I'm hoping that someone can provide me with some results or point me in the right direction.
I'm working with finite fields; really, I'm just doing arithmetic modulo a prime $p$. I'm taking elements to powers, so I believe this deals with the multiplicative group in particular. Now I basically require that there are at least $m$ elements of a certain order (or greater). We can call this order $n$. I'm wondering if there's a fairly simple and/or easy way to get an estimate of $p$, like how great $p$ must be. The idea is, I want to work with a prime that's big enough to contain $m$ elements of order $n$, but preferably not much larger than the minimum prime that does so.
Extra Credit I'd like an easy way to find the $m$ elements of order $n$. I'm really looking for the simplest way to accomplish both of these goals.
MAIN GOAL I'm trying to ensure that $p$ doesn't need to be astronomically large compared to $m$ and $n$.