〈2〉-exponents of shift register transformations nonlinearity dipgraphs

The matrix-graph approach is used to estimate the set of essential and non-linear variables of coordinate functions of the product of transformations of vector spaces. For essential variables, estimates are obtained by multiplying binary mixing matrices (or digraphs) of multiplied transformations, for non-linear variables - by multiplying ternary non-linearity matrices of multiplied transformations or their corresponding non-linearity digraphs, the arcs of which are labeled by the numbers of the set {0,1, 2}. For degrees of a given transformation, the area of non-trivial estimates is limited: for a set of essential variables, by the exponential of the mixing matrix (digraph); for a set of nonlinear variables, the 〈2〉-exponent of the matrix (digraph) of nonlinearity. For the class of transformations of binary shift registers, an attainable estimate of 〈2〉-exponents is obtained, expressed in terms of the length of the shift register and the set of numbers of essential and nonlinear variables of the feedback function. For register transformations whose non-linearity digraph has a loop, an exact formula for the 〈2〉-exponent is obtained. The results can be used to evaluate the nonlinearity characteristics of cryptographic functions built on the basis of iterations of register transformations.