Discrete logarithms and primitive roots: Algorithms, properties, and typical solution methods

Abstract

In mathematics, the logarithm, log_ab, where a(0,1)(1,) and b>0, is always defined as the real number x, such that a^x=b. Moreover, in the field of number theory, a similar concept called the discrete logarithm can be defined as follows: For a given positive integer m(m2), let aN^(+ ) with (a,m)=1, and r is the primitive root of m, x=ind_r a if r^xa (mod m). Here, x is the discrete logarithm. The Discrete Logarithm Problem, which is a famous problem in number theory, is formulized as: For a positive integer b and a prime number p, and a is the primitive root of p, the goal is to find the exact value of i, such that a^ib (mod p), in other words, it is targeted at finding the exact value of ind_a b. The goal of this research is to give several solutions to the Discrete Logarithm Problem, so firstly, some background concept like order and primitive root will be introduced with the proof of some foundational theories of these two concepts, then this essay will give two methods that can solve the Discrete Logarithm Problem called Shanks' Babystep-Giantstep Algorithm and Pohlig-Hellman Discrete Logarithm Algorithm.