Weak Decisional Diffie-Hellman Problem Is this problem still hard?
Given $$(g,g^a,g^b,c)$$ decide if $c=a\cdot b$?
If there is an adversary that solves the standard Decisional Diffie-Hellman Problem then it can solve my new problem. But I can't understand that my new problem still hard or not.
Did anyone see this problem or similar to my problem? Can anyone help me?
Yes it is. It can be formally reduced to the hardness of the decisional square Diffie-Hellman assumption, which states that distinguishing $(g,g^a,g^{a^2})$ from random is hard (this is a well established assumption).
It follows in a relatively simple way from the answer I wrote here to a related question. I can let you work out the details in case you want to play a bit with these reductions. In case you cannot figure out the formal reduction, just ask in the comment and I will elaborate.