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New explicit constructions of differentially 4-uniform permutations via special partitions of $$\mathbb{F}_{2^{2 k}}$$. (English) Zbl 1408.94957
Summary: In this paper, we further study the switching constructions of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2 k}}$$ from the inverse function and propose several new explicit constructions. In our constructions, we first partition the finite field $$\mathbb{F}_{2^{2 k}}$$ into some minimal subsets that are closed under both mappings $$x \mapsto \frac{1}{x^{- 1} + 1}$$ and $$x \mapsto \omega x$$, where $$\omega \in \mathbb{F}_{2^{2 k}}^\ast$$ is of order 3. Then, by utilizing some properties of such subsets to extend differentially 4-uniform permutations over the subfield $$\mathbb{F}_4$$ or $$\mathbb{F}_{2^4}$$ to that over $$\mathbb{F}_{2^{2 k}}$$, we give new constructions of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2 k}}$$ for the cases $$k$$ odd, $$k / 2$$ odd and $$k / 2$$ even respectively. As compared to previous constructions, our new constructions explicitly give large numbers (at least $$2^{2^{2 k - 2} - 1}$$) of functions.

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 12E20 Finite fields (field-theoretic aspects)
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