I have $n$ polynomial values at points $[1,..,n]$. I don't know its degree, it can be anything from $0$ to $n-1$. I need to evaluate polynomial values at points $[n+1,..,n+m]$.
In my case this is practical programming problem, $n$ and $m$ and quite big. I think I have to use Fast Fourier Transform because of its speed. I am working in modulo prime ring, so it's Number Theoretical Transform, but I think it doesn't change anything, since they have the same properties.