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

2026-02-25

AI Summary

This study addresses the threat of c-differential cryptanalysis by focusing on permutation polynomials over finite fields that exhibit perfect c-nonlinearity (PcN). It provides the first complete characterization of PcN permutations via their difference distribution tables (DDTs), establishing necessary and sufficient conditions and proposing an efficient verification algorithm with time complexity O(p²ⁿ). The work demonstrates the fundamental incompatibility between PcN and almost perfect nonlinear (APN) properties, proves that monomial permutations satisfy an “all-or-nothing” derivative permutation behavior, and derives explicit algebraic criteria along with tight nonlinearity bounds for quadratic permutations. Furthermore, it constructs the first class of affine transformations preserving c-differential uniformity, systematically integrating finite field algebra, permutation polynomial theory, and computational optimization techniques.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• permutation polynomials
• perfect c-nonlinearity
• c-differential attacks
• difference distribution table
• finite fields

Innovation

• perfect c-nonlinearity
• difference distribution table
• permutation polynomials
• c-differential uniformity
• finite fields