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
Abstract:
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.
Keywords:nonlinearitry, differential characteristic, linear characteristic, transitivity, primitivity.