The nonlinearity index for a piecewise-linear substitution of the additive group of the field $\mathbb F_{2^n}$

In this paper, we give a lower bound on the nonlinearity of permutations on a field $\mathbb F_{2^n}$ with restrictions to cosets of $H$ in $\mathbb F_{2^n}^*$, $H<\mathbb F_{2^n}^*$, $|H|=l$, $l\cdot r=2^n-1$, being the maps $x\mapsto A_jx$, $A_j\in\mathbb F_{2^n}^*$, $j=0,\dots,r-1$. Nonlinearity spectra of this permutations are found in the cases $r=3,5$.