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.

Do not confuse it with Berlekamp's algorithm for factoring polynomials over finite fields.

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

1. Let the stream be ${\displaystyle s_{0},s_{1},s_{2}...s_{n}}$ 
2. Initialise three arrays ${\displaystyle b}$ , ${\displaystyle c}$  and ${\displaystyle t}$  of the length of the stream to be zeroes, except ${\displaystyle b_{0}\leftarrow 1,c_{0}\leftarrow 1}$ ; initialise ${\displaystyle N\leftarrow 0,L\leftarrow 0,m\leftarrow -1}$ 
3. While ${\displaystyle N}$  is less than ${\displaystyle n}$ :
   • Let ${\displaystyle d}$  be ${\displaystyle s_{N}+c_{1}s_{N-1}+c_{2}s_{N-2}+...+c_{L}s_{N-L}}$ 
   • If ${\displaystyle d}$  is zero, then ${\displaystyle c}$  is already a polynomial which annihilates the portion of the stream from ${\displaystyle N-L}$  to ${\displaystyle N}$ ; increase ${\displaystyle N}$  by 1 and continue.
   • If ${\displaystyle d}$  is 1, then:
     • let ${\displaystyle t}$  be a copy of ${\displaystyle c}$ 
     • Set ${\displaystyle c_{N-m}\leftarrow c_{N-m}\oplus b_{0},c_{N-m+1}\leftarrow c_{N-m+1}\oplus b_{1},...}$  up to ${\displaystyle c_{n-1}\leftarrow c_{n-1}\oplus b_{n-N+m-1}}$  (where ${\displaystyle \oplus }$  is the Exclusive or operator)
     • If ${\displaystyle L\leq {\frac {N}{2}}}$ , set ${\displaystyle L\leftarrow N+1-L}$ , set ${\displaystyle m\leftarrow N}$ , and let ${\displaystyle b\leftarrow t}$ ; otherwise leave ${\displaystyle L}$ , ${\displaystyle m}$  and ${\displaystyle b}$  alone
     • Increment ${\displaystyle N}$  and continue
4. At the end of the algorithm, ${\displaystyle L}$  is the length of the minimal LFSR for the stream, and we have ${\displaystyle c_{L}s_{a}+c_{L-1}s_{a+1}+c_{L-2}s_{a+2}...=0}$  for all ${\displaystyle a}$ .

External links

• Berlekamp–Massey algorithm at PlanetMath.
• Weisstein, Eric W. "Berlekamp–Massey Algorithm". MathWorld.
• Implementation in Mathematica
• Applet Berlekamp–Massey algorithm