Adapted spectral-differential method for construction of differentially 4-uniform piecewise-linear permutations, orthomorphisms, and involutions of the field $\mathbb{F}_{2^{n}}$

We propose a method to construct permutations of the field $\mathbb{F}_{2^{n}}$ with low value of the $\Delta$-uniformity such that their restrictions to the cosets of the group $\mathbb{F}^{\times}_{2^{n}}$ by some of its subgroup $H$ are linear. Using the proposed method, we construct a large number of new CCZ-nonequivalent differentially 4-uniform permutations, orthomorphisms, and involutions over the field $\mathbb{F}_{2^{n}}$ with $n=6,8$.