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

Ranit Dutta, Pantelimon Stanica, Bimal Mandal

TL;DR

The first class of affine transformations preserving c-differential uniformity is identified and tight nonlinearity bounds revealing fundamental incompatibility between PcN and APN properties are derived.

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.

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

TL;DR

The first class of affine transformations preserving c-differential uniformity is identified and tight nonlinearity bounds revealing fundamental incompatibility between PcN and APN properties are derived.

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.
Paper Structure (9 sections, 28 theorems, 76 equations, 1 table)

This paper contains 9 sections, 28 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Context and related work
  4. Organization
  5. Preliminaries
  6. Characterization and structural properties
  7. Invariance under affine transformations
  8. Nonlinearity of P$c$N functions
  9. Conclusion

Key Result

Theorem 2.1

Let $F$ be a permutation polynomial over $\mathbb{F}_{p^n}$. For any $a, b, c \in \mathbb{F}_{p^n}$ with $c \neq 0$, the outer $c$-differential entries of $F$ correspond to the inner $c$-differential entries of its inverse $F^{-1}$. Specifically, Consequently, $F$ is P$c$N (with respect to outer $c$-differential) if and only if $F^{-1}$ is P$c$N with respect to the inner $c$-differential.

Theorems & Definitions (65)

  • Theorem 2.1: SDM25
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4: LJ78PS20
  • Theorem 3.5
  • proof
  • ...and 55 more