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 u bar right arrow f(g)(u), where f(g)(u) equivalent to g(u) (mod p) and 0 <= f(g) (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.
cycle dynamical system discrete logarithm