On generalizations of differential uniform permutations over finite fields based on 2-to-1 mappings

The design of S-boxes, or substitution boxes, in block ciphers relies on measures such as the Difference Distribution Table (DDT) and differential uniformity to gauge the function’s resistance to differential cryptanalysis. S-boxes with low differential uniformity are strong candidates for S-box design due to their robust resistance to differential attacks, leading to the creation of historical families of Almost Perfect Nonlinear (APN) and Perfect Nonlinear (PN) functions with the lowest differential uniformity. Recently, the concept of c-DDT and c-differential uniformity has been introduced, opening the door to c-Almost Perfect Nonlinear (APcN) and c-Perfect Nonlinear (PcN) functions, potentially expanding differential cryptanalysis. Functions with low c-differential uniformity, notably APcN and PcN, have gained significant attention, leading to new connections with other objects. In this paper, we explore new classes of perfect c-nonlinear and almost perfect c-nonlinear functions over finite fields of arbitrary characteristics. We extend the methods used in the AGW (Akbary, Ghioca, Wang) criterion and AGW-like criterion to design perfect c-nonlinear and almost perfect c-nonlinear functions over finite fields by considering linearized polynomials and linear structures.