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 Möbius function in the multiplicative group of \BBFq is closer to being optimal in ambiguity than the inverse function in the additive group of \BBFq. We note that the inverse function over \BBF₂⁸ 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.