Book contents
- Frontmatter
- Contents
- List of figures
- List of tables
- Acknowledgements
- 1 Introduction
- PART I ALGEBRAICALLY DEFINED SEQUENCES
- PART II PSEUDO-RANDOM AND PSEUDO-NOISE SEQUENCES
- PART III REGISTER SYNTHESIS AND SECURITY MEASURES
- PART IV ALGEBRAIC BACKGROUND
- Appendix A Abstract algebra
- Appendix B Fields
- Appendix C Finite local rings and Galois rings
- Appendix D Algebraic realizations of sequences
- Bibliography
- Index
Appendix A - Abstract algebra
from PART IV - ALGEBRAIC BACKGROUND
Published online by Cambridge University Press: 05 February 2012
- Frontmatter
- Contents
- List of figures
- List of tables
- Acknowledgements
- 1 Introduction
- PART I ALGEBRAICALLY DEFINED SEQUENCES
- PART II PSEUDO-RANDOM AND PSEUDO-NOISE SEQUENCES
- PART III REGISTER SYNTHESIS AND SECURITY MEASURES
- PART IV ALGEBRAIC BACKGROUND
- Appendix A Abstract algebra
- Appendix B Fields
- Appendix C Finite local rings and Galois rings
- Appendix D Algebraic realizations of sequences
- Bibliography
- Index
Summary
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.
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- Algebraic Shift Register Sequences , pp. 407 - 431Publisher: Cambridge University PressPrint publication year: 2012