How to sign multiple data sets with a single signature, so that it can be validated for single data set without requiring the rest of the data?

Question

Let's say Alice has three secrets: SecretA, SecretB, SecretC

Alice shows these three secrets to Bob and gets a single signature from him that signs all three secrets together. Something like

Signature = Sign(Hash(SecretA, SecretB, SecretC))

Of course some care needs to be taken here with the hash function to avoid second preimage attacks.

Now Alice would like to prove to Carol that Bob has seen/signed SecretA, without revealing SecretB and SecretC to her (Bob is not around anymore to ask, but Carol trusts Bob's certificate).

Is there some form of hashing & signatue that would allow this? So that she can generate some proof using all three secrets plus the signature, but that will require to reveal only SecretA to Carol (plus maybe some data derived from SecretB and SecretC, which does not allow to recreate these secrets however)

Answer 1

Yes, if Bob creates his signature over hash(SecretA) || hash(SecretB) || hash(SecretC), like so:

Signature = Sign( hash(SecretA) || hash(SecretB) || hash(SecretC) )

and hash(SecretA), hash(SecretB), and hash(SecretC) are made public.

Then, Alice can prove to Carol that Bob signed SecretA by taking a hash of SecretA, and concatenating it with hash(SecretB) and hash(SecretC) [remember hash(SecretB) and hash(SecretC) are public, so Alice and Carol do not need to know SecretB and SecretC] , then verifying the signature over the above using Bob's public key.