Multiplicative structure of values of the Euler function

We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain "smoothing" effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of "strong primes", which are commonly encountered in cryptography. We also consider the problem of obtaining upper and lower bounds for the number of positive integers n less than or equal to x for which the value of the Euler function phi(n) is a perfect square and also for the number of n less than or equal to x such that phi(n) is squarefull. We give similar bounds for the Carmichael function lambda(n).