Abstract algebra and number theory provide the mathematical basis for many of the constructions used in modern communications. Finite fields play an especially important role, particularly in the design of sequence generators with various critical properties. In this appendix we describe the basic algebraic structures that are involved in these constructions, generally without proofs. There are many fine textbooks available on abstract algebra, both in general and about specific aspects [4, 45, 77, 90, 95, 96, 97, 98, 124, 131, 135, 156, 187].

Group theory

Basic properties

A group is a set G with an associative binary operation ⋆ (meaning that (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c) for all a, b, c ∈ G), an identity element e ∈ G (meaning that e ⋆ a = a ⋆ e = a for all a ∈ G), and inverses (meaning that for any a ∈ G there exists b ∈ G such that a ⋆ b = e). From these axioms it follows that the identity e is unique, that the inverse, b = a-1 is uniquely determined by a, and that b ⋆ a = e as well. The group G is commutative or Abelian if a ⋆ b = b ⋆ a for all a, b ∈ G. It is common to use multiplicative notation, writing ab for a ⋆ b and a-1 for the inverse of a ∈ G.