Estimating Nonlinearity Characteristics for Iterative Transformations of a Vector Space

Abstract

We present theoretical foundations for the matrix-graphic approach (MGA) to the estimation of characteristics of the sets of essential and nonlinear variables of the composition of transformations of an $n$-dimensional vector space over a field. The ternary nonlinearity matrix corresponds to a transformation, where the $i$th row and the $j$th column of the matrix contain $0$, $1$, or $2$ if and only if the $j$th coordinate function of the transformation depends on the $i$th variable fictitiously, or linearly, or nonlinearly, $0\leq i,j < n$. MGA is based on the inequality according to which the nonlinearity matrix of the product of transformations is at most (the inequality is elementwise) the product of the nonlinearity matrices of the transformations. We define the multiplication for ternary matrices. The properties are studied of the multiplicative monoid of all ternary matrices of order $n$ without zero rows and columns and of the monoid $\mathbf{\Gamma}_n$ bijectively corresponding to it of all $n$-vertex digraphs with edges labeled with $0$, $1$, and $2$, where each vertex has nonzero indegree and outdegree. The iteration depth (number of multipliers) for transformations is estimated with the use of MGA in which the four types of the nonlinearity of transformations can be achieved, where each or some of the coordinate functions of the product of transformations can depend nonlinearly on all or at least some variables. We present the results of research on the nonlinearity of iterations of round substitution of the block ciphers DES and “Magma.”