Paper
Permutation Polynomials Under Multiplicative-Additive Perturbations: Characterization via Difference Distribution Tables
Authors
Ranit Dutta, Pantelimon Stanica, Bimal Mandal
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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2602.21632v1</id>\n <title>Permutation Polynomials Under Multiplicative-Additive Perturbations: Characterization via Difference Distribution Tables</title>\n <updated>2026-02-25T06:55:59Z</updated>\n <link href='https://arxiv.org/abs/2602.21632v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2602.21632v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.IT'/>\n <published>2026-02-25T06:55:59Z</published>\n <arxiv:primary_category term='cs.IT'/>\n <author>\n <name>Ranit Dutta</name>\n </author>\n <author>\n <name>Pantelimon Stanica</name>\n </author>\n <author>\n <name>Bimal Mandal</name>\n </author>\n </entry>"
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