Nonsingularity of Grain-like cascade FSRs via semi-tensor product

In this paper, Grain-like cascade feedback shift registers (FSRs) are regarded as two Boolean networks (BNs), and the semi-tensor product (STP) of the matrices is used to convert the Grain-like cascade FSRs into an equivalent linear equation. Based on the STP, a novel method is proposed herein to investigate the nonsingularity of Grain-like cascade FSRs. First, we investigate the property of the state transition matrix of Grain-like cascade FSRs. We then propose their sufficient and necessary nonsingularity condition. Next, we regard the Grain-like cascade FSRs as Boolean control networks (BCNs) and further provide a sufficient condition of their nonsingularity. Finally, two examples are provided to illustrate the results obtained in this paper.

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