In this paper we develop specific techniques for asymmetric groups. We introduce a new computational assumption, under which we can recover all the aggregation results of Groth-Sahai proofs known in the symmetric setting. We adapt the arguments of membership in linear spaces of (G) over cap (m) to linear subspaces of (G) over cap (m) x (G) over cap (n). In particular, we give a constant-size argument that two sets of Groth-Sahai commitments, defined over different groups (G) over cap, (H) over cap open to the same scalars in Z(q), a useful tool to prove satisfiability of quadratic equations in Z(q). We then use one of the arguments for subspaces in (G) over cap (m) x (H) over cap (n) and develop new techniques to give constant-size QA-NIZK proofs that a commitment opens to a bit-string. To the best of our knowledge, these are the first constant-size proofs for quadratic equations in Z(q) under standard and falsifiable assumptions. As a result, we obtain improved threshold Groth-Sahai proofs for pairing product equations, ring signatures, proofs of membership in a list, and various types of signature schemes.