Discrete Logarithm - Definition

Definition

In general, let G be a finite cyclic group with n elements. We assume that the group is written multiplicatively. Let b be a generator of G; then every element g of G can be written in the form g = bk for some integer k. Furthermore, any two such integers k₁ and k₂ representing g will be congruent modulo n. We can thus define a function

(where Z_n denotes the ring of integers modulo n) by assigning to each g the congruence class of k modulo n. This function is a group isomorphism, called the discrete logarithm to base b.

The familiar base change formula for ordinary logarithms remains valid: If c is another generator of G, then we have