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I'm actually searching some particular primitive compatible with Groth-Sahai commitment.

I would like to know a signature scheme (on group elements), such that there exists an algorithm $\mathtt{SigCom}$ such that it takes as input the commitment key $\mathtt{ck}$ , the signing key $\mathtt{sk}$ and a commitment $\mathtt{c}$ and outputs a committed signature $\mathtt{c}_\sigma$ with a (Groth-Sahai) proof $\pi$ that certifies the signature committed is the signature of the value (in the commitment of the input).

I've found this article. It does exactly what I want except that the value are on a precise type $\left(G^m, H^m\right)$, and it's too restrictive for me.

If you know a more general result (or something that fit with two group elements without any relation between them), it would be good for me.

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  • $\begingroup$ see this paper aswell: link.springer.com/chapter/10.1007/978-3-642-14623-7_12 It has the same message space so it might still be too restrictive though. Maybe you could clarify your restrictions? $\endgroup$
    – joakimb
    Jun 19 '21 at 7:51
  • $\begingroup$ I would like to have something very general which can consider each vector of group elements as message. $\endgroup$
    – Ievgeni
    Jun 29 '21 at 9:23
  • $\begingroup$ Without having studied the details of the paper I linked, it does not seem that any party actually has to know neither $m$ nor $H^m$, which should then satisfy your requirements. Also, the authors state that the scheme can be instantiated for $G_1 = G_2$, which seems in line with your general setting. $\endgroup$
    – joakimb
    Jun 30 '21 at 9:05
  • $\begingroup$ $H$ is fixed, then only few vectors can be signed. For example $(G, H^2)$ can't be signed. $\endgroup$
    – Ievgeni
    Jun 30 '21 at 9:07
  • $\begingroup$ you are not actually signing $H^m$ in section 4 of the linked paper, the value only exists. But instead, have a look at section 5 of the linked paper. It proposes a scheme for signing vectors of group elements, which seems to be what you need $\endgroup$
    – joakimb
    Jun 30 '21 at 9:15
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The paper Structure-Preserving Signatures and Commitments to Group Elements presents a commitment scheme for committing to (vectors of) group elements, and a signature scheme which outputs signatures (on vectors of group elements) as group elements. Composing these two should be straightforward for your needs.

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  • $\begingroup$ No, because, Sigcom takes as input a commitment, then it's not a simple composition. $\endgroup$
    – Ievgeni
    Jul 1 '21 at 11:11

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