Groth-Sahai proofs and the hidden-bits model
The hidden bits model is an abstraction of the CRS where the prover is able to reveal some random bits and not others (and where these are guaranteed to be uniformly distributed). The Groth-Sahai approach indeed uses a trapdoor in the CRS but its not the same as the hidden bits approach since the trapdoor is used in a different way (in the hidden bits model the trapdoors are the preimages of each block which enable revealing the bits).