Computing Multiplicative Order and Primitive Root in Finite Cyclic Group

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 $\mathrm{\mid }G\mathrm{\mid }$, it is called generator or primitive root. This paper describes the algorithms for computing multiplicative order and primitive root in ${\mathbb{Z}}_p^{\ast }$, we also present a logarithmic improvement over classical algorithms.