Mark Goresky

On the Distinctness of Decimations of ℓ-Sequences

Maximal length Feedback with Carry Shift Register sequences (or l- sequences) have several remarkable statistical properties. Among them is the property that the arithmetic correlations between any two cyclically distinct decimations are precisely zero. It is open, however, whether all such pairs of decimations are indeed cyclically distinct. In this paper we show that a d-fold decimation of an l-sequence with prime connection number q is cyclically distinct from the original l-sequence if (1) d = -1, (2) d = (q + 1)/2, for q ≡ 1 mod 8 or (3) d < q/(c log(q)4) for a certain constant c.

Algebraic Shift Register 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.

Algebraic Shift Register Sequences: PSEUDO-RANDOM AND PSEUDO-NOISE SEQUENCES

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.

Algebraic Shift Register Sequences: List of figures

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.

Algebraic Shift Register Sequences: ALGEBRAIC BACKGROUND

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.

Algebraic Shift Register Sequences: ALGEBRAICALLY DEFINED SEQUENCES

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.

Algebraic Shift Register Sequences: Index

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.

Algebraic Shift Register Sequences: Acknowledgements

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.

Algebraic Shift Register Sequences: Frontmatter

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.

Feedback with carry shift registers: 2-adic models and summation combiners

Pseudorandom sequences with a variety of statistical properties are important in many-areas of communications and computing. Sequences such as m-sequences, nonlinear feedback shift register sequences, and summation combiner sequences, have been successfully studied largely because there is an algebraic framework based on finite fields for analysing them. We describe a new type of feedback register, feedback with carry shift registers (FCSRs). These easily implemented devices relate summation combiner sequences, arithmetic codes and 1/q sequences. We describe an algebraic framework, based on algebra over the 2-adic numbers, in which FCSR sequences can be analyzed. This gives a method for cracking the summation combiner which has been suggested for generating cryptographically secure binary sequences.<<ETX>>