Variations of Diffie-Hellman Problem

This paper studies various computational and decisional DiffieHellman problems by providing reductions among them in the high granularity setting. We show that all three variations of computational DiffieHellman problem: square Diffie-Hellman problem, inverse Diffie-Hellman problem and divisible Diffie-Hellman problem, are equivalent with optimal reduction. Also, we are considering variations of the decisional Diffie-Hellman problem in single sample and polynomial samples settings, and we are able to show that all variations are equivalent except for the argument DDH ⇐ SDDH. We are not able to prove or disprove this statement, thus leave an interesting open problem.