But that's not really due to using a small $e$ rather, it is due to not applying a proper padding. Actually, with RSA as you describe, there is a problem with a very small $e$: if you use $e = 3$ and encrypt the very same message $m$ with three distinct public keys, then an attacker can recover $m$. On the other hand, there is no problem in having a small $e$, down to $e = 3$. The accepted wisdom is that trying to get a $d$ much smaller than $n$ is a bad idea for security.
On a more general basis, if the size of $d$ is lower than 0.29 times the size of $n$ (in bits) then there exists an efficient key recovery attack. If it is very small then an attacker can simply try values for $d$ exhaustively. $n$ is public (by construction) so $d$ must be kept private at all costs. Im trying to encrypt a short message using RSA algorithm in C and decrypt the message using a python skript. That being said, $d$ is the 'private exponent' and knowledge of $d$ and $n$ is sufficient to decrypt messages. First I must state that a secure RSA encryption must use an appropriate padding, which includes some randomness.
