The equivalence of the computational Diffie–Hellman and discrete logarithm problems in certain groups

Whether the discrete logarithm problem can be reduced to the Diffie–Hellman problem is a celebrated open question. The security of Diffie–Hellman key exchange and other cryptographic protocols rests on the assumed difficulty of the computational Diffie–Hellman problem; such a reduction would show that this is equivalent to assuming that computing discrete logarithms is hard. What is known is that a near-reduction exists for general groups, assuming that a conjecture about the existence of smooth numbers in an interval is true. Given access to a Diffie–Hellman oracle, and a small amount of additional information (this being the parameters of certain elliptic curves with smooth order), it is possible to compute discrete logarithms using a polylogarithmic number of calls to the oracle.