On the difference characteristics of piecewise-linear permutations of the field $\mathbb{F}_{2^{n}}$

We obtain upper and lower estimates for the difference characteristics of permutations of the field $\mathbb{F}_{2^{n}}$ whose restrictions to cosets of the group $\mathbb{F}^{\times}_{2^{n}}$ by its subgroup $H$, $|H|=l$, $l\cdot r=2^{n}-1$, are mappings of the form $x\to c_{i}x$, $c_{i}\in\mathbb{F}^{\times}_{2^{n}}$, $i=0,\dots,r-1$.