The concepts of ambiguity and deficiency for a bijection on a finite Abelian group were recently introduced. In this paper, we present some further fundamental results on the ambiguity and deficiency of functions; in particular, we note that they are invariant under the well-known Carlet-Charpin-Zinoviev-equivalence, we obtain upper and lower bounds on the ambiguity and deficiency of differentially k-uniform functions, and we give a lower bound on the nonlinearity of functions that achieve the lower bound of ambiguity and deficiency. In addition, we provide an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a Dembowski-Ostrom (DO) polynomial, and using this technique, we find exact values for known cases of DO permutations with few terms. We also derive exact values for the ambiguities and deficiencies of DO permutations obtained from trace functions. The key relationship between the above polynomials is that they all have linearized difference map.

In this paper, we extend previously known results on the complexities of normal elements. Using algorithms that exhaustively test field elements, we are able to provide the distribution of the complexity of normal elements for binary fields with degree extensions up to 39. We also provide current results on the smallest known complexity for the remaining degree extensions up to 512 by using a combination of constructive theorems and known exact values. We give an algorithm to exhaustively search field elements by using Gray codes, which allows us to reuse previous computations. We compare this with a standard method. We analyze this algorithm and show both experimentally and asymptotically that the Gray code optimization gives substantial savings. The total computation of the distribution of the complexity of normal elements for degrees up to 39 in our experiments allows us to draw several conjectures. In particular, our data provides remarkable evidence for the conjecture that the complexity of normal elements follows a normal distribution. Finally, we conjecture that there is no linear bound on the minimum complexity with respect to the degree of the extension.