I want to create a NIZK proof of a multi-exponentiation via Groth-Sahai proofs. In the lectures of Jens Groth in a winter school of Bar-Ilan University, he shows a way to transform multi-exponentiations to paring product equations. Multi-exponentiation

My problem is as follows. Let me take a pairing in the first form ($e(A_j,h^{y_j})$) I need to commit to $h^{y_j}$ as I understand. But I don't know how to prove that I commit the right exponentiation. I think that I just have another exponentiation to prove, since $y_j$ is also a variable -not a constant. Is there a different way to create proofs for multi-exponentiation equations or am I missing something?



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