Pseudo-random sequences are essential ingredients of every modern digital communication system including cellular telephones, GPS, secure internet transactions and satellite imagery. Each application requires pseudo-random sequences with specific statistical properties. This book describes the design, mathematical analysis and implementation of pseudo-random sequences, particularly those generated by shift registers and related architectures such as feedback-with-carry shift registers. The earlier chapters may be used as a textbook in an advanced undergraduate mathematics course or a graduate electrical engineering course; the more advanced chapters provide a reference work for researchers in the field. Background material from algebra, beginning with elementary group theory, is provided in an appendix.
• Self-contained chapters make the book accessible to a wide range of students • Complete proofs are provided and exercises are included in chapters covering basic material • Provides a unified approach to a wide range of pseudo-random sequences
Contents
1. Introduction; Part I. Algebraically Defined Sequences: 2. Sequences; 3. Linear feedback shift registers and linear recurrences; 4. Feedback with carry shift registers and multiply with carry sequences; 5. Algebraic feedback shift registers; 6. d-FCSRs; 7. Galois mode, linear registers, and related circuits; Part II. Pseudo-Random and Pseudo-Noise Sequences: 8. Measures of pseudo-randomness; 9. Shift and add sequences; 10. M-sequences; 11. Related sequences and their correlations; 12. Maximal period function field sequences; 13. Maximal period FCSR sequences; 14. Maximal period d-FCSR sequences; Part III. Register Synthesis and Security Measures: 15. Register synthesis and LFSR synthesis; 16. FCSR synthesis; 17. AFSR synthesis; 18. Average and asymptotic behavior of security measures; Part IV. Algebraic Background: A. Abstract algebra; B. Fields; C. Finite local rings and Galois rings; D. Algebraic realizations of sequences; Bibliography; Index.
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