Primitive root

A primitive root $s$ of a prime number $p$ is a number whose powers generate $1,\dots ,p-1$.

So $s^1\text{ mod }p,s^2\text{ mod }p,\dots ,s^{p-1}\text{ mod }p$ are distinct, i.e. a permutation of $1$ through $p-1$.

Hence $\forall b\in \mathbb{Z}.\exists i\in \{1,\dots ,p-1\}.b=s^i\text{ mod }p$.