Consider the group $G = (\mathbb{Z}/m\mathbb{Z})^*$, where $m$ is such that $G$ is cyclic. Let $g\in G$ be some fixed generator, and let $a_1,\dots,a_{\varphi(m)}$ be an enumeration of the elements of $G$ in the "obvious way" --- specifically that when they're all viewed as elements of $\{1,\dots, \varphi(m)\}$, we have that $a_i < a_j\iff i<j$.
I'm interested in any concrete statements regarding how $\log_g a_i$ (the discrete logarithm with respect to $g$) behave. A (strong) example of this might be something like:
For any interval $[i, j]$, the elements $\log_g a_i,\dots, \log_g a_j$ are equidistributed throughout the group $G$.
I don't expect this to be true, but I expect some result that the discrete logarithm ``mixes elements well'' (as it's my understanding that a property along these lines is required for the discrete logarithm to be computationally difficult, which it's thought that it is).
- Can this intuition be formalized at all?
- Are there any provable results along these lines?