Besides being interesting fundamental mathematical objects in their own right, linearly recurrent sequences have proved to be useful in many applications, including pseudo-random number generation, error correcting codes, private key cryptosystems, radar ranging, code division multiple access communications, and many other areas. They provide a fast and simple method of generating statistically random sequences. Moreover, many of their properties can be analyzed using various algebraic structures. The primary algebraic tools used to analyze linearly recurrent sequences are polynomials, power series, and trace functions on finite fields. The results in this section are all classical, many of them having been known for over 100 years. However we have organized this section in a slightly unusual way (from the modern perspective) in order to better illustrate how they are parallel to the FCSR and AFSR theory which will be described in later chapters.

There are many ways to describe the output sequence of an LFSR, each of which has its merits. In this chapter we discuss the matrix presentation (Section 3.2), the generating function presentation (Theorem 3.5.1), the algebraic presentation (Proposition 3.7.1), the trace representation (Theorem 3.7.4), and the sums of powers representation (Theorem 3.7.8).

Definitions

In this section we give the definitions and describe the basic properties of linear feedback shift registers and linearly recurrent sequences. Throughout this chapter we assume that R is a commutative ring (with identity denoted by 1).