Permutation Polynomials Under Multiplicative-Additive Perturbations: Characterization via Difference Distribution Tables

TL;DR

This paper characterizes permutation polynomials with perfect c-nonlinearity (PcN) over finite fields, providing efficient verification methods, algebraic characterizations for quadratic cases, and insights into their structural properties relevant for cryptography.

Contribution

It introduces the first characterization of PcN permutation polynomials via difference distribution tables and explores their algebraic and structural properties.

Findings

PcN polynomials characterized by DDT conditions

Verification of PcN property improved to O(p^{2n}) time

Identified affine transformations preserving c-differential uniformity

Abstract

We investigate permutation polynomials F over finite fields F_{p^n} whose generalized derivative maps x -> F(x + a) - cF(x) are themselves permutations for all nonzero shifts a. This property, termed perfect c-nonlinearity (PcN), represents optimal resistance to c-differential attacks - a concern highlighted by recent cryptanalysis of the Kuznyechik cipher variant. We provide the first characterization using the classical difference distribution table (DDT): F is PcN if and only if Delta_F(a,b) Delta_F(a,c^{-1}b) = 0 for all nonzero a,b. This enables verification in O(p^{2n}) time given a precomputed DDT, a significant improvement over the naive O(p^{3n}) approach. We prove a strict dichotomy for monomial permutations: the derivative F(x + alpha) - cF(x) is either a permutation for all nonzero shifts or for none, with the general case remaining open. For quadratic permutations, we provide explicit algebraic characterizations. We identify the first class of affine transformations preserving c-differential uniformity and derive tight nonlinearity bounds revealing fundamental incompatibility between PcN and APN properties. These results position perfect c-nonlinearity as a structurally distinct regime within permutation polynomial theory.