I know, it is what I meant to say. I just assumed that like in RSA keys, if you start with one public key and change just the exponent of it in order to change the hash, then it's private keys of all of these new keys would have the same factors and be able to calculate the private exponent for all these public keys.
Assuming that during key generation factors are not discarded and just private exponent kept.
I made a lot of assumptions here based on my understanding of RSA keys and how you could generate different hashes for public keys with same module by just changing the public exponent, so that the owner of private keys could generate a private exponent corresponding to the new public exponent. I assumed it is a similar process and that all of these public keys would be (visibly) connected in such a way. Am I correct?