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. $G=\{1,g,g^2,\mathrm{\dots },g^{p-2}\}$.For a number $x\in G$ we want to know the unique number $0\le n\le p-2$ with

$$x=g^n.$$

This number $n$ is called the discrete logarithm or index of $x$ to the basis $g$ and is denoted as ${\mathrm{ind}}_g(x)$. For $x,y\in G$ it satisfies the following properties:

${\mathrm{ind}}_g(xy)$$={\mathrm{ind}}_g(x)+{\mathrm{ind}}_g(y);$

${\mathrm{ind}}_g(x^{-1})$$=-{\mathrm{ind}}_g(x);$

${\mathrm{ind}}_g(x^k)$$=k\cdot {\mathrm{ind}}_g(x).$

Furthermore, for a pair $g,h$ of distinct primitive roots, we also have, for any $x\in G$:

${\mathrm{ind}}_h(x)$$={\mathrm{ind}}_h(g)\cdot {\mathrm{ind}}_g(x);$

$1$$={\mathrm{ind}}_g(h)\cdot {\mathrm{ind}}_h(g);$

${\mathrm{ind}}_g(-1)$$={\displaystyle \frac{p-1}{2}}.$

It is a difficult problem to compute the discrete logarithm, while powering is very easy. Therefore this is of some interest to cryptography.