Analysis of Periodic Feedback Shift Registers
This work provides a theoretical framework for analyzing periodic orbits in a class of nonlinear feedback shift registers, which is relevant for researchers in dynamical systems and finite field theory.
This paper develops methods for analyzing periodic orbits of linear feedback shift registers with periodic coefficients, enabling feasible computation of orbit lengths for a class of nonlinear FSRs. It advances the theory of Periodic Finite State systems by developing a finite-field Floquet theory and analyzing trajectory structure through shift-invariant linear systems.
This paper develops methods for analyzing periodic orbits of states of linear feedback shift registers with periodic coefficients and estimating their lengths. These shift registers are among the simplest nonlinear feedback shift registers (FSRs) whose orbit lengths can be determined by feasible computation. In general such a problem for nonlinear FSRs involves infeasible computation. The dynamical systems whose model includes such FSRs are termed as Periodic Finite State systems (PFSS). This paper advances theory of such dynamical systems. Due to the finite field valued coefficients, the theory of such systems turns out to be radically different from that of linear continuous or discrete time periodic systems with real coefficients well known in literature. A special finite field version of the Floquet theory of such periodic systems is developed and the structure of trajectories of the PFSS is analyzed through that of a shift invariant linear system after Floquet transformation. The concept of extension of a dynamical system is proposed for such systems whenever the equivalent shift invariant system can be obtained over an extension field.