Modern cryptography is concerned with the feasibility or infeasibility of securely realizing a task. The answer to whether or not a task can be securely realized depends on the assumed power of the adversary. The obtained security holds in a strong mathematical sense, meaning that breaking a secure cryptographic protocol is either impossible or it would require to solve some computational problem which is believed to be hard to solve. The most basic computational assumption made in cryptography is the existence of one-way functions. One-way functions are functions that are easy to compute but are hard to invert on almost all inputs. By just assuming the existence of one-way functions it is already possible to realize many cryptographic tasks, but there are some tasks that can only be realized by assuming the existence of trapdoor functions (also called backdoored functions). Trapdoor functions are easy