A LFSR is set of rules to alter a set of bits. They are useful to psudeo random number generators, and as key generators for stream ciphers.
All LFSRs are cyclical in nature, and after a set amount of time will repeat back into themselves. The initial state of the bits in the LFSR is called the seed.
The maximum period for a n bit shift register is
2^n - 1
An LFSR can be generalized as a recurrence relationship where
A tap is where a bit is read and fed back into itself.
An LFSR generates values based on a linear expression modulous 2, therefore we can reverse engineer the state of the LFSR based on a sequence we are given. This can be done using the Berlekamp-Massey algorithm.
So first we will start with a simpler version. If we have a sequence and we know the number of bits in the LSFR, we can create a matrix of the values. If S_{i} is the i th value out of an LSFR, we can solve the following
Sa = x
Where S is a matrix of the outputted values formatted below
A has the coefficents of the LFSR
and x has values of the bit string, as formatted below.
Assume 4 bits -- ---- -- -- -- | s0 s1 s2 s3 || a0 | | s4 | | s1 s2 s3 s4 || a1 | = | s5 | | s2 s3 s4 s5 || a2 | | s6 | | s3 s4 s5 s6 || a3 | | s7 | -- ---- -- -- --
Note that
Given this, we can find the coefficents by solving
a = S^-1 * x
Once we do this, it will give us all of the coefficents! Everywhere there is a 1 a tap will be located there and all of these values are XORed and placed onto the back of the register.
To make this the Berlekamp-Massey algorithm, we first start and assume the number of bits n is 1, check if it makes the right seuqnece, and if not we increase n and try again. That all there is to it!