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Berlekamp–Massey Algorithm [chapter]

David Naccache, Friedrich L. Bauer, Ebru Celikel Cankaya, David Elliott Bell, Burt Kaliski, Anne Canteaut, CISSP Aaron Estes, Jonathan K. Millen, Berk Sunar, Bodo Möller, Burt Kaliski, Nary Subramanian (+38 others)
2011 Encyclopedia of Cryptography and Security  
doi:10.1007/978-1-4419-5906-5_335 fatcat:flnglbmjyzcqpbzx5zgtsahhwy

Generalised Berlekamp–Massey algorithm

M.-H. Cheng
2002 IEE Proceedings - Communications  
It further enables the presentation of a generalised Berlekamp Massey algorithm, including the conventional algorithm and the inversion-free algorithm as two special cases.  ...  The BerlekampMassey algorithm is revisited and proven again by using the matrix representation.  ...  Introduction The Berlekamp-Massey algorithm (BMA) [I. 21 is an efficient method for determining the error-locator polynomial in decoding the Reed-Solomon and BCH codes.  ... 
doi:10.1049/ip-com:20020413 fatcat:qkkluv2d3zhqdjztikkvk6dzdu

The Berlekamp-Massey Algorithm revisited

Nadia Ben Atti, Gema M. Diaz–Toca, Henri Lombardi
2006 Applicable Algebra in Engineering, Communication and Computing  
We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence.  ...  Next, we introduce our modified Berlekamp-Massey Algorithm in pseudocode (Algorithm 2): Algorithm 2 Modified Berlekamp-Massey Algorithm Input: n ∈ N.  ...  That proves that our modification of Berlekamp-Massey Algorithm is right.  ... 
doi:10.1007/s00200-005-0190-z fatcat:qzlaz2af3zamloghnncwjoca5a

On the matrix berlekamp-massey algorithm

Erich Kaltofen, George Yuhasz
2013 ACM Transactions on Algorithms  
We analyze the Matrix Berlekamp/Massey algorithm, which generalizes the Berlekamp/Massey algorithm [Massey 1969] for computing linear generators of scalar sequences.  ...  The Matrix Berlekamp/Massey algorithm computes a minimal matrix generator of a linearly generated matrix sequence and has been first introduced by Rissanen [1972a] , Dickinson et al. [1974], and Coppersmith  ...  Matrix Berlekamp/Massey algorithm will compute a minimal generator.  ... 
doi:10.1145/2500122 fatcat:w4xakdv63zglhkdaa5wp5kw3wu

The Berlekamp-Massey Algorithm via Minimal Polynomials [article]

Graham H. Norton
2010 arXiv   pre-print
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.  ...  1 Introduction The Berlekamp-Massey (BM) Algorithm The BM algorithm determines a linear recurrence of least order L ≥ 0 which generates a given (finite) sequence s of length n ≥ 1 over a field F, [13  ... 
arXiv:1001.1597v3 fatcat:xpsbkzry2zamhb5xo7z6afd3se

A fraction free Matrix Berlekamp/Massey algorithm

Erich Kaltofen, George Yuhasz
2013 Linear Algebra and its Applications  
We describe a fraction free version of the Matrix Berlekamp/Massey algorithm. The algorithm computes a minimal matrix generator of linearly generated square matrix sequences over an integral domain.  ...  For scalar sequences, the matrix algorithm specializes to a more efficient algorithm than the algorithm currently in the literature.  ...  Fraction Free Matrix Berlekamp/Massey Algorithm We begin by describing the fraction free Matrix Berlekamp/Massey algorithm.  ... 
doi:10.1016/j.laa.2013.06.016 fatcat:5j7qpgywwzhnleooa6ckcrdltm

New serial architecture for the Berlekamp-Massey algorithm

Hsie-Chia Chang, C.B. Shung
1999 IEEE Transactions on Communications  
We propose a new efficient serial architecture to implement the Berlekamp-Massey algorithm, which is frequently used in BCH and Reed-Solomon decoders.  ...  An inversionless Berlekamp-Massey algorithm is adopted which not only eliminates the finite-field inverter but also introduces additional parallelism.  ...  of the inversionless Berlekamp-Massey algorithm, which is shown in Fig. 2 .  ... 
doi:10.1109/26.764911 fatcat:jnuyylpphrdovn24mgzyhqhg2e

Extension of the Berlekamp-Massey algorithm to N dimensions

Shojiro Sakata
1990 Information and Computation  
INTRODUCTION In the previous paper (Sakata, 1988) , we gave an extension of the BerlekampMassey algorithm (Berlekamp, 1968; Massey, 1969) to two dimensions.  ...  we restrict W(u) to be the class of minimal polynomial sets F composed of "manic" polynomials f in "reduced normal form" in the sense that for any .fe F, Tfs {Deg(f) ) u d(F), where such reduction can BERLEKAMP-MASSEY  ... 
doi:10.1016/0890-5401(90)90039-k fatcat:gtwevola2vblbmnwjmcrnuhdji

