A Novel Method of Searching Primitive Roots Modulo Fermat Prime Numbers

Abstract

Primitive root is a fundamental concept in modern cryptography as well as in modern number theory. Fermat prime numbers have practical uses in several branches of number theory. As of today, there is no simple general way to compute the primitive roots of a given prime, though there exists methods to find a primitive root that are faster than simply trying every possible number. We prove the equivalence between the primitive roots and the quadratic nonresidues modulo Fermat prime numbers. Therefore, the problem of searching primitive roots is transformed into solving the quadratic residues modulo Fermat primes, which is a much easier problem, having very simple solutions. Theoretical analysis and experimental results verify our conclusion.