Security of RSA variants with $e=n-2$ and $e=n$

Question

RSA as defined by PKCS#1v2.2 allows public exponent $e=n-2$. And textbook RSA was born with $e=n$ (see second bullet here).

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?

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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 $\max(p,q)-1$.

Answer 1

Rather obvious, but does answer one of the questions: both can be reduced to the security of RSA with $e' = a_n n (n-2)$ for any (reasonably sized) public integer $a$ which can be a function of $n$. For example, given $c \equiv m^n$ for a random unknown $m$, we solve $c^{n-2} \equiv m^{n(n-2)}$ for $m$. With probability at least $1/gcd(e,\lambda(n))$ the answer will be $m$.