Generalized cycle joining method and its application to the construction of long-period Galois NFSRs

Nonlinear feedback shift registers (NFSRs) are used in many recent stream ciphers as their main building blocks. One security criterion for the design of a stream cipher is to assure its used NFSR has a long period. As the period of a Fibonacci NFSR is equal to its largest cycle length, a common way to get a maximum-period Fibonacci NFSR is to join the cycles of an original Fibonacci NFSR into a maximum cycle. Nevertheless, so far only the maximum-period Fibonacci NFSRs with stage numbers no greater than 33 have been found. Considering that Galois NFSRs may have higher implementation efficiency than Fibonacci NFSRs, this paper first generalizes the cycle joining method for Fibonacci NFSRs to Galois NFSRs and establishes some conditions for maximum-period Galois NFSRs. It then reveals the cycle structure of some cascade connections of two Fibonacci NFSRs. Based on both, the paper constructs some long-period Galois NFSRs including maximum-period Galois NFSRs with stage numbers up to 41. Finally, it analyzes their hardware implementation via the technology mapping obtained by synthesizing the NFSRs with Synopsys Design Compiler L\(-\)2016.03-Sp1 using the TSMC 90nm CMOS library, and the results show that they have good hardware performance.