Berlekamp–Massey algorithm

The Berlekamp–Massey algorithm is an algorithm for finding the shortest linear feedback shift register (LFSR) for a given output sequence. Equivalently, it is an algorithm for finding the minimal polynomial of a linearly recurrent sequence.

The algorithm was invented by Elwyn Berlekamp in 1968. Its connection to linear codes was observed by James Massey the following year. It became the key to practical application of the now ubiquitous Reed–Solomon code.

The algorithm

#Let $s\_0,\; s\_1,\; s\_2\; ...\; s\_n$ be the bits of the stream.
#Initialise three arrays $b$, $c$ and $t$ each of length $n$ to be zeroes, except $b\_0\; leftarrow\; 1,\; c\_0\; leftarrow\; 1$; assign $N\; leftarrow\; 0,\; L\; leftarrow\; 0,\; m\; leftarrow\; -1$.
#While $N$ is less than $n$:
#*Let $d$ be $s\_N\; +\; c\_1s\_\{N-1\}\; +\; c\_2s\_\{N-2\}\; +\; ...\; +\; c\_Ls\_\{N-L\}$.
#*If $d$ is zero, then $c$ is already a polynomial which annihilates the portion of the stream from $N-L$ to $N$; increase $N$ by 1 and continue.
#*If $d$ is 1, then:
#** Let $t$ be a copy of $c$.
#** Set $c\_\{N-m\}\; leftarrow\; c\_\{N-m\}\; oplus\; b\_0,\; c\_\{N-m+1\}\; leftarrow\; c\_\{N-m+1\}\; oplus\; b\_1,\; ...$ up to $c\_\{n-1\}\; leftarrow\; c\_\{n-1\}\; oplus\; b\_\{n-N+m-1\}$ (where $oplus$ is the Exclusive or operator).
#** If $L\; le\; frac\{N\}\{2\}$, set $L\; leftarrow\; N+1-L$, set $m\; leftarrow\; N$, and let $b\; leftarrow\; t$; otherwise leave $L$, $m$ and $b$ alone.
#**Increment $N$ and continue.

At the end of the algorithm, $L$ is the length of the minimal LFSR for the stream, and we have $c\_Ls\_a\; +\; c\_\{L-1\}s\_\{a+1\}\; +\; c\_\{L-2\}s\_\{a+2\}\; ...\; =\; 0$ for all $a$.

External links

* [http://planetmath.org/encyclopedia/BerlekampMasseyAlgorithm.html Berlekamp–Massey algorithm] at PlanetMath.
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* [http://modp.com/release/mma_lfsr/ Implementation in Mathematica]
* [http://www.informationsuebertragung.ch/indexAlgorithmen.html Applet Berlekamp–Massey algorithm]