An observation associated with asymmetric cryptography is that some functions are (believed to be) easy to perform in one direction but difficult to invert. Furthermore, if there exists some 'trapdoor' information that allows the inverse operation to be computed quickly then the problem becomes a candidate for a public key cryptography scheme.
Classic trapdoor problems, made famous by RSA, include the factoring problem and the discrete log problem. Around the same time that RSA was published, Rabin invented a public key cryptosystem grounded on finding discrete square roots (this was later proved to be at least as difficult as factoring).
Other candidates have cropped up over the years. KNAPSACK (soon after RSA), Elliptic Curve "Logarithms" with specific parameters, and Lattice Shortest Basis Problems are examples of problems whose trapdoor problems used in other published schemes. It is also easy to see that such problems must reside somewhere in NP.
This exhausts my knowledge of trapdoor functions. It also seems to exhaust the list on Wikipedia as well.
I am hoping that we can get a community wiki list of languages that admit trapdoors and relevant literature. The list will be useful. Evolving demands of cryptography also change which trapdoor functions can be the basis of cryptosystems. The explosion of storage on computers makes schemes with large keysizes possible. The perpetual-looming spectre of Quantum Computing invalidates schemes that can be broken with an oracle for finding hidden abelian subgroups. Gentry's Fully Homomorphic Cryptosystem works only because we've discovered trapdoor functions that respect homomorphisms.
I am especially interested in problems that are not NP-Complete.