What is the purpose of calculating the N modulus in the encryption phase of asymmetric cryptography? I was reading a paper entitled : "Mathematical Underpinnings of Asymmetric Cryptography". I came across this 

If $N$ is the product of two large prime numbers $p$ and $q$, say 300 digits each, and $e$ is just coprime with $(p-1)(q-1)$ so it can be a relatively smaller number, my problem is that if I am to raise a certain numerical representation of my text, say $M$, to the power of $e$, it will still be smaller than $N$, therefore $M^e \bmod N = M^e$. So the modulus is not doing anything. I realize that you can just add $kN$ ($k$ being an relative integer), but that's missing the whole point.

What am I missing?
What am I missing?


Not much; you've just come across one of the reasons why we never use "raw RSA".  Instead, whenever we use RSA (say, for encryption), we always pad out the message so that, after padding, it's not much smaller than $N$.  This prevents the message from being less than $N^{1/e}$ (which, as you point out, would be completely insecure), and it also prevents subtler attacks based on the homomorphic properties of raw RSA (namely, $RSA(x)RSA(y) = RSA(xy)$