Two New Measures for Permutations: Ambiguity and Deficiency

We introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite Abelian groups of the same size. Then, we study the optimum lower bounds of these measures for permutations of an Abelian group. A construction of permutations, by modifying some permutation functions over finite fields, is given. Their ambiguity and deficiency is investigated; most of these functions are APN permutations. We show that, when they are not optimal, the Mobius function in the multiplicative group of F-q is closer to being optimal in ambiguity than the inverse function in the additive group of. We note that the inverse function over F-28 is used in AES. Finally, we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the APN function employed in the SAFER cryptosystem.