Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$ is just $a^k$ mod $q$ and $c_k$ is the remainder term modulo $q$. We can also write:
$$c_k = \lfloor{a^k/q}\rfloor$$
modulo $q$.
We know that $b_k$ is unbiased up to the order of $a$. That is, it takes on every value exactly once equally often modulo $q$. We also know that $c_k$ is biased (for example it may take on $0$ multiple times if $a$ is small compared to $q$).
I'm trying to answer roughly how biased the sequence is in general. Is it basically random except for the initial stretch? Are there any special cases for which it is highly biased?
Any useful references would be helpful.
Note: By bias I technically mean the Shannon entropy of the sequence modulo $q$ considered as a random variable in $k$ up to $q-1$.
