Berlekamp–Massey algorithm: Difference between revisions

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{{Short description|Algorithm on linear-feedback shift registers}}

{{distinguish|Berlekamp's algorithm}}

[[File:Berlekamp–Massey algorithm.png|thumb|right|Berlekamp–Massey algorithm]]

The '''Berlekamp–Massey algorithm''' is an [[algorithm]] that will find the shortest [[linear -feedback shift register]] (LFSR) for a given binary output sequence. The algorithm will also find the [[Minimal polynomial (field theory)|minimal polynomial]] of a linearly [[Recurrence relation|recurrent sequence]] in an arbitrary [[field (mathematics)|field]]. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse.<ref>{{Harvnb|Reeds|Sloane|1985|p=2}}</ref> Reeds and Sloane offer an extension to handle a [[ring (mathematics)|ring]].<ref>{{Citation |last1=Reeds |first1=J. A. |last2=Sloane |first2=N. J. A. |author-link2=N. J. A. Sloane |journal=SIAM Journal on Computing |volume=14 |issue=3 |pages=505–513 |year=1985 |title=Shift-Register Synthesis (Modulo n) |url=http://neilsloane.com/doc/Me111.pdf |doi=10.1137/0214038 |citeseerx=10.1.1.48.4652 }}</ref>

[[Elwyn Berlekamp]] invented an algorithm for decoding [[BCH code|Bose–Chaudhuri–Hocquenghem (BCH) codes]].<ref>{{Citation

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|url= http://crypto.stanford.edu/~mironov/cs359/massey.pdf

|doi= 10.1109/TIT.1969.1054260

|s2cid= 149442779003708

}}</ref><ref>{{Citation

|last1= Ben Atti

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|url= http://hlombardi.free.fr/publis/ABMAvar.html

|doi= 10.1007/s00200-005-0190-z

|arxiv= 2211.11721

|s2cid= 14944277

|s2cid= 14944277

}}</ref> Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm),<ref>{{Harvnb|Massey|1969|p=124}}</ref> but it is now known as the Berlekamp–Massey algorithm.

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:<math> d = d - (d/b)b = d - d = 0.</math>

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 the 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 samplePseudocode ==

The algorithm from {{Harvtxt|Massey|1969|p=124}} for an arbitrary field:

<!-- Notes: notation changes from Massey:

Massey Here

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T(D) T(x) polynomial

-->

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr">

<syntaxhighlight lang=C>

polynomial(field ''K'') s(x) = ... <span class="cm">/* coeffs are s_js_j; output sequence as N-1 degree polynomial) */</span>

<span class="cm">/* connection polynomial */</span>

polynomial(field K) C(x) = 1; <span class="cm">/* coeffs are c_jc_j */</span>

polynomial(field K) B(x) = 1;

int L = 0;

int m = 1;

field K b = 1;

int n;

<span class="cm">/* steps 2. and 6. */</span>

<span class="k">for</span> (n = 0; n < N; n++) {

/* steps 2. and 6. */

<span class="cm">/* step 2. calculate discrepancy */</span>

for (n = 0; n < N; n++) {

field K d = s_n + {{math|&sum;{{su|p=L|b=i=1}} c_i s_{n - i}}} <!--Σi=1Lci⋅sn−i;-->

/* step 2. calculate discrepancy */

}

field K d = s_n + \Sigma_{i=1}^L c_i * s_{n-i};

<span class="k">if</span> (d == 0) {

<span class="cm">/* step 3. discrepancy is zero; annihilation continues */</span>

if (d == 0) {

/* step 3.m discrepancy= ism zero+ 1; annihilation continues */

} <span class="k">else</span> <span class="k">if</span> (2 * L <= n) {

m = m + 1;

} else if (2 * L <span class="cm">/* n)step {5. */</span>

<span class="cm">/* steptemporary 5.copy of C(x) */</span>

/* temporary copypolynomial(field ofK) CT(x) */= C(x);

polynomial(field K) T(x) = C(x);

C(x) = C(x) - d b⁻¹ x^m B(x);

C(x) = C(x)L -= dn + b^{-1} x^m- B(x)L;

L = nB(x) + 1 -= LT(x);

B(x) b = T(x)d;

b m = d1;

} <span m class="k">else</span> 1;{

<span class="cm">/* step 4. */</span>

} else {

C(x) = C(x) - d b⁻¹ x^m B(x);

/* step 4. */

C(x) = C(x)m -= d b^{-1} x^m B(x)+ 1;

m = m + 1;}

}

<span class="k">return</span> L;

}

</div>

return L;

</syntaxhighlight>

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.

{{sxhl|2=c|1=<nowiki/>

/* ... */

<syntaxhighlight lang=C>

for (n = 0; n < N; n++) {

/* ... */

/* if odd step number, discrepancy == 0, no need to calculate it */

for (n = 0; n < N; n++) {

/* if odd step number, discrepancy((n&1) =!= 0, no need to calculate it) */{

if ((n&1) ! m = 0)m {+ 1;

m = m + 1continue;

continue;}

/* ... }*/

}}

/* ... */

</syntaxhighlight>

==See also==

* [[Reed–Solomon error correction]]

* [[Reeds–Sloane algorithm]], an extension for sequences over integers mod&nbsp;''n''

* [[Nonlinear-feedback shift register]] (NLFSR)

* [[NLFSR]], Non-Linear Feedback Shift Register

==References==

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==External links==

* {{springer|title=Berlekamp-Massey algorithm|id=p/b120140}}

* [https://web.archive.org/web/20120716181541/http://planetmath.org/encyclopedia/{{PlanetMath|BerlekampMasseyAlgorithm.html |Berlekamp–Massey algorithm] at [[PlanetMath]].}}

* {{MathWorld|urlname=Berlekamp-MasseyAlgorithm|title=Berlekamp–Massey Algorithm}}

* [https://code.google.com/p/lfsr/ GF(2) implementation in Mathematica]