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There already exist interactive proof protocols with logarithmic communication for proving that a secret multi-exponent $x ∈ Z^{n}_{q}$ for a public multi-exponentiation $P = \textbf{g}^{\textbf{x}} ∈ G$ is mapped to a given public value y under an arbitrary but given group homomorphism $f : Z_{q}^{n} → G_{T}$, where $\textbf{x} ∈ Z_{q}^{n}$ and $\textbf{g} ∈ G^n$ . For example, https://eprint.iacr.org/2020/753.pdf.

I was wondering if the same thing was possible where the map is not a homomorphism, but we allow for linear communication in group elements, and we do not require that x be in the form of a secret multi-exponent (Although it wouldn't hurt).

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