Revisiting products of the form X times a linearized polynomial L(X)

For a q-polynomial L over a finite field \(\mathbb {F}_{q^n}\), we characterize the differential spectrum of the function \(f_L:\mathbb {F}_{q^n} \rightarrow \mathbb {F}_{q^n}, x \mapsto x \cdot L(x)\) and show that, for \(n \le 5\), it is completely determined by the image of the rational function \(r_L :\mathbb {F}_{q^n}^* \rightarrow \mathbb {F}_{q^n}, x \mapsto L(x)/x\). This result follows from the classification of the pairs (L, M) of q-polynomials in \(\mathbb {F}_{q^n}[X]\), \(n \le 5\), for which \(r_L\) and \(r_M\) have the same image, obtained in Csajbók et al. (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.