The elliptic-curve version of "from two public halves, recover the shared secret."
The idea:Diffie-Hellman with curve points instead of big numbers: Alice sends aP, Bob sends bP, and each multiplies what they received by their own secret to reach the same point abP. The bet is that seeing aP and bP doesn't let anyone else reach abP.
Why it matters: Keeps elliptic-curve Diffie-Hellman key exchange secret.
The details
On the map: on the curve
Given aP and bP on a curve, compute abP. Easy if ECDLP is easy. It is the assumption behind elliptic-curve Diffie-Hellman key exchange (KE)[1, 2].
This assumption gives key exchange on its own, so it needs more than one-way functions and falls if public-key encryption is impossible.
TD
This assumption gives trapdoor functions on its own (computable by anyone, invertible only with a secret), so it falls if trapdoor functions don't exist. Schemes built on it need not use a trapdoor: ElGamal doesn't.
References
Whitfield Diffie and Martin E. Hellman. New Directions in Cryptography. Democratizing Cryptography, 1976. doi:10.1145/3549993.3550007
Ueli M. Maurer and Stefan Wolf. The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms. SIAM Journal on Computing, 1999. doi:10.1137/s0097539796302749