Linear Feedback shift register over integers The Solina's paper Generalized Mersenne Numbers contains a Linear Feedback shift register I am not able to understand. Here it is:



It is supposed to be a normal Linear Feedback shift register but it doesn't seem to follow the "normal" lfsr rule. Is anyone able to explain the logic it follows?
I'm not sure I understood your question correctly, but what Solinas is doing is shifting the input polynomial by multiplying it by $t$ and reducing modulo $f(t)$

To compute the reduction it subtracts $f(t)=t^3-t+1$. 
So, for example, after shifting the first line (where shifting means multiplication by $t$) what you obtain is: $t^3$ from which you subtract the modulus, so $t^3-t^3+t-1=t-1$ which is what you obtain in the second row.

And so on for the other rows.

In the table, the top row identifies the constant term, degree-1 term, and degree-2 term of the polynomials, while the second row (-1,1,0) identifies the value used for the modular reduction. The third row is the initial value of the LFSR which is $t^2$ by Solinas' construction.