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 found this article : https://eprint.iacr.org/2002/014.pdf 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 amazing.


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