Variant of the Decisional Bilinear Diffie Hellman problem I am working on a cryptographic scheme and I need to rely on the following problem, which I have nicknamed the "Hybrid Decisional Bilinear Diffie Hellman (hDBDH)" problem:


  Let $e: \mathbb G_1 \times \mathbb G_1 \rightarrow \mathbb G_T$ be an efficient bilinear pairing. Given the tuple $(g,g^x,e(g,g)^y,Q)$ as input, the problem is to decide whether $Q = e(g,g)^{xy}$


First, has this problem already been defined? Now, I want to prove that this problem is equivalent to the DBDH problem. I have the feeling that they are equivalent, but so far I only have that $DBDH \leq hDBDH$:


  Proof: From a hDBDH solver, we can construct a DBDH solver in the following way:
  
  Input: DBDH tuple = $(g, g^a, g^b, g^c, Q)$
  
  Output: Return $hDBDH_{\mathsf{solver}}(g,g^a,e(g^b,g^c),Q)$


Now I want to prove the other direction. Any ideas of how a DBDH oracle could be used to solve the hDBDH problem? Any ideas of why it is not possible?
Your problem is stronger than the DBDH problem because any oracle that responds to your query can also respond to DBDH. I think that you must try to reduce this problem to DDH. This means an input $(e(g,g),P=e(g,g)^x,R=e(g,g)^y,Q)$ and the decision whether $Q=e(g,g)^{xy}$.