Nonsingularity of nonlinear feedback shift registers

In this paper, the multi-valued nonlinear feedback shift register is studied and a new approach is present to analyze its nonsingularity and number of cycles. First, using the semi-tensor product of matrices, the nonlinear feedback shift register is expressed in an algebraical form, based on which several necessary and sufficient conditions are given for the nonsingularity. Then, a new method is established to determine the number of cycles with different lengths for arbitrarily given feedback shift register. Finally, an illustrative example is studied to support our new results.