Studying the isomorphism of NFSRs via a general framework of bijections

Nonlinear feedback shift registers (NFSRs) are used in many recent stream ciphers as their main building blocks. Two NFSRs are said to be isomorphic if their state diagrams are isomorphic, and to be equivalent if their sets of output sequences are equal. So far, numerous work has been done on the equivalence of NFSRs with same bit number, but much less has been done on their isomorphism. Actually, the equivalence problem of NFSRs with same bit number can be transformed to their isomorphism problem. The latter can be solved if the bijection between their states and its inverse can be explicitly expressed, which are quite hard to get in general. This paper studies the isomorphism of NFSRs by building a general framework for bijections. It first gives basic bijections. It then presents a unified formula for bijections, and discloses that any bijection can be expressed as a composite of finite basic bijections, setting up a general framework for bijections. Based on the general framework, the paper discloses in theory how to obtain all Galois NFSRs that are isomorphic to a given NFSR, and then reveals the bijections between the states of the previous types of Galois NFSRs and their own equivalent Fibonacci NFSRs. Finally, it proposes a new type of Galois NFSRs that are isomorphic and further equivalent to Fibonacci NFSRs, covering and improving most previous types of Galois NFSRs known to be equivalent to Fibonacci NFSRs.