Generating a Diffie Hellman tuple without "being able to know" one of the discrete logs involved
It looks as though the case with a fixed $g$ has shown up under the name "knowledge of exponent assumption(s)"--Googling this phrase will turn up a few papers which discuss/use it. In particular I notice "Statistically Hiding Sets" by Prabhakaran and Xue claims to show the assumption holds in the generic group model. There may be more recent information out there, but that should serve as a starting point at least. Long story short, as far as I can tell: It's a nonstandard assumption, but it's not obviously false and some people have built things with it.