Modified Berlekamp-Massey Algorithm for Approximating the k-Error Linear Complexity of Binary Sequences

Alecu, A.; Sălăgean, Ana

doi:10.1007/978-3-540-77272-9_14

Cited by 8 publications

(11 citation statements)

References 9 publications

(19 reference statements)

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“…As for the second equality, 1 ′ = 0 since e 0 = 1 > 0 and 1 ′ + 1 − L ′ 1 = 1 = deg(µ (1) ). Suppose inductively that n ≥ 2 and that (i) is true for 1 1) . We have to show that…”

Section: The Recursive Theoremmentioning

confidence: 98%

“…We prove (i) by induction on n. For n = 1, µ (1) = x − ∆ 1 • ε and max{e 0 , 0} + L 0 = 1 = deg(µ (1) ). As for the second equality, 1 ′ = 0 since e 0 = 1 > 0 and 1 ′ + 1 − L ′ 1 = 1 = deg(µ (1) ). Suppose inductively that n ≥ 2 and that (i) is true for 1 1) .…”

Section: The Recursive Theoremmentioning

confidence: 99%

“…When s is understood, we will write µ (j) for µ(s (j) ). A minimal polynomial of s (1) is clear by inspection, so we could use n = 1 as the basis of the recursion, but with slightly more work, we will see that we can use n = 0 as the basis.…”

Section: The Recursive Theoremmentioning

confidence: 99%

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The Berlekamp-Massey Algorithm via Minimal Polynomials

Norton

2010

Preprint

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We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain D. This theorem is relative to any element of D. The ingredients are: the arithmetic of Laurent polynomials over D, a recursive 'index function' and simple mathematical induction. Taking reciprocals gives a 'Berlekamp-Massey theorem' i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any element of D. The recursive theorem readily yields the iterative minimal polynomial algorithm due to the author and a transparent derivation of the iterative Berlekamp-Massey algorithm.We give an upper bound for the sum of the linear complexities of s which is tight if s has a perfect linear complexity profile. This implies that over a field, both iterative algorithms require at most 2⌊ n 2 4 ⌋ multiplications.

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“…As for the second equality, 1 ′ = 0 since e 0 = 1 > 0 and 1 ′ + 1 − L ′ 1 = 1 = deg(µ (1) ). Suppose inductively that n ≥ 2 and that (i) is true for 1 1) . We have to show that…”

Section: The Recursive Theoremmentioning

confidence: 98%

Section: The Recursive Theoremmentioning

confidence: 99%

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The Berlekamp-Massey Algorithm via Minimal Polynomials

Norton

2010

Preprint

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“…In the literature, this length of the LFSR is referred to as the linear complexity [22]. The Berlekamp-Massey algorithm is an efficient method of determining the linear complexity of a sequence [23]. The forward unpredictability can be confirmed by the linear complexity property.…”

Section: Linear Complexitymentioning

confidence: 99%

Well Balanced Multi-value Sequence and its Properties Over Odd Characteristic Field

Ali

Kodera²,

Rabbi

et al. 2019

Adv. sci. technol. eng. syst. j.

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The authors propose a well balanced multi-value sequence (including a binary sequence). All the sequence coefficients (except the zero) appear almost the same in number, thus, the proposed sequence is so called the well balanced sequence. This paper experimentally describes some prominent features regarding a sequence, for instance, its period, autocorrelation, and cross-correlation. The value of the autocorrelation and cross-correlation can be explicitly given by the authors formulated theorems. In addition, to ensure the usability of the proposed multi-value sequence, the authors introduce its flexibility by making it a binary sequence. Furthermore, this paper also introduces a comparison in terms of the linear complexity and distribution of bit patterns properties with their previous works. According to the comparison results, the proposed sequence holds better properties compared to our previous sequence.

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“…This approach is particularly appealing since there exists an efficient procedure (it is so called the Berlekamp-Massy algorithm [23]) for finding the shortest LFSR. This length is referred as the linear complexity associated with the sequence.…”

Section: Linear Complexitymentioning

confidence: 99%

Multi-Value Sequence Generated over Sub Extension Field and Its Properties

Ali¹,

Kodera²,

Kusaka³

et al. 2019

JIS

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Pseudo-random sequences with long period, low correlation, high linear complexity, and uniform distribution of bit patterns are widely used in the field of information security and cryptography. This paper proposes an approach for generating a pseudo-random multi-value sequence (including a binary sequence) by utilizing a primitive polynomial, trace function, and k-th power residue symbol over the sub extension field. All our previous sequences are defined over the prime field, whereas, proposed sequence in this paper is defined over the sub extension field. Thus, it's a new and innovative perception to consider the sub extension field during the sequence generation procedure. By considering the sub extension field, two notable outcomes are: proposed sequence holds higher linear complexity and more uniform distribution of bit patterns compared to our previous work which defined over the prime field. Additionally, other important properties of the proposed multi-value sequence such as period, autocorrelation, and cross-correlation are theoretically shown along with some experimental results.

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Modified Berlekamp-Massey Algorithm for Approximating the k-Error Linear Complexity of Binary Sequences

Cited by 8 publications

References 9 publications

The Berlekamp-Massey Algorithm via Minimal Polynomials

The Berlekamp-Massey Algorithm via Minimal Polynomials

Well Balanced Multi-value Sequence and its Properties Over Odd Characteristic Field

Multi-Value Sequence Generated over Sub Extension Field and Its Properties

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