The Berlekamp-Massey Algorithm revisited

@article{Atti2005TheBA,
  title={The Berlekamp-Massey Algorithm revisited},
  author={Nadia Ben Atti and Gema Mar{\'i}a D{\'i}az-Toca and Henri Lombardi},
  journal={Applicable Algebra in Engineering, Communication and Computing},
  year={2005},
  volume={17},
  pages={75-82}
}
We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation. 

Simple closed form Hankel transforms based on the central coefficients of certain Pascal-like triangles

We study the Hankel transforms of sequences related to the central coefficients of a family of Pascal-like triangles. The mechanism of Riordan arrays is used to elucidate the structure of these

A Fast Parallel Sparse Polynomial GCD Algorithm

We present a parallel GCD algorithm for sparse multivariate polynomials with integer coefficients. The algorithm combines a Kronecker substitution with a Ben-Or/Tiwari sparse interpolation modulo a

ON LINEAR COMPLEXITY OF GENERALIZED SHRINKING-MULTIPLEXING GENERATOR

It is shown that the linear complexity of the GSMG based on LFSRs is greater than the linear complex of the Shrinking Generator.

On the Resistance of Boolean Functions Against Algebraic Attacks Using Univariate Polynomial Representation

  • P. Rizomiliotis
  • Computer Science, Mathematics
    IEEE Transactions on Information Theory
  • 2010
A framework to assess the resistance of Boolean functions against the new algebraic attacks, including fast algebraic attack, is provided and a new infinite family of balanced Boolean functions described by their univariate polynomial representation is introduced.

Testing, Selection, and Implementation of Random Number Generators

Abstract : An exhaustive evaluation of state-of-the-art random number generators with several well-known suites of tests provides the basis for selection of suitable random number generators for use

A New Class of Rateless Codes Based on Reed–Solomon Codes

This paper proposes a new class of erasure codes based on Reed- Solomon codes that unlike other Reed-Solomon-based erasurecodes are rateless and also, unlike other rateless codes, guarantee zero overhead even for small k, and has a reasonable computational complexity when k is not too large.

Constructing de Bruijn sequences by concatenating cycles of feedback shift registers

This thesis introduces sufficient conditions for when an ordering of universal cycles for disjoint sets can be concatenated to obtain a universal cycle for the union of those sets, and proves the validity of three new de Bruijn sequences.

Enhancing Networking Cipher Algorithms with Natural Language

This paper summarizes how languages can be integrated into symmetric encryption as a way to assist in the encryption of vulnerable streams that may be found under attack due to the natural frequency distribution of letters or words in a local language stream.

References

SHOWING 1-10 OF 11 REFERENCES

Continued fractions and Berlekamp's algorithm

The algorithm for developing the rational approximations is based on continued fraction techniques and is virtually equivalent to an algorithm employed by Berlekamp for decoding BCH codes.

On the equivalence between Berlekamp's and Euclid's algorithms

  • J. Dornstetter
  • Computer Science, Mathematics
    IEEE Trans. Inf. Theory
  • 1987
It is shown that Berlekamp's iterative algorithm can be derived from a normalized version of Euclid's extended algorithm. Simple proofs of the results given recently by Cheng are also presented.

A simple Hankel interpretation of the Berlekamp-Massey algorithm

On the continued fraction and Berlekamp's algorithm

  • U. Cheng
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1984
The sequence D(k), k \geq 0 , in Berlekamp's algorithm provides the information about when Berlecamp's algorithm completes one iterative step of the continued fraction, and this happens when D(K) ; and when D (k) \neq D( k + 1) , it implies that Berle Kampan's algorithm begins the next iterative steps of the started fraction.

Continued fractions and linear recurrences

Let to, t1, t2, be a sequence of elements of a field F. We give a continued fraction algorithm for tox + tlx2 + t2x3 + * . . If our sequence satisfies a linear recurrence, then the continued fraction

A Method for Solving Key Equation for Decoding Goppa Codes

New Techniques for the Computation of Linear Recurrence Coefficients

  • V. Pan
  • Computer Science, Mathematics
  • 2000
Algorithms are shown that simplify the solution and lead to further improvements of the latter bound, by factors ranging from order of log n, for c=0 and cn (in which case the overhead constant drops dramatically), to order of min (c, log n), for 2?c?n; the algorithms use Las Vegas type randomization in the case of 2.

Shift-register synthesis and BCH decoding

  • J. Massey
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1969
It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback

A computational introduction to number theory and algebra

This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications.

Algebraic coding theory

  • E. Berlekamp
  • Computer Science
    McGraw-Hill series in systems science
  • 1968
This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering