Multiplicative Structure of Values of the Euler Function

This is a preprint of a book chapter published in High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS (2004). © American Mathematical Society.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 ≤ x for which the value of the Euler function φ (n) is a perfect square and also for the number of n ≤ x such that φ (n) is squarefull. We give similar bounds for the Carmichael function λ (n)

Finding the group structure of elliptic curves over finite fields

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Multiplicative structure of values of the Euler function

Dedicated to Hugh Williams on the occasion of his sixtieth birthday. Abstract. 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 ≤ x for which the value of the Euler function ϕ(n) is a perfect square and also for the number of n ≤ x such that ϕ(n) is squarefull. We give similar bounds for the Carmichael function λ(n).