I need to know, please:
(1) Is there anyway to pick uniformly at random permutation polynomial in a field of prime order?
(2) Are there many permutation polynomials in a field?
(3) In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d ?
***Indeed has the permutation polynomial used in this way in cryptography to generate uniformly random value? If yes, where?