Improved Transformation Algorithms for Generalized Galois NLFSRs

The transformation algorithm for Nonlinear Feedback Shift Registers (NLFSRs) converts NLFSRs between Fibonacci and Galois configurations. Up to now, three types of Galois NLFSRs namely Type-I, Type-II and Type-III Galois NLFSRs have been discovered to be equivalent to Fibonacci NLFSRs in existing works. However, either no transformation algorithm has been proposed or the proposed algorithm has very high complexity for these Galois NLFSRs. More importantly, the common issue is that the output sequence is assumed to be generated by the first stage of the NLFSR. The sequences generated by other stages are not considered. In this paper, we develop a compensation method to address all these issues. Based on this unified method, we propose Fibonacci-to-Galois and Galois-to-Fibonacci transformation algorithms for the three types of Galois NLFSRs. Moreover, we discover a new type of Galois NLFSRs (Type-IV) that can be transformed to Fibonacci NLFSRs and propose transformation algorithms based on the same compensation method. No matter what the output function is, the output sequences are the same before and after being transformed by any of the proposed transformation algorithms.