The motivation for this problem is that many security systems use one-way functions: mathematical operations that are fast to compute, but hard to reverse.
exchanged as part of the protocol, and the two parties both compute the shared key
A fast means of solving the DHP would allow an eavesdropper to violate the privacy of the Diffie–Hellman key exchange and many of its variants, including ElGamal encryption.
The problem has survived scrutiny for a few decades and no "easy" solution has yet been publicized.
In fact, significant progress (by den Boer, Maurer, Wolf, Boneh and Lipton) has been made towards showing that over many groups the DHP is almost as hard as the DLP.
There is no proof to date that either the DHP or the DLP is a hard problem, except in generic groups (by Nechaev and Shoup).
The most significant variant is the decisional Diffie–Hellman problem (DDHP), which is to distinguish gxy from a random group element, given g, gx, and gy.