Combinatorial properties of differentially $2$-uniform substitutions

A combinatorial approach to the investigation and methods of construction of differentially $2$-uniform substitutions of the vector space over the finite field $F_2$ is proposed. Necessary and sufficient conditions for the family of sets associated with a differentially $2$-uniform substitution to be a symmetric block design are given. It is shown that a substitution is differentially $2$-uniform if and only if it is a solution of a similarity equations system connecting a family of translations with a family of unequal weights involutions. We suggest methods of construction of differentially $2$-uniform substitutions by means of the Cayley table of an additive group of finite field $F_{2^m}$.