Revisiting products of the form \(\it X\) times a linearized polynomial \(\textit {L(X)}\)

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