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
- 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
- PART IV ALGEBRAIC BACKGROUND
- Bibliography
- Index
10 - m-sequences
from PART II - PSEUDO-RANDOM AND PSEUDO-NOISE SEQUENCES
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
- 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
- PART IV ALGEBRAIC BACKGROUND
- Bibliography
- Index
Summary
Basic properties of m-sequences
Let R be a finite commutative ring and let a be a periodic sequence of elements in R. Let T be the period of a.
Definition 10.1.1 The sequence a is an m-sequence (over the ring R) of rank r (or degree r or span r) if it can be generated by a linear feedback shift register with r cells, and if every nonzero block of length r occurs exactly once in each period of a.
In other words, the sequence a is the output sequence of an LFSR that cycles through all possible nonzero states before it repeats. The second condition in the definition also says that a is a punctured de Bruijn sequence. See Section 8.2.4. In this section we recall standard results about m-sequences which have been known since the early 1900s [41] and which may be found in Golomb's book in the binary case [61, 62].
Proposition 10.1.2Suppose a is an m-sequence of rank r over a (finite commutative) ring R, generated by an LFSR with connection polynomial q(x) ∈ R[x] of degree r. Then the following hold.
1. The ring R is a field and the connection polynomial q(x) is a primitive polynomial.
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- Algebraic Shift Register Sequences , pp. 208 - 229Publisher: Cambridge University PressPrint publication year: 2012