Period of the power generator and small values of Carmichael's function

Consider the pseudorandom number generator where we are given the modulus $m$, the initial value $u_0=\vartheta$and the exponent $e$. One case of particular interest is when the modulus $m$ is of the form $pl$, where $p,l$ are different primes of the same magnitude. It is known from work of the first and third authors that for moduli $m=pl$, if the period of the sequence $(u_n)$ exceeds $m^{3/4+\varepsilon}$, then the sequence is uniformly distributed. We show rigorously that for almost all choices of $p,l$ it is the case that for almost all choices of $\vartheta,e$, the period of the power generator exceeds $(pl)^{1-\varepsilon}$. And so, in this case, the power generator is uniformly distributed.

We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto.

The principal tool is an estimate related to the Carmichael function $\lambda(m)$, the size of the largest cyclic subgroup of the multiplicative group of residues modulo $m$. In particular, we show that for any $\Delta\ge(\log\log N)^3$, we have $\lambda(m)\ge N\exp(-\Delta)$for all integers $m$ with $1\le m\le N$, apart from at most $N\exp\left(-0.69\left(\Delta\log \Delta\right)^{1/3}\right)$ exceptions.