Title: Iteration of Quadratic Polynomials Over Finite Fields,
Abstract: For a finite field of odd cardinality $q$, we show that the sequence of
iterates of $aX^2+c$, starting at $0$, always recurs after $O(q/\log\log q)$
steps. For $X^2+1$ the same is true for any starting value. We suggest that the
traditional "Birthday Paradox" model is inappropriate for iterates of $X^3+c$,
when $q$ is 2 mod 3.