## 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

Zbl 1366.94481

### Software:

SPECK; SIMECK; PICARO; SIMON
Full Text:

### References:

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