Analysis of Periodic Feedback Shift Registers

TL;DR

This paper introduces new methods for analyzing the behavior and orbit lengths of linear feedback shift registers with periodic coefficients, advancing the theory of periodic finite state systems over finite fields.

Contribution

It develops a finite field Floquet theory and analyzes the structure of periodic feedback shift registers, which are simpler nonlinear systems with computable orbit lengths.

Findings

Finite field Floquet theory for PFSS is established.

Orbit lengths of certain shift registers can be feasibly computed.

Structure of trajectories analyzed through shift invariant systems.

Abstract

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.