Primitive Roots and Discrete Logarithms

A primitive root module a prime $p$ is an integer $r$ in ${\mathbb{Z}}_p$ such that every non-zero element of ${\mathbb{Z}}_p$ is a power of $r$.

We can say here $r$ can generate the space of ${\mathbb{Z}}_p$

An important fact is that there is a primitive root module $p$ for every prime $p$. Suppose this primitive root is $r$, and $a$ is an integer between $1$ and $p-1$ inclusive. If $r^e\mathrm{mod}p=a$ and $0\le e\le p-1$, we say that $e$ is the discrete logarithm of $a$ module $p$ to the base $r$ and we write ${\log }_{r}a=e$