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The problem 'given a group and the elements $g, g^x, g^y$, find $g^{xy}$' is known as the Diffie-Hellman problem (or, more precisely, the computational Diffie-Hellman problem). As for how difficult it is, well, we typically work with groups where this is difficult (by design; we often need to assume that the DH problem is hard). As for how it might be ...

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