The Discrete Logarithm Problem

Let G be a finite cyclic group, and let a be a generator for G. Then $$G = \{ {\alpha ^i}|0 \leqslant i\# G\} $$ , where #G is the order of G. The discrete logarithm (logarithm) of an element β to the base α in G is an integer x such that α x = β. If x is restricted to the interval 0 ≤ x < #G then the discrete logarithm of β to the base α is unique. We typically write x = log α β.