Diffie-Hellman-style puzzles adapted to pairing curves: work out a combined secret from pieces you can see but can't take apart.
The idea: With a pairing you can check a Diffie-Hellman relationship that you can't compute. Given g^a and h^a, a pairing confirms they share the same hidden a, but producing h^a yourself, from h and without a, is believed hard. That asymmetry is a signature scheme: a is the private key, h is the hash of a message, and h^a is the signature, which anyone can check with a pairing.
Why it matters: The basis of short BLS signatures and identity-based encryption.
The details
On the map: co-CDH, BDH, SXDH
Diffie-Hellman-style assumptions adapted to pairing groups:
co-CDH: given g^a in one group, compute h^a in the other[1].
Bilinear DH: compute e(g, g)^(abc) from g^a, g^b and g^c[2].
SXDH: decisional Diffie-Hellman stays hard in both G1 and G2.
They underlie BLS signatures and identity-based encryption, and they give key exchange (Joux's one-round three-party key exchange, 2000)[3], hence the KE tag.
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
Dan Boneh, Ben Lynn and Hovav Shacham. Short Signatures from the Weil Pairing. Journal of Cryptology, 2004. doi:10.1007/s00145-004-0314-9
Dan Boneh and Matt Franklin. Identity-Based Encryption from the Weil Pairing. Advances in Cryptology — CRYPTO 2001. doi:10.1007/3-540-44647-8_13
Antoine Joux. A One Round Protocol for Tripartite Diffie–Hellman. Algorithmic Number Theory, 2000. doi:10.1007/10722028_23