# Is there an interactive proof protocol for proving that a preimage in a map $g: Z_{q}^{n} → G_{T}$ is mapped to a given public value, say y?

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).