

Contribution

It generalizes previous results on power functions by determining spectra for all gcd conditions using a novel approach with algebraic curves.

Findings

Determined differential spectrum for general gcd conditions.

Calculated boomerang spectrum for the power mappings.

Introduced a new method applicable to Niho type functions.

Abstract

Let $f(x)=x^{s(p^m-1)}$ be a power mapping over $\mathbb{F}_{p^n}$, where $n=2m$ and $\gcd(s,p^m+1)=t$. In \cite{kpm-1}, Hu et al. determined the differential spectrum and boomerang spectrum of the power function $f$, where $t=1$. So what happens if $t\geq1$? In this paper, we extend the result of \cite{kpm-1} from $t=1$ to general case. We use a different method than in \cite{kpm-1} to determine the differential spectrum and boomerang spectrum of $f$ by studying the number of rational points on some curves. This method may be helpful for calculating the differential spectrum and boomerang spectrum of some Niho type power functions.