Is there an Asymetric encryption scheme, or public key encryption, where multiple recipients are able to access the data but in order for one of the recipients to leak the key to the data they would need to share their private key?

I believe this to be true for most one-to-one methods of public key encryption (correct me if I'm mistaken) where if something is encrypted using a public key the corresponding private key would need to be shared to decrypt the encrypted data and could used to identify the corresponding public key.

However, I am curious if such guarantees could be made in a Asymetric encryption mechanism with multiple recipients.

An example is GPG with multiple recipients. I believe the way this is accomplished is by encrypting a common synchronous key with each of the public keys specified so I don't think this could make the guarantee that the private key would need to be shared in order to grant permission to decrypt the data to another party since the synchronous key likely could just be shared.

Another example I identified is An efficient public key traitor tracing scheme but it's not clear to me weather the private key would need to be shared in order to be considered a "traitor".

The use case I see for this is if something of value could be accessed using the private key then the party holding the private key would be motivated to keep it secrete. It doesn't guarantee they will but it will put something at stake.

Stated formally what looking for is; Given $A, B, C \in \{pub,priv\}$, a set of functions for encrypting/decrypting data $x$ follow $f_0:x,seed,A_{pub},B_{pub},C_{pub}\rightarrow x_{encrypted}$ and $f_1:x_{encrypted}, A_{priv} \lor B_{priv} \lor C_{priv} \rightarrow x$ then the private key of $A$, $B$, or $C$ would need to be revealed in order for data encrypted by $f_0$.

Tying value to the private key and encrypting in this manor strikes me as a useful mechinism for incentivizing the owners of $A,B,C$ to keep their private keys private as they a continuous data broadcast with encrypted data for their eyes only. Obviously the parties could just relay the data but that's another topic for another post and what I want to focus on here is a shared key mechanism that that can guarantee a high probably that the raw private key will need to be shared in order to actively decrypt the encrypted data broadcast.


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