On the way of constructing differentially $2\delta$-uniform permutations over $\mathbb{F}_{2^{2m}}$

The paper studies new ways of constructing differentially $2\delta$-uniform bijections over $\mathbb{F}_{2^{2m}}$, $m \ge 3$, that are based on $TU$-construction. Some well known results on the constructing differentially $4$-uniform permutations over $\mathbb{F}_{2^{2m}}$ are generalized in this work. The core idea is to use $TU$-construction and differentially $\delta$-uniform bijections to construct $2^t \cdot \delta$-uniform permutations. A generalized method for constructing $2m$-bit differentially $4$-uniform permutations is proposed, and new constructions of differentialy $6$ and $8$-uniform permutations are introduced.