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 $f_L:{\mathbb{F}}_{q^n}\to {\mathbb{F}}_{q^n},x\mapsto x\cdot L(x)$ and show that, for $n\le 5$, it is completely determined by the image of the rational function $r_L:{\mathbb{F}}_{q^n}^{\ast }\to {\mathbb{F}}_{q^n},x\mapsto L(x)/x$. This result follows from the classification of the pairs $(L,M)$ of $q$-polynomials in ${\mathbb{F}}_{q^n}[X]$, $n\le 5$, 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 $$ of differential uniformity bounded above by $$ and scattered $$-polynomials and show that, for odd values of $$, we can construct CCZ-inequivalent functions $$ with bounded differential uniformity from a given function $$ fulfilling certain properties.