Statistical properties of binary nonautonomous shift registers with internal XOR

The statistical and algebraic properties of binary nonautonomous shift registers and shift registers with internal XOR are compared, during which the state vector is summed with its one-step shift. The isomorphism of transition graphs of these automata is proved. It is shown that, with a Bernoulli random input, the stationary distribution of the register states with internal XOR is uniform. The form of the probability function of these registers is obtained. It is shown that, under certain conditions on the output function, registers with internal XOR are not Cesaro-hereditary. The authors show input sequences that possess the property of stability of the relative frequencies of arbitrary multigrams, while output sequences do not have this property.