Update for 15-12-21 15:00

tech/LFSR.wiki

@@ -22,3 +22,42 @@ 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 
+- The S matrix is _n_ bits squared
+- All other matrices are _n_ tall
+- You need 2*n - 1 sample bits
+
+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!