Smooth Carmichael and Square Euler

Igor Shparlinski
Macquarie, Australia; visiting Royal Holloway and Oxford

We outline some recent results about arithmetic properties of the values of the Carmichael $\lambda$ and Euler $\phi$ functions.

For example, we study how often $\lambda(n)$ is smooth (i.e. free of large prime factors) and show that this doesn't happen too often (but more frequently than for a random integer). We also study how often $\phi(n)$ is a perfect square and show that surprisingly enough this is quite common (much more frequent than for a random integer).

We also show relevance of these and several related questions to cryptography. Several useful (but rather simple) number theoretic techniques, used in the proofs, will be outlined.

The talk is based on joint work (in progress) with Bill Banks, John Friedlander and Carl Pomerance.