Trapdoor functions are a fundamental part of public key cryptography. An example of the most common trapdoor is
- Prime Factorization, used in cryptosystems such as RSA
How are these trapdoor functions found?
There are some theoretical constructions of TDP and you can find two examples in the technical report by Oded Goldreich. It is concerned with enhancements of trapdoor permutations, why they are required and two constructions of the enhanced TDP (in appendix) based on RSA and Rabin's collection.