Generalization of the DL-assumption in bilinear group pair When thinking about a pairing-based cryptographic scheme, I encountered the following problem. Let $e \colon G_1, G_2 \to G_T$ be a Type 3 pairing. Then:


  Given $P, zP \in G_1$ and $Q, zQ \in G_2$, compute $z$.


I haven't found this problem in the literature anywhere, so I am not aware of any name of it. It clearly implies the discrete logarithm problems in $G_1$ and $G_2$, but those two do not seem to imply this problem on their own or combined. On the other hand, it is implied by the co-CDH* as well as the q-SDH assumptions - but those are stronger than I need.

I'm wondering the following two things:


Has this problem appeared anywhere in the literature?
Is it implied by any other common assumptions?


It seems very natural to assume the hardness of this problem in a Type 3 cryptographic scheme. Usually the point of working with a Type 3 pairing is exactly to have such a tuple, and use the pairing to decide if the fourth element indeed equals $zQ$ or some other element. So if this problem is not hard and $z$ is private, then you're in trouble.
Usually the point of working with a Type 3 pairing is exactly to have
  such a tuple, and use the pairing to decide if the fourth element
  indeed equals $zQ$ or some other element. So if this problem is not hard
  and $z$ is private, then you're in trouble.


From your explanation, I don't think you require a Discrete Log-like assumption. You are saying that the pairing is used to check if some element is indeed $zQ$ or some random value. This seem to imply that $P$, $zP$, and $Q$ are known to an adversary, and you want to ensure that an adversary cannot produce a value that passes the check you make with the pairing.

Assuming I understood this right, then this seems to be a fit for the Computational co-Diffie Hellman problem (co-CDH): given $P$, $zP \in G_1$, and $Q \in G_2$, the co-CDH problem is to output is $zQ \in G_2$. This output is what you are checking with the pairing, so it seems to me that the co-CDH problem is what you are looking for.

In this case, a Discrete Log-like assumption may be overkill, since an adversary wouldn't need to find $z$. It suffices to compute $zQ$.