A simple Hankel interpretation of the Berlekamp-Massey algorithm

Edmund Jonckheere, Chingwo Ma
1989 Linear Algebra and its Applications  
A simple interpretation of the Berlekamp-Massey algorithm in the light of the Hankel matrix is presented.  ...  Next, the Hankel system of equations that yields the updated connection polynomial is solved via the natural LU factorization of the Hankel matrix, which itself leads to the Berlekamp-Massey algorithm  ...  An earlier successful attempt to make the Berlekamp-Massey algorithm more understandable is due to Imamura and Yoshida [3] .  ... 
doi:10.1016/0024-3795(89)90032-3 fatcat:ds4bnn4ijrfi7kb7bwhglgz4qy


S. M. Ratseev, A. D. Lavrinenko, E. A. Stepanova
2021 Vestnik of Samara University Natural Science Series  
Algorithms for decoding the listed codes based on the Berlekamp Massey algorithm are given.  ...  An optimized Berlekamp Massey algorithm is also given for the case ofa field of characteristic 2.  ...  On the Berlekamp -Massey algorithm and its application...Пусть на приемном конце получен вектор:v = (1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1), в котором до двух ошибок (как видно, стирающих символов  ... 
doi:10.18287/2541-7525-2021-27-1-44-61 fatcat:isxj6654ircz3ga55arhxejxlu

The Berlekamp-Massey Algorithm and the Euclidean Algorithm: a Closer Link [article]

Maria Bras-Amorós, Michael E. O'Sullivan
2009 arXiv   pre-print
This results in a new, simpler, and compacter presentation of the Berlekamp-Massey algorithm.  ...  The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation.  ...  From the Euclidean to the Berlekamp-Massey algorithm In this section we will see that the previous algorithm and the Berlekamp-Massey algorithm are essentially the same.  ... 
arXiv:0908.2198v1 fatcat:e3x47pxazzcufif7z6dugixdke

Universal Reed-Solomon decoders based on the Berlekamp-Massey algorithm

Zhiyuan Yan, Dilip V. Sarwate
2004 Proceedins of the 14th ACM Great Lakes symposium on VLSI - GLSVLSI '04  
Universal field inversions and divisions are not discussed in this paper since the an inversion-less Berlekamp-Massey algorithm is used to solve the key equation.  ...  Furthermore, the scaling factors cancel out in Forney's formula, used to evaluate error values, and thus correct error values can be obtained from the universal Berlekamp-Massey algorithm.  ...  Sarwate and Shanbhag proposed a reformulated Berlekamp-Massey algorithm and designed high-speed RS decoder architectures based on the reformulated algorithm [10] .  ... 
doi:10.1145/988952.988965 dblp:conf/glvlsi/YanS04 fatcat:oojsaizm2jefzis5qc26dexkc4

Berlekamp—Massey Algorithm, Continued Fractions, Pade Approximations, and Orthogonal Polynomials

S. B. Gashkov, I. B. Gashkov
2006 Mathematical Notes  
The Berlekamp-Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Padé approximations to the Laurent series over an arbitrary field with singularity at infinity.  ...  THE BERLEKAMP-MASSEY ALGORITHM (BMA) Let us show that the BMA simultaneously calculates both the elements a n (z) of the continued fraction in the case of an arbitrary series f (z) and the sequence Q n  ...  Suppose that Λ i = c (i) L i (f ) x L i (f ) + · · · + c (i) 1 x + c (i) 0 , c (i) 0 = 1, i = 1, . . . , k ; BERLEKAMP-MASSEY ALGORITHM 45 then, by definition, s i=1 c (k) i f j−i = f j for j = s, . .  ... 
doi:10.1007/s11006-006-0004-z fatcat:jbjqtepeb5bfdni7chdhco34hi

Analysis of the Berlekamp-Massey Linear Feedback Shift-Register Synthesis Algorithm

F. G. Gustavson
1976 IBM Journal of Research and Development  
An analysis of the Berlekamp-Massey Linear Feedback Shift-Register (LFSR) Synthesis Algorithm is provided which shows that an input string of length 12 requires C ( i 2 ) multiplication/ addition operations  ...  Furthermore, we exhibit a connection between step 1 of the Ling-Palermo algorithm and the LFSR Algorithm, and the LFSR Algorithm turns out to be computationally superior.  ...  The LFSR algorithm, which is essentially the same as the Berlekamp Iterative Algorithm [ 31, was developed for encoding and is more general and computationally faster than the BOIC scheme.  ... 
doi:10.1147/rd.203.0204 fatcat:a4jas7k2srdh3hvkiqkpsquqha

Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm

Mark Giesbrecht, Erich Kaltofen, Wen-shin Lee
2002 Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02  
THE FRACTION-FREE BERLEKAMP/ MASSEY ALGORITHM The Berlekamp/Massey algorithm [18] processes a sequence of elements a0, a1, . . . from a field K.  ...  By computing the corresponding Berlekamp/Massey quantities of the subresultant GCD algorithm, we obtain our fraction-free Berlekamp/Massey algorithm, which outputs an integral multiple of the minimal generator  ... 
doi:10.1145/780506.780519 dblp:conf/issac/GiesbrechtKL02 fatcat:stbmtl4tcbewllqivxe5mfpuj4
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