Ambiguity and deficiency of permutations from finite fields

The concepts of ambiguity and deficiency for a given bijection on a finite Abelian group were recently introduced [13]. In this work we investigate the ambiguity and deficiency of some well-known polynomials which satisfy D_n(x+y, xy) = xⁿ+yⁿ for every x, y ϵ F_q and n ϵ N, as well as linearized polynomials and Dembowski-Ostrom polynomials (DO polynomials). For some specific values of n (related to q) these polynomials generate permutations on F_q. We derive explicitly the ambiguity and deficiency of some of them. Numerical results on the ambiguity and deficiency of the others are also provided. Some of these polynomials are almost perfect nonlinear (APN) functions.