  Given a prime $p$, we consider the dynamical system generated by repeated
exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where
$f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. 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.

Number Theory