Berlekamp-Massey algorithm is an algorithm that will find the shortest linear feedback shift register (LFSR) for a given binary output sequence. Here we present. ‘Berlekamp-Massey theorem’ i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any. Often, L is something we want to know in addition to the coefficients. This is where the Berlekamp–Massey algorithm comes in, as it also determines L.
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LFSRs have been used in the past as pseudo-random number generators for use in stream ciphers due to their simplicity. Unfortunately, an LFSR is a linear system, which makes cryptanalysis easy. Once the LFSR is known, who whole output stream is known.
In that case the Linear Predictor tries to determine the next number in alggorithm sequence using a linear combination of previous samples. This is usually done on the reals, and we don’t expect the Linear Predictor to perfectly predict the next sample, but we choose prediction coefficients that minimise the error between predicted and seen values.
The LFSR operates in the same way, but no longer operates on the real numbers; it operates instead on a finite field, usually GF 2.
Additionally, we assume an LFSR perfectly generates our binary sequence. Unfortunately finite fields can be difficult to explain to people who have not algogithm abstract algebra, I have found A Book of Abstract Algebra by Pinter to be a good introductory book on the topic. In the equation above the values of are the bitstream we are trying to predict, and the values of are the coefficients of the LFSR.
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The equation above means bdrlekamp together a subset of the previous L samples to generate the next sample in the sequence”. In this example we will add new bits onto the right hand side of the bit string, with the oldest bits on the left. Hopefully the example above illustrates how we combine masse last L bits of the bit string to masset the next bit.
This can be continued for arbitrarily long sequences, though eventually the sequence will repeat. Previously we have taken an LFSR and several initialisation bits and produced a bit string. Now we are going to invert the process; we will start with a bit string and try to build an LFSR that generates it. The algorithm used to achieve this is called the Berlekamp—Massey algorithm.
We will start with a slightly different algorithm, then move onto Berlekamp—Massey.
Our first try at solving this problem will rely on the linear nature of the problem, and we will also assume we know L beforehand. We will set up a matrix equation:. Where is the matrix containing our bit string, contains the coefficients of our LFSR, and contains more values of our bit string. Using the bit string we generated in the example abovewe will construct our masset and solve for:.
Now to solve, we just invert and solve for:. So we have recovered the vector of coefficients that we used for our example!
One of the drawbacks with this algorithm, though, is that we needed to know L before we started.
Often, L is something we want to know in addition to the coefficients. This is where the Berlekamp—Massey algorithm comes in, as it also determines L. If it matches, we are done, otherwise it increases L and modifies the coefficients so there everything matches, then tries again.
We’d like to fix it! Leave a comment on the page and we’ll take a look. We will set up a matrix equation: Using the bit string we generated in the example abovewe will construct our matrices and solve for: Now to solve, we just invert and solve for: We can do all this in matlab using the following commands: Further reading We recommend these books if you’re interested in finding out more.
A Book of Abstract Algebra: