The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$

In this paper, we give lower and upper bounds on the differential uniformity of substitutions over the field $\mathbb{F}_{2^{n}}$ with restrictions to cosets of $H$ in $\mathbb{F}^{\times}_{2^{n}}$, $H<\mathbb{F}^{\times}_{2^{n}}$, $|H|=l$, $l\cdot r=2^{n}-1$, being the maps $x\mapsto c_{i}x$, $c_{i}\in\mathbb{F}^{\times}_{2^{n}}$, $i=0,\dots,r-1$.