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.

翻译：本文研究有限域F_{p^n}上的置换多项式F，其广义导数映射x -> F(x + a) - cF(x)对所有非零位移a均保持置换性质。该性质被称为完美c非线性（PcN），代表了针对c差分攻击的最优抵抗性——这一特性因近期对Kuznyechik密码变体的密码分析而备受关注。我们首次利用经典差分分布表（DDT）给出刻画：F是PcN的充要条件为对所有非零a,b满足Delta_F(a,b) Delta_F(a,c^{-1}b) = 0。基于预计算的DDT，该判据可在O(p^{2n})时间内完成验证，较原始的O(p^{3n})方法有显著改进。我们证明了单项式置换的严格二分性：导数F(x + alpha) - cF(x)要么对所有非零位移都是置换，要么对所有位移均非置换，一般情形仍有待研究。对于二次置换，我们给出了显式的代数刻画。我们发现了首类保持c差分一致性的仿射变换，并推导出紧致的非线性度界限，揭示了PcN性质与APN性质之间的本质不兼容性。这些成果将完美c非线性确立为置换多项式理论中结构独特的研究范式。