This is the most general, but also the most abstract chapter on sequence generators in the book. In this chapter we describe the theory of algebraic feedback shift registers (AFSRs). This is a “shift register” setting for the generation of pseudo-random sequences, which includes as special cases the theory of linear feedback shift register (LFSR) sequences, feedback with carry shift register (FCSR) sequences, function field sequences, and many others. The general framework presented here first appeared in 1999 [120] (following an intermediate generalization of FCSRs [105]). It “explains” the similarities between these different kinds of sequences. Many of the interesting properties of these sequences simply reflect general properties of any AFSR sequence. However, because this chapter is fairly abstract, the rest of the book has been written so as to be largely independent of this chapter, even at the expense of sometimes having to repeat, in a special case, a theorem or proof which is fully described in this chapter.

Definitions

The ingredients used to define an algebraic shift register are the following.

1. An integral domain R (see Definition A.2.1).

2. An element π ∈ R.

3. A complete set of representatives S ⊂ R for the quotient ring R/(π). (This means that the composition S → R → R/(π) is a one to one correspondence. See Section 5.2.)