On maximum length nonlinear feedback shift registers using a Boolean network approach

Nonlinear feedback shift registers (NFSRs) is very popular in many applications such as cryptography and communications. The NFSRs, especially, the maximum length NFSRs, have been of interest over the past decades. The sequences generated by the maximum length NFSRs is more cryptographically secure than the other sequences. However, the theory of NFSRs is not well-understood due to its complexity and lack of efficient algebraic tools to deal with the involved nonlinear problems. This paper continues to address this research using a Boolean network approach, which is a theoretically useful tool to deal with the nonlinear problems induced by NFSRs. A Boolean network is an autonomous system that evolves as an automaton through Boolean functions. Viewing an NFSR as a Boolean network, we first give its Boolean network representation in a linear system, which is characterized with a transition matrix. Based on the representation, some sufficient and necessary conditions are then given for the maximum length NFSRs.