Short cycles in repeated exponentiation modulo a prime

Given a prime p, we consider the dynamical system generated by repeated exponentiations modulo p, that is, by the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u \mapsto f_g(u)}$$\end{document}, where fg(u) ≡ gu (mod p) and 0 ≤ fg(u) ≤ p − 1. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.