The equivalence transformation between Galois NFSRs and Fibonacci NFSRs

Abstract

The transformation between Galois type and Fibonacci type of nonlinear feedback shift registers (NFSRs) is investigated. First, the algebraic forms of Galois NFSRs and Fibonacci NFSRs are proposed respectively by viewing the NFSR as a Boolean network. Then, the observability of NFSRs is defined and the observability matrix is constructed, based on which, some necessary and sufficient conditions for the transformation between an n-stage Galois NFSR and a Fibonacci NFSR with the same stage or stage less than n are obtained. Finally, two examples are provided to depict the theoretical results.

Research Status of Nonlinear Feedback Shift Register Based on Semi-Tensor Product

“…2 . The literature [75,76] gave a series of necessary and sufficient conditions for equivalence from the perspective of observability matrix and output tuple, respectively.…”

“…In [76] the authors investigated the equivalence transformation between Galois NFSRs and Fibonacci NFSRs based on observability matrix based on STP. According to the definition of observability of sequence generators, the NFSR-based stream ciphers should avoid unobservable Galois NFSRs from the security viewpoint and select observable ones.…”

“…These results are more novel than previous conclusions and are very helpful for the design of practical stream cipher algorithms and decoding algorithms. According to the specific research object, the main achievements can be roughly divided into the modeling of NFSRs [65][66][67], the analysis of the structure of NFSRs [68][69][70][71][72][73][74][75][76][77][78][79][80][81], and the study of the properties of NFSRs [82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97]. This paper will give an overview of the latest achievements in NFSR research based on STP from these three aspects.…”

Semi‐tensor product‐based algebra‐logic mixed representation and fault diagnosis for a class of gate networks

“…Besides, based on the Boolean function of gate networks, an algebraic representation, that is, a linear form of the product of a matrix and a vector, was proposed through the technology of semi-tensor product of matrices, which was an original theory first proposed by Cheng and Qi in 2009 [22]. Thereafter, extensive superior work has been done on the applications of logic systems based on their algebraic representations, such as state estimation [23,24], detectability, and observability [25][26][27] as well as control of Boolean networks [28][29][30][31][32], synchronization of Boolean networks [33,34], games [35][36][37], fault detection of digital circuits [38][39][40], and transformation of two feedback shift registers [41].…”