On Galois NFSRs with Terminal Bits

Abstract

Nonlinear feedback shift registers (NFSRs) are generally classified as Fibonacci NFSRs and Galois NFSRs according to their implementation configurations. Some Galois NFSRs can be equivalent to Fibonacci ones in the sense that their sets of output sequences are equal. Finding the characterization of those equivalent NFSRs is helpful to the design of NFSR-based stream ciphers. Moreover, one of their design's security criteria is to assure their used NFSRs are nonsingular. This paper considers the Galois NFSRs with terminal bits, which have the first several bits involved only shifts and have been used in many stream ciphers such as Grain and Trivium. The paper first gives a special class of such Galois NFSRs and reveals its relation with Fibonacci ones with respect to their sets of output sequences. It then presents a necessary and sufficient condition for an n- stage Galois NFSR with terminal bit equivalent to an n-stage Fibonacci NFSR. Based on this condition, the paper enumerates those n-stage Galois NFSRs with the same terminal bit that are equivalent to a given n-stage Fibonacci NFSR. Finally, the paper gives a necessary/sufficient condition for the nonsingularity of Galois NFSRs with terminal bits.