On the period of the linear congruential and power generators

Abstract

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+ε(n)}$ for any monotone function $ε(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-ε}$ for any $ε>0$. Moreover, the period is unconditionally greater than $n^{1/2+δ}$, for some fixed $δ>0$, for a positive proportion of $n$ in the above mentioned sets. These bounds are related to lower bounds on the multiplicative order of an integer $e$ modulo $p-1$, modulo $λ(pl)$, and modulo $λ(m)$ where $p,l$ range over the primes, $m$ ranges over the integers, and where $λ(n)$ is the order of the largest cyclic subgroup of $(\Z/n\Z)^\times$.