Linking Bitcoin addresses based on a private key - is that possible I have seen a claim as follows:


  [We] apply algorithms to detect which bitcoin addresses are sharing the same private keys. In most cases, this leads to a high-level assertion of certain addresses to be owned and controlled by a single party


is that possible?

What I'm thinking is one private key per one public key 
As far as i understand, HD wallets use a seed to create many private/public key pairs, but no two public key shares the same private keys.

Any pointers would be appreciated.
You are correct in thinking that there is only one private key for one public key (or rather that is one of the assumptions that ECDSA relies on). While technically it is possible for a Bitcoin address to have multiple public keys and therefore multiple private keys, this is extremely unlikely to occur as it requires a hash collision. Of course there are guaranteed to be addresses that have multiple public-private keypairs associated with them because there are more keypairs than addresses.

In practice, it is impossible for such a collision to occur so, for all intents and purposes, an address has only one public and private key. It is the same for BIP 32 HD wallets since the seed is used to generate the private keys for each address. There is no known mathematical relation between two child private keys so it is impossible to determine whether two public keys are derived from the same BIP 32 HD seed.

However, BIP 32 is not the only way to derive private keys as basically any string of 256 bits can be a private key. So people may be using other methods to generate their private keys, and those may have mathematical relationships between each other. So it is possible that this claim you have quoted refers to these other ways of generating private keys.



Alternatively this person is simply incorrect and doesn't understand cryptography, or they are trying to scam you.