Such a scheme includes factorization as an additional barrier in case discrete logarithm modulo primes is broken and so why is this not popular and defined in standards?
Actually, it would not be an "additional barrier", instead, it would be an additional avenue of attack. After all, the standard attacks against a discrete log problem still work ...
Nigel Smart has written Cryptography Made Simple. If you have institutional access, the ebook can be downloaded for free from SpringerLink.
To quote the book regarding prerequisites:
The background I assume is what one could expect of a third or fourth year undergraduate
in computer science. One can assume that such students have already met the basics of ...
Johannes Buchmann, Introduction To Cryptography, Springer, 2nd Ed, 2004
is very nice and pitched squarely at undergraduates. You can see a preview here
Its contents are:...
The correct form is:
The integer $a \mod Z_n$ has multiplicative inverse iff $gcd(n,a)=1$
Here, you are working on exponents, so you must consider the modulo as $\phi(n)$ not $n$.
Therefore, here you can find the inverse of $e$ iff $gcd(e, \phi(e))=1$. This guarantees that you be able to find the inverse of $e$ using extended Euclidean algorithm.