On a relationship between linear and differential characteristics of binary vector spaces mappings and diffusion characteristics over blocks of imprimitivity systems of translation group of the binary vector space

We examine relationships between the nonlinearity parameters of mappings $f\colon V_{n} \to V_{m} $ of binary vector spaces $V_{n} =\mathrm{GF}(2)^n $, ${V_{m} =\mathrm{GF}(2)^{m} }$, diffusion properties of imprimitivity systems of the translation group $V_{n}^{+} $ of space $V_{n} $, and also (for $m=n$ and $f\in S(V_{n} )$) transitivity and primitivity properties of the groups $\langle W^{+} ,f\rangle $, where $W^{+} $ is the translation group of the subspace $W<V_{n} $. It is shown that, in some methods of cryptoanalysis of block cipher algorithms, in fact, insufficient diffusion of blocks of the imprimitivity system of the group $V_{n}^{+} $ is used.