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It is a major open research question whether such a scheme exists, and how to construct one (see, for example, Open Problem 9.10). Of course, we do have schemes like (hashed) ElGamal, which are based on the conjectured hardness of the (computational or decisional) Diffie-Hellman problem. But it is unknown whether either of these problems is equivalent to the ...


Your problem is stronger than DBDH problem. Because any oracle that response to your problem, can response to DBDH. I think that you must try to reduce this problem to DDH. This mean that input $(e(g,g),P=e(g,g)^x,R=e(g,g)^y,Q)$ and decide whether $Q=e(g,g)^{xy}$.

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