5
In practice, in order to pick a uniformly random $x\in \mathbb{Z}^*_q$, we reduce that task to the problem of picking uniformly random and independent bits. On a modern computer system, there will be some operating system service supplying such bits, e.g. /dev/urandom on many Unix derivatives (we'll get back to the problem of how such bits have been obtained ...
3
This is already answered here for 1 iteration, and here for many iterations, but since the latter answer presents an heuristic argument I'll leave here Lemma 4 of Functional Graphs and Their Applications in Generic Attacks on Iterated Hash Constructions which gives a good approximation based on Theorem 2 of Random Mapping Statistics, and (3.10) of On the ...
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