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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. (English) Zbl 1382.11093

Summary: Quadratic differentially 4-uniform \((n, n - 1)\)-functions are given in [C. Carlet and Y. Alsalami, Adv. Math. Commun. 9, No. 4, 541–565 (2015; Zbl 1366.94481)] 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 [loc. cit.], 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 \((x,x_{n})\mapsto (x_{n}+1)x^{2^{n}-2}+x_{n} \alpha x^{2^{n}-2}\), \(x \in \mathbb{F}_{2^{n-1}}\), \(x_{n}\in \mathbb {F}_{2}\), and \(\alpha \in \mathbb {F}_{2^{n-1}}\setminus \mathbb{F}_{2}\), where \(\mathrm{Tr}_{1}^{n-1}(\alpha )=\mathrm{Tr}_{1}^{n-1}(\frac {1}{\alpha })=1\). 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 \(L\) \(F\), 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.

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
94D10 Boolean functions

Citations:

Zbl 1366.94481

Software:

SPECK; SIMECK; PICARO; SIMON
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References:

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