arXiv subjects

Bimal Mandal

Publications and source records attributed to Bimal Mandal.

3 recordsLinked to original sources

arXiv2026

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

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.

cs.IT↗

arXiv2025

Extended c-differential distinguishers of full 9 and reduced-round Kuznyechik cipher

This paper introduces {\em truncated inner $c$-differential cryptanalysis}, a technique that enables the practical application of $c$-differential uniformity to block ciphers. While Ellingsen et al. (IEEE Trans. Inf. Theory, 2020) established the notion of $c$-differential uniformity by analyzing the equation $F(x\oplus a) \oplus cF(x) = b$, a key challenge remained: the outer multiplication by $c$ disrupts the structural properties essential for block cipher analysis, particularly key addition. We address this challenge by developing an \emph{inner} $c$-differential approach where multiplication by $c$ affects the input: $(F(cx\oplus a), F(x))$, thereby returning to the original idea of Borisov et al. (FSE, 2002). We prove that the inner $c$-differential uniformity of a function $F$ equals the outer $c$-differential uniformity of $F^{-1}$, establishing a duality between the two notions. This modification preserves cipher structure while enabling practical cryptanalytic applications. We apply our methodology to Kuznyechik (GOST R 34.12-2015) without initial key whitening. For reduced rounds, we construct explicit $c$-differential trails achieving probability $2^{-84.0}$ for two rounds and $2^{-169.7}$ for three rounds, representing improvements of 5.2 and 4.6 bits respectively over the best classical differential trails. For the full 9-round cipher, we develop a statistical truncated $c$-differential distinguisher. Through computational analysis involving millions of differential pairs, we identify configurations with bias ratios reaching $1.7\times$ and corrected p-values as low as $1.85 \times 10^{-3}$. The distinguisher requires data complexity $2^{33}$ chosen plaintext pairs, time complexity $2^{34}$, and memory complexity $2^{16}$.

cs.CR↗

arXiv2023

Rayleigh quotients of Dillon's functions

The Walsh--Hadamard spectrum of a bent function uniquely determines a dual function. The dual of a bent function is also bent. A bent function that is equal to its dual is called a self-dual function. The Hamming distance between a bent function and its dual is related to its Rayleigh quotient. Carlet, Danielsen, Parker, and Sole studied Rayleigh quotients of bent functions in ${\mathcal PS}_{ap}$, and obtained an expression in terms of a character sum. We use another approach comprising of Desarguesian spreads to obtain the complete spectrum of Rayleigh quotients of bent functions in $\mathcal{PS}_{ap}$.

cs.DM↗