Are these variants essentially as secure as (textbook) RSA with fixed $e$? With random $e$? Can we reduce the security of one to the security of the other, or of another variant of RSA?
Note: as in RSA with fixed $e$, we choose $p$ and $q$ large random distinct secret primes. And further:
- for $e=n-2$, it must hold $\gcd[q-2,p-1]=1$ and $\gcd[p-2,q-1]=1$ ;
- for $e=n$, if must hold $\min(p,q)$ does not divide $\max(p,q)-1$.