Paper 2024/1688

Revisiting Products of the Form $X$ Times a Linearized Polynomial $L(X)$

Abstract

For a $q$-polynomial $L$ over a finite field $\mathbb{F}_{q^n}$, we characterize the differential spectrum of the function $f_L\colon \mathbb{F}_{q^n} \rightarrow \mathbb{F}_{q^n}, x \mapsto x \cdot L(x)$ and show that, for $n \leq 5$, it is completely determined by the image of the rational function $r_L \colon \mathbb{F}_{q^n}^* \rightarrow \mathbb{F}_{q^n}, x \mapsto L(x)/x$. This result follows from the classification of the pairs $(L,M)$ of $q$-polynomials in $\mathbb{F}_{q^n}[X]$, $n \leq 5$, for which $r_L$ and $r_M$ have the same image, obtained in [B. Csajbok, G. Marino, and O. Polverino. A Carlitz type result for linearized polynomials. Ars Math. Contemp., 16(2):585–608, 2019]. For the case of $n>5$, we pose an open question on the dimensions of the kernels of $x \mapsto L(x) - ax$ for $a \in \mathbb{F}_{q^n}$. We further present a link between functions $f_L$ of differential uniformity bounded above by $q$ and scattered $q$-polynomials and show that, for odd values of $q$, we can construct CCZ-inequivalent functions $f_M$ with bounded differential uniformity from a given function $f_L$ fulfilling certain properties.