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Alice has a hidden message $g^{a \alpha}=(g^a)^\alpha$ where $g^a$ is the message, and a signature $s$ on $g^a$ from Charlie. She sends both $g^{a \alpha}$ and $s$ to Bob. She later wants to prove to Bob that she knows an $\alpha$ such that $s$ is a signature of $(g^{\alpha a})^{\alpha^{-1}}$ without disclosing neither $g^a$ nor $\alpha$.

Does such a PoK exist?

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    $\begingroup$ The message is $g^{\alpha a}$ or $g^a$? the signing key is $\alpha$? what's the corresponding verification key? What's the corresponding statement about the witness $g^a$? $\endgroup$
    – X.H. Yue
    Commented Jun 18 at 4:14
  • $\begingroup$ @X.H.Yue the message is $g^a$, and the signing key is not mentionned. We assume that $g^a$ has been signed by another entity, Charlie. We could write that $sk_C$ is the private key of Charlie that produced the signature $s$ which is only the appendix of signature (that is, without the message) $\endgroup$
    – Adam54
    Commented Jun 18 at 7:07
  • $\begingroup$ The solution of your question I think is the verifiable encryption, which provides a proof to convince the verifier that the message embedding in the ciphertext is well-formed. In your case, the message is the message $g^a$ and the corresponding signature $s$, the relationship between $g^a$ and $s$ is the verification algorithm of the signature scheme. Therefore, you can prove that I have a valid signature on some message meeting the mentioned relationship with PoK. $\endgroup$
    – X.H. Yue
    Commented Jun 19 at 2:53
  • $\begingroup$ @X.H.Yue Thank you very much for the reference. Are there some schemes which enable exactly this verification ? Are there popular approaches based on standard assumptions? $\endgroup$
    – Adam54
    Commented Jun 19 at 16:02
  • $\begingroup$ I recommend researching group signature schemes based on the Groth-Sahai proof system. In group signatures, members are required to prove in zero knowledge that they possess a valid signature issued by the group manager. I believe this topic will be beneficial for your research. $\endgroup$
    – X.H. Yue
    Commented Jun 20 at 6:38

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