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This lecture by Dan Boneh should answer all your questions and more: https://www.youtube.com/watch?v=F4x2kQTKYFY ps. I saw you also had a question about NIZK proofs, so you should check out the full list of videos https://www.youtube.com/playlist?list=PLXF_IJaFk-9C4p3b2tK7H9a9axOm3EtjA which has almost 3 hours of lectures on Groth-Sahai proofs by Jens ...

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The problem can be simplified to the following problem, since the standard argument doesn't really take into account that you can't generate all the polynomials of the given maximum degree : Assume that we have sampled a random point $\vec{x}\in \mathbb{F}_p^n$. We let an adversary adaptively choose polynomials of degree at most $d$ and after each choice ...

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Yes, sure, in some cases. In the following I assume that we speak about pairings as used in cryptography. Lets look at the trivial case, i.e., when your equation reduces to $1_{G_T}=1_{G_T}$ where $1_{G_T}$ is the identity in your target group $G_T$. Then any assignment of your variables that makes at least one input element of each pairing the element ...

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