discrete logarithm

Let $p$ be a prime. We know that the group $G:=(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic. Let $g$ be a primitive root of $G$ , i.e. $G=\{1,g,g^2,\dots,g^{p-2}\}$ . For a number $x\in G$ we want to know the unique number $0\leq n\leq p-2$ with $$x=g^n.$$ This number $n$ is called the discrete logarithm or index of $x$ to the basis $g$ and is denoted as $\operatorname{ind}_g(x)$ . For $x,y\in G$ it satisfies the following properties:

Furthermore, for a pair $g,h$ of distinct primitive roots, we also have, for any $x\in G$ :
It is a difficult problem to compute the discrete logarithm, while powering is very easy. Therefore this is of some interest to cryptography.