In RSA: Suppose I have already chosen p,q and constructed phi(n).
Now my question is if it’s important whether I encrypt with e or with d? I don’t think so, e.g. because when encrypting I take plainoneofthe2numbers = cipher and then decrypting I take ciphertheotherofthe2numbers = plain, which is the same as taking plainoneofthe2numberstheotherofthe2numbers, which is the same as plainoneofthe2numbers * theotherofthe2numbers and since in multiplication both sides can be switched there doesn’t seem to be any possible difference whether e or d was “oneofthe2numbers”.
However our lecturer said that e is to encrypt and d is to encrypt (might have said so for pedagogical reasons), so now I’m unsure. In addition his slides say that when signing a message the roles are reversed. So that you encrypt/sign with d, but decrypt/verify with e. However if there really is no mathematical difference in their role in the decryption/encryption process this differentiation wouldn’t really be needed. (I realize that when signing “key-knowledge” is reversed, meaning the key to encrypt is secret, and the one used to decrypt is public, but my question only concerns the mathematical aspect.) Thanks for any answers
EDIT: It has been said now by both commenters that knowing d and n, with the secret key being a small e it would be possible to factor N. Can somebody explain how? Aside from brute-forcing phi(n) and then going from there? (Pls note that I never suggested using a small e as the private key, but was just interested in the mentioned hypothetical.)