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I'm a bit lost. I know that ZKBoo can be used for proving $y = C(x)$ for any boolean or arithmetic circuit $C$. However, I'm unsure how to combine several circuits when we want to do more complex proofs.

Suppose we have $X = \mathsf{Hash}(x)$, $Y = \mathsf{Hash}(y)$ and $Z = \mathsf{Hash}(z)$ and a publicly known prime $p$. Assume we want to prove that $x\cdot y = z\bmod p$, and we already have a circuit for modular multiplication $C_p$ and a circuit for the hash function $C_\mathsf{hash}$. How can we prove the relation between pre-images?

Shall we construct another circuit that combines modular multiplication and hash computation $C_{all}$, or is it possible to use $C_\mathsf{hash}$ and $C_p$? My main confusion is about the share/reconstruct part. In the ZKBoo proof, the reconstruction happens at the end, and all $3$ values are revealed along with selected $2$ traces. However, if we naively combine two circuits, the reconstruction result after the first circuit should remain secret, which makes me think that a naive approach will not work.

Do you know any works about combining several circuits for ZKBoo proofs? I also would appreciate any good article about the modular multiplication (or addition) circuit.

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