Does the set of automorphisms of a cyclic group exhibit some sense of randomness? I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or  notions from probability theory, which I welcome.
Let $\{x\}$ denote the fractional part of $x$, $(p,q)=1$ and $p\leq q$. 
If $n\in\mathbb{Z}_q$, then the set of $\phi(q)$ functions defined by 
$$f_p(n)=q\{pn/q\}$$  
form the complete set of automorphisms of $\mathbb{Z}_q$. 
As we vary $p$ through $\mathbb{Z}_q^{\times}$, it is not apparent to me that there is  prescribed order in which the elements of $\mathbb{Z}_q$ reappear, neither do I see a rule not involving the function $\{x\}$.   
Say we choose some large $q$, then is it the case that order in which the elements reappear exhibits some sense of randomness, and in what sense if so? 
 A: It appears that this is equivalent to the question of the complexity of the discrete logarithm problem, so I think this question is in general an open problem and ought to be retagged and treated as such. 
As Maarty Isaacs commented $$f_p(n)=np\mod q.$$ To say that the order in which the elements reappear is of a random nature is to say that $np\mod q$ is of high complexity, which is the same as the complexity of its inverse. 
In other words, given a cyclic group $G$, a generator $g$ and an element $f$, the question is: 

How difficult is it to find an integer $n$ such that $f=g^n$?

EDIT: This is not correct. Firstly, the complexity of the inverse (i.e. the discrete logarithm problem) is not necessarily the same. In fact, from what I can gather, this is part of the reason why discrete logs are useful in cryptography at present. Secondly, when the modulus is large (compared to $p$), one gets a string of arithmetic progressions that eventually include the whole set, which is rather predictable. So the only case that may still work is when the modulus is small, but frankly this doesn't look very interesting either!  
