I was reading the Boneh-Franklin IBE paper, and it seemed rather conspicuous to me that they
didn't address the question of how to find primes $p$ and $q$ satisfying what they need (on page 19).
Since one can efficiently generate factored integers with an almost uniform distribution,
it would be enough for there to exist a noticeable density of primes $p$ satisfying the required
condition, i.e., one does not need to worry about efficiently finding the $q$ given such a $p$.
Is there an effective lower bound on the density of primes $p$ such that $\:p+1\:$ has a "large" prime factor? $\;\;$ (for whatever meaning of "large")
What if one additionally requires that $\;\; p \equiv 2 \pmod 3 \;\;$?