Algebraic Shift Register Sequences

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.

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.

Mark Goresky is a member of the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.
Andrew M. Klapper received the A.B. degree in mathematics from New York University, New York, NY, in 1974, the M.S. degree in applied mathematics from SUNY, Binghamton, Binghamton, NY, in 1975, the M.S. degree in mathematics from Stanford University, Stanford, CA, in 1976 and the Ph.D. degree in mathematics from Brown University, Providence, RI, in 1982. His thesis, in the area of arithmetic geometry, concerned the existence of canonical subgroups in formal group laws. From 1981 to 1984 he was a postdoc in the Department of Mathematics and Computer Science at Clark University. From 1984 to 1991 he was an Assistant Professor in the College of Computer Science at Northeastern University. From 1991 to 1993 he was an Assistant Professor in the Computer Science Department at the University of Manitoba. Currently he is a Professor in the Department of Computer Science at the University of Kentucky. He was awarded a University Research Professorship for 2002–2003. His past research has included work on algebraic geometry over p-adic integer rings, computational geometry, modeling distributed systems, structural complexity theory and cryptography. His current interests include statistical properties of pseudo-random sequences based on abstract algebra, with applications in cryptography and CDMA and covering properties of codes. Dr Klapper is a member of the Information Theory Society and is a Senior Member of the IEEE. He was the general chair of Crypto '98 and of SETA 2008. He was the Associate Editor for Sequences for the IEEE Transactions on Information Theory from 1999 to 2002. He is currently an Associate Editor of the journals Applied Mathematics of Communications and Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences.