On the difference characteristics of piecewise-linear permutations of the field F2n $\mathbb{F}_{2^{n}}$

Abstract We obtain upper and lower estimates for the difference characteristics of permutations of the field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> </m:msub> </m:math> $\mathbb{F}_{2^{n}}$ whose restrictions to cosets of the group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mo>×</m:mo> </m:mrow> </m:msubsup> </m:math> $\mathbb{F}^{\times}_{2^{n}}$ by its subgroup H , | H | = l , l · r = 2 n − 1, are mappings of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo stretchy="false">→</m:mo> <m:msub> <m:mi>c</m:mi> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:msub> <m:mi>c</m:mi> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mo>×</m:mo> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo>−</m:mo> <m:mn>1.</m:mn> </m:math> $x\to c_{i}x,\, c_{i}\in\mathbb{F}^{\times}_{2^{n}},\, i=0,\dots,r-1.$