The discrete logarithm problem

DLP in $(\mathbb{Z}_m, +)$ can be solved with extended Euclid algorithm

$(\mathbb{Z}^{*}_p, \cdot)$ is the only group where DLP is hard

Raising an element of a group to an integer power is harder than DLP in the same group

There are groups where DLP can be solved effectively

DLP in $(\mathbb{Z}^{*}_p, \cdot)$ is useful in cryptographic applications