Ambiguity and deficiency of permutations from finite fields

Abstract

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 Dn(x+y, xy) = x n +y n for every x, y ∈ Fq and n ∈ 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 Fq. 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.

On difference maps and their cryptographic applications

“…Reversed Dickson polynomials which are known to be perm utation polynomials of are given in Table 3.1. An an analogous table for reversed Dickson permutation polynomials defined over odd characteristics appears in Table 3 In Proposition 8.2.5, we state the ambiguity and deficiency of the reversed Dickson polynomials over F 2» th a t appear in [54]. The coefficients of the reversed Dickson polynomial are given first in Lemma 8.2.3.…”

“…This section gives the opening steps to calculate the ambiguity and deficiency of reversed Dickson perm utation polynomials over finite fields. The ambiguity and deficiency of two reversed Dickson polynomials appears in the proceedings [54]. to be a perm utation polynomial of F9.…”

“…New results on the ambiguity and deficiency of specific polynomials appear in the proceedings paper [54]. In this paper, the authors consider monomials (or power functions), Linearized poly nomials, Dickson polynomials and Dembowski-Ostrom polynomials.…”

Deficiency and Ambiguity of Low Degree Permutation Polynomials

“…where a ∈ G [47], [48], [49], [50]. In [47], they are defined for a general map between finite Abelian groups.…”