On construction and (non)existence of \(c\)-(almost) perfect nonlinear functions (Q2662058)

The paper investigates a generalizations of Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions by introducing a notion of \(c\)-differential uniformity: Given a \(p\)-ary \((n, m)\)-function \(f : \mathbb{F}_p^n\rightarrow \mathbb{F}_p^m\), and \(c\in\mathbb{F}_p^m\), the \(c\)-derivative of \(f\) with respect to \(a\) is the function \(_cD_af(x) = f(x + a)- cf(x)\). For an \((n, n)\)-function \(f\), and \(a, b \in F_p^n\), let \(_c\Delta_f (a, b) := |\{x \in F_p^n : f(x + a)- cf(x) = b\}|\), and \(_c\Delta_f :=\max\{_c\Delta_f (a, b) : a, b\in F_p^n, (a, c) \neq (0, 1)\}\). We say that \(f\) is differentially \((c, \delta)\)-uniform if \(_c\Delta_f\le\delta\in\mathbb{N}\). Authors focus on the special cases \(\delta = 1\) and \(\delta = 2\), i.e. PcN and APcN functions. For quadratic PcN and APcN functions authors provide their characterizations and a correspondence between Dembowski-Ostrom polynomials and APcN maps. They show that it is possible to construct several classes of APcN and PcN functions using the AGW criterion [\textit{A. Akbary} et al., Finite Fields Appl. 17, No. 1, 51--67 (2011; Zbl 1281.11102)]. Paper concludes with nonexistence results for some exceptional monomial APcN and PcN functions using connections with algebraic curves and Galois theory tools.