Assume that $G$ is any cyclic group where the discrete log problem is hard, such as the elliptic curve group. Let $g$ be some generator of $G$.
The problem is as follows: Given $(g, g^x)$ for unknown $x$, output any pair of the form $(g^y, xy)$ for $y \neq 0$.
This seems awfully close to the discrete log problem but I could not find any reference for it, nor prove the equivalence myself.
Some things are clear: That algorithm cannot know $y$, since it cannot know $x$ (because the discrete log problem is hard). Also, the algorithm cannot use the same $y$ for different $x$, since that would also reveal $y$, and thereby, $x$.
For this case, we may assume that the Decision Diffie-Hellman problem in $G$ is hard. However, a hardness proof for non-DDH-hard groups will be nicer.