Linear-Feedback Shift Register in Mod 3 I have an exercise for a crypto class that I'm working on and trying to understand. In the problem I have a Linear-Feedback shift register working in mod 3 with digits {0, 1, 2}.  

The LFSR is using a recurrence relation of degree 2 which looks like this
$Z_{i+2}=C_0Z_i+C_1Z_{i+1}$ 

I also have part of the keystream $S=...11022...$

I am looking to find $C_0$ &amp; $C_1$ along with
the three keystream numbers that follow and precede $S$

Unfortunately I have no examples to work off so I'm confused on how i'd start even with finding $C_0$ &amp; $C_1$?
Given any LFSR equation $$Z_{i+L}=C_0 Z_i+C_1 Z_{i+1}+\cdots+C_{L-1} Z_{i+L-1},$$ over any field, you can rewrite it as a linear system
$$
\left(
\begin{array}{cccc} 
Z_{i} &amp; Z_{i+1} &amp; \cdots &amp; Z_{i+L-1} \\
Z_{i+1} &amp; Z_{i+2} &amp; \cdots &amp; Z_{i+L} \\
 \vdots &amp; &amp; &amp; \vdots \\
Z_{i+L-1} &amp; Z_{i+L} &amp; \cdots &amp; Z_{i+2L-2}\\
\end{array}
\right)
~
\left(\begin{array}{c}
C_0 \\
C_1 \\
\vdots\\
C_{L-1}
\end{array}\right)
=
\left(
\begin{array}{c}
Z_{i+L} \\
Z_{i+L+1}\\
\vdots\\
Z_{i+2L-1}
\end{array}
\right)
$$

and solve it by using linear algebra, since the right hand vector and the matrix entries are obtained from the observed keystream sequence $(Z_t)$. You will need $2L$ keystream symbols to obtain the coefficients uniquely. 

In this case there are faster ways of solving such a system, such as the Berlekamp Massey algorithm, which recursively finds the solution with the smallest $L$, given the keystream $Z_t.$ See the answer to the question berlekamp massey to construct minimal lfsr? for more details.