Mixing properties of $2$-cascade generators

The important properties of the dependence of gamma signs on all signs in the initial state of a gamma generator are called the mixing properties of the generator. It is known that if the mixing properties of a generator are good, then the transition graph of the generator is primitive or local primitive. In this paper, mixing properties are evaluated for the following $2$-cascade generators constructed of Linear Feedback Shift Register (LFSRs): generator based on shift register series, generator of $1$–$2$ steps, and generator of intermittent steps. Namely, for these generators, some necessary and sufficient conditions for local primitiveness or quasiprimitiveness are given and upper bounds for appropriate local exponents or quasiexponents depending on the parameters of LFSR are obtained. For many values of parameters, the bounds are close to the sum of lengths of LFSRs in the generator.