Period of the Power Generator and Small Values of Carmichael's Function

Consider the pseudo-random number generator u sub n = u sup e sub (n-1) (mod m), 0= u sub n = m - 1 , n = 1,2,...., where we are given the modulus m, the initial value u sub 0 = v 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 sub n) exceeds m sup (3/4+epsilon), 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, v, e, the period of the power generator exceeds (pl) sup (1-epsilon). And so, in this case, the power generator is uniformly distributed.