Abstract. This paper studies various computational and decisional Diffie-Hellman problems by providing reductions among them in the high granularity setting. We show that all three variations of computational Diffie-Hellman 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. Keywords: Diffie-Hellman problem, Square Diffie-Hellman problem, Inverse Diffie-Hellman problem, Divisible Diffie-Hellman problem

The security of many cryptographic constructions relies on assumptions related to Discrete Logarithms (DL), e.g., the Di#e-Hellman, Square Exponent, Inverse Exponent or Representation Problem assumptions. In the concrete formalizations of these assumptions one has some degrees of freedom o#ered by parameters such as computational model, problem type (computational, decisional) or success probability of adversary. However, these parameters and their impact are often not properly considered or are simply overlooked in the existing literature.