The iterated Carmichael \lambda-function and the number of cycles of the power generator

Abstract

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of \lambda(\lambda(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of \lambda. We deduce that for almost all n, the psuedorandom number generator described above has at least exp((1+o(1))(log log n)^2 log log log n) disjoint cycles. In addition, we show that this expression is accurate for almost all n under the assumption of the Generalized Riemann Hypothesis for Kummerian fields. We also consider the number of iterations of \lambda it takes to reduce an integer n to 1, proving that this number is less than (1+o(1))(log log n)/log 2 infinitely often and speculating that log log n is the true order of magnitude almost always.Comment: 28 page