Assume we have a shortened $(n=18, k=12, t=3)$ Reed Solomon code in $GF(2^{8})$.Let $\alpha$ be a primitive element of $GF(2^{8})$. Consider the primitive polynomial given by: $p(D) = D^{8} + D^{4} + D^{3} + D^{2} + 1$. Show that $p(D)$ is a primitive polynomial with root $\alpha$ for the field $GF(2^{8})$.
To start of with we've never worked with a shortened RS code. I'm assuming $(n=18, k=12, t=3)$ comes from $(n=255, k=249, t=3)$ where we just removed 237 information bits?
Now I know I could construct a table for every element from 0 to $\alpha^{253}$ using the "primitive polynomial" p(D), and if every element has a different representation we know p(D) was in fact the primitive polynomial but.. Well.. Doing that that many times will take A LOT OF TIME. Is there a more efficient way?