Adapted spectral-differential method for construction of differentially 4-uniform piecewise-linear permutations, orthomorphisms, and involutions of the field F2n$\mathbb{F}_{2^{n}}$

Abstract We propose a method to construct 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> $\mathbb{F}_{2^{n}}$ with low value of the Δ -uniformity such that their restrictions to the 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 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 <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> $\mathbb{F}_{2^{n}}$ with n = 6, 8.