On mixing and nonlinear properties of modified additive generators

The paper presents the research results related to mixing and nonlinear properties of modified additive generators (MAG) based on shift registers of length 8 over the set of $32$-dimensional binary vectors. The local characteristics are obtained for three kinds of feedback taps and two variants of modifying transformation. These characteristics are as follows: a) the local $(0,256)$-exponent of the mixing matrix $M$ that means the smallest natural number $\gamma_0$ such that the columns in the matrix $M^ t$ with numbers $0,1,\dots,31$ are positive for $t\ge\gamma_0$; b) the index of 0-perfectness that means the smallest number of rounds after that each output coordinate function in the 0th block essentially depends on all the bits of the initial state of MAG; c) the index of $0$-strong nonlinearity that means the smallest number of rounds after that each coordinate function in the 0th block is nonlinear. All obtained values (from 8 to 29) are summarized and presented in the table. These results can be used for the construction of cryptographic algorithms based on MAG, in particular, the key schedule algorithms for iterative block ciphers, which provide a complex nonlinear dependence of bits in the encryption and round keys on each other.