A theorem in cryptography

Question

My teacher gave us to read the following paper, But we don't know many of the terminology. Can you please tell me the terminology like:

1. Uniform random function on a group $G$
2. $G^m$; are they represent $m$ copies of $G$?

Any help will be appreciated. Thanks!

Answer 1

There are $(\#G)^{\#G}$ functions from a group $G$ to itself; for each $g\in G$, one has to make a choice of what the output $u(g)$ is. A uniform random function is a random choice of one of these $(\#G)^{\#G}$ functions where each function is equally likely.

$G^m$ refers to the set of ordered $m-$tuples $(g_1,g_2,\dots,g_m)$, where each $g_i\in G$ for $i=1,2,\dots,m$.