Adjacencies Between the Cycles of a Shift Register with Characteristic Polynomial (1 + x)<sup>n</sup>

Hemmati, F.; Schilling, D. L.; Eichmann, George

doi:10.1109/tc.1984.5009342

Cited by 7 publications

(3 citation statements)

References 5 publications

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“…1. The number of conjugate pairs between any two cycles in [17] is shown to be ≤ 2 but without any method to determine them. A complete adjacency graph is not provided.…”

Section: Discussionmentioning

confidence: 99%

“…In the literature, studies on the case of characteristic polynomials with repeated roots have been quite limited, e.g., q(x) = (x + 1) n in [17] and q(x) = (x + 1) a p(x) with p(x) having no repeated roots or is primitive done, respectively, in [22,25]. Prior studies looked into cases with (x + 1) b | q(x) for b ∈ N because their cycle structures and adjacency graphs had been well-established.…”

Section: The Case Of Repeated Rootsmentioning

confidence: 99%

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On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Chang

Ezerman

Ling

et al. 2018

Des. Codes Cryptogr.

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We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial f (x). We study in detail the cycle structure of the set Ω ( f (x)) that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order n ≈ 20. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.Keywords Binary periodic sequence · LFSR · de Bruijn sequence · cycle structure · adjacency graph · cyclotomic number Mathematics Subject Classification (2010) 11B50 · 94A55 · 94A60 1 Introduction A binary de Bruijn sequence of order n has period N = 2 n in which each n-tuple occurs exactly once in each period. There are 2 2 n−1 −n of them [5]. Some of their earliest applications are in communication systems. They are generated in a deterministic way, yet satisfy the randomness criteria in [14, Ch. 5] and are balanced, containing the same number of 1s and 0s. In cryptography, they have been used as a source of pseudo-random numbers and in keysequence generators of stream ciphers [29, Sect. 6.3]. In computational molecular biology, one of the three assembly paradigms in DNA sequencing is the de Bruijn graph assemblers model [31, Box 2]. Some roles of de Bruijn sequences in robust positioning patterns are discussed by Bruckstein et al. in [4]. They have numerous applications, e.g., in robotics,

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“…1. The number of conjugate pairs between any two cycles in [17] is shown to be ≤ 2 but without any method to determine them. A complete adjacency graph is not provided.…”

Section: Discussionmentioning

confidence: 99%

Section: The Case Of Repeated Rootsmentioning

confidence: 99%

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Chang

Ezerman

Ling

et al. 2018

Des. Codes Cryptogr.

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“…Nonsingular LFSR can be used to generate de Bruijn cycles by the cycle-joining method [1]. The applications of this method requires the full knowledge of the cycle structures and adjacency graphs of the original LFSR [14,15,16,17,18]. However, the cycle-joining method does not work to construct de Bruijn cycles from singular LFSR since the state diagrams don't only contains disjoint cycles.…”

Section: Introductionmentioning

confidence: 99%

State Diagrams of a Class of Singular LFSR and Their Applications to the Construction of de Bruijn Cycles

Wang,

Sun,

Zhang

2018

Preprint

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Universal circuit matrix for adjacency graphs of feedback functions

Żurawiecki¹

1994

Discrete Mathematics

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Adjacencies Between the Cycles of a Shift Register with Characteristic Polynomial (1 + x)ⁿ

Cited by 7 publications

References 5 publications

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

State Diagrams of a Class of Singular LFSR and Their Applications to the Construction of de Bruijn Cycles

Universal circuit matrix for adjacency graphs of feedback functions

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Adjacencies Between the Cycles of a Shift Register with Characteristic Polynomial (1 + x)n

Cited by 7 publications

References 5 publications

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

State Diagrams of a Class of Singular LFSR and Their Applications to the Construction of de Bruijn Cycles

Universal circuit matrix for adjacency graphs of feedback functions

Contact Info

Product

Resources

About

Adjacencies Between the Cycles of a Shift Register with Characteristic Polynomial (1 + x)ⁿ