In the case of the discrete logarithm (with the application of elliptic curve cryptography that maps a finite cyclic group to the cyclic subgroup of the elliptic curve defined by the generator point) we know that the map is a group isomorphism. However so far no one was able to provide an easy to compute inverse map.
Though I am pessimistic that I get a positive reply I was wondering if examples are known where we can proof that it is impossible to analytically construct an easy to compute inverse map?