While the equivalence between RSA and integer factorization is still an open question, the Rabin (exponent 2) trapdoor permutation is tightly equivalent to fact

While the equivalence between RSA and integer factorization is still an open question, the Rabin (exponent 2) trapdoor permutation is tightly equivalent to factoring. Furthermore, for most groups the DHP is polynomially equivalent to the DLP. The requirement for this to be true is that there exists an elliptic curve with smooth order modulo the Diffie-Hellman group's order. Such smooth-order curves are hard to actually find for large groups, exponentially so (this is a fine example of the chasm between uniform and nonuniform reductions); but for elliptic curves groups used in practice, it is possible to find them. In other words, an easy way to break the DHP in smallish elliptic curve groups would lead to ECDLP solving with only polynomial overhead.