Computing Multiplicative Order and Primitive Root in Finite Cyclic Group

TL;DR

This paper introduces improved algorithms for computing the multiplicative order and primitive roots in finite cyclic groups, specifically in the multiplicative group modulo a prime, with enhanced efficiency over classical methods.

Contribution

The paper presents new algorithms with logarithmic improvements for calculating multiplicative order and primitive roots in finite cyclic groups.

Findings

Algorithms achieve logarithmic speedup over classical methods

Effective computation of primitive roots in prime fields

Enhanced efficiency in multiplicative order calculations

Abstract

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.