Suppose I have a message $M$ for which I generate an RSA-2048 digital signature as follows:
$H = H(M)$, $H(M)$ being the SHA-256 of the message $M$
$S = H^d \bmod N$
Assume $N = pq$ is properly generated and $d$ is the RSA private key. And I verify the signature as follows:
$S^e \bmod N = H'$
where $H'$ is the SHA-256 of the message to be authenticated. Assume $e$ is the RSA public key.
Since I've not used any padding then are there any flaws with the above approach? What if $e = 3$? What if $e = 2^{16}+1=65537$?
Your guidance is much appreciated.