  Multiplicative order of an element $a$ of group $G$ is the least positive
integer $n$ such that $a^n=e$, where $e$ is the identity element of $G$. If the
order of an element is equal to $|G|$, it is called generator or primitive
root. This paper describes the algorithms for computing multiplicative order
and primitive root in $\mathbb{Z}^*_{p}$, we also present a logarithmic
improvement over classical algorithms.
