The Berlekamp–Massey algorithm is an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
that will find the shortest
linear-feedback shift register
In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state.
The most commonly used linear function of single bits is exclusive-or (XOR). Thus, an LFSR is most often a ...
(LFSR) for a given binary output sequence. The algorithm will also find the
minimal polynomial of a linearly
recurrent sequence in an arbitrary
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse. Reeds and Sloane offer an extension to handle a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
Elwyn Berlekamp
Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10. ...
invented an algorithm for decoding
Bose–Chaudhuri–Hocquenghem (BCH) codes.
James Massey
James Lee Massey (February 11, 1934 – June 16, 2013) was an American information theorist and
cryptographer, Professor Emeritus of Digital Technology at ETH Zurich. His notable work includes the application of the Berlekamp–Massey algorithm ...
recognized its application to linear feedback shift registers and simplified the algorithm. Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm),
but it is now known as the Berlekamp–Massey algorithm.
Description of algorithm
The Berlekamp–Massey algorithm is an alternative to the
Reed–Solomon Peterson decoder for solving the set of linear equations. It can be summarized as finding the coefficients Λ
''j'' of a polynomial Λ(''x'') so that for all positions ''i'' in an input stream ''S'':
:
In the code examples below, ''C''(''x'') is a potential instance of ''Λ''(''x''). The error locator polynomial ''C''(''x'') for ''L'' errors is defined as:
:
or reversed:
:
The goal of the algorithm is to determine the minimal degree ''L'' and ''C''(''x'') which results in all
syndromes
:
being equal to 0:
:
Algorithm:
''C''(''x'') is initialized to 1, ''L'' is the current number of assumed errors, and initialized to zero. ''N'' is the total number of syndromes. ''n'' is used as the main iterator and to index the syndromes from 0 to ''N''−1. ''B''(''x'') is a copy of the last ''C''(''x'') since ''L'' was updated and initialized to 1. ''b'' is a copy of the last discrepancy ''d'' (explained below) since ''L'' was updated and initialized to 1. ''m'' is the number of iterations since ''L'', ''B''(''x''), and ''b'' were updated and initialized to 1.
Each iteration of the algorithm calculates a discrepancy ''d''. At iteration ''k'' this would be:
:
If ''d'' is zero, the algorithm assumes that ''C''(''x'') and ''L'' are correct for the moment, increments ''m'', and continues.
If ''d'' is not zero, the algorithm adjusts ''C''(''x'') so that a recalculation of ''d'' would be zero:
:
The ''x
m'' term ''shifts'' B(x) so it follows the syndromes corresponding to ''b''. If the previous update of ''L'' occurred on iteration ''j'', then ''m'' = ''k'' − ''j'', and a recalculated discrepancy would be:
:
This would change a recalculated discrepancy to:
:
The algorithm also needs to increase ''L'' (number of errors) as needed. If ''L'' equals the actual number of errors, then during the iteration process, the discrepancies will become zero before ''n'' becomes greater than or equal to 2''L''. Otherwise ''L'' is updated and algorithm will update ''B''(''x''), ''b'', increase ''L'', and reset ''m'' = 1. The formula ''L'' = (''n'' + 1 − ''L'') limits ''L'' to the number of available syndromes used to calculate discrepancies, and also handles the case where ''L'' increases by more than 1.
Code sample
The algorithm from for an arbitrary field:
polynomial(field K) s(x) = ... /* coeffs are s_j; output sequence as N-1 degree polynomial) */
/* connection polynomial */
polynomial(field K) C(x) = 1; /* coeffs are c_j */
polynomial(field K) B(x) = 1;
int L = 0;
int m = 1;
field K b = 1;
int n;
/* steps 2. and 6. */
for (n = 0; n < N; n++)
return L;
In the case of binary GF(2) BCH code, the discrepancy d will be zero on all odd steps, so a check can be added to avoid calculating it.
/* ... */
for (n = 0; n < N; n++) {
/* if odd step number, discrepancy 0, no need to calculate it */
if ((n&1) != 0) {
m = m + 1;
continue;
}
/* ... */
See also
*
Reed–Solomon error correction
*
Reeds–Sloane algorithm, an extension for sequences over integers mod ''n''
*
Nonlinear-feedback shift register (NLFSR)
References
External links
*
Berlekamp–Massey algorithmat
PlanetMath
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.
*
GF(2) implementation in Mathematica*
Online GF(2) Berlekamp-Massey calculator
{{DEFAULTSORT:Berlekamp-Massey Algorithm
Error detection and correction
Cryptanalytic algorithms
Articles with example code