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Constructions with high algebraic degree of differentially 4-uniform (n, n-1)-functions and differentially 8-uniform (n, n-2)-functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2018年第4期10卷 611-628页

作者： Alsalami, Yousuf Khalifa Univ Sci & Technol Dept Elect & Comp Engn Abu Dhabi U Arab Emirates

Quadratic differentially 4-uniform (n, n - 1)-functions are given in Carlet J. Adv. Math. Commun. 9(4), 541-565 (2015) where a question is raised of whether non-quadratic differentially 4-uniform (n, n - 1)- functions exist. In this paper, we give highly nonlinear differentially 4-uniform (n, n - 1)-functions of optimal algebraic degree for both n even and odd. Using the approach in Carlet J. Adv. Math. Commun. 9(4), 541-565 (2015), we construct these functions using two APN (n - 1, n - 1)-functions which are EA-equivalent Inverse functions satisfying some necessary and sufficient conditions when n is even. We slightly generalize the approach to construct differentially 4-uniform (n, n - 1)-functions from two differentially 4-uniform (n - 1, n - 1)-functions satisfying some necessary conditions. This allows us to derive the differentially 4-uniform (n, n - 1)-functions (x, x(n)) bar right arrow (x(n) + 1)x(2n-2) + x(n)alpha x(2n-2), x is an element of F2n-1, x(n) is an element of F-2, and alpha is an element of F2n-1 \F-2, where Tr-1(n-1) (alpha) = Tr-1(n-1) (1/alpha) = 1. These (n, n - 1)-functions are balanced whatever the parity of n is and are then better suited for use as S-boxes in a Feistel cipher. We also give some properties of the Walsh spectrum of these functions to prove that they are CCZ-inequivalent to the differentially 4-uniform (n, n - 1)-functions of the form L circle F, where F is a known APN (n, n)-function and L is an affine surjective (n, n - 1)-function. Finally, we also give two new constructions of differentially 8-uniform (n, n - 2)-functions from EA-equivalent Cubic functions and from EA-equivalent Inverse functions.

关键词： Block ciphers S-boxes Vectorial Boolean functions APN functions differentially 4-uniform functions

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