How quickly can we find an element of (at least) multiplicative order at least $p$, where $p \in \mathbb{N}$?
The complete question is that we start with a number system of $s$ elements; for example if we are working with the quaternions, $s=4$. We take all components modulo $m \in \mathbb{N}$. So, for example, if we are working modulo 5, we would start with a number:
$$16 + 17i + 18j + 19k \equiv 1 + 2i + 3j + 4k \bmod 5$$
Again, I would like to know how quickly can we find an element of multiplicative order $p$ or greater?
Essentially, the multiplicative order is the amount of times we multiply a number by itself to get the number 1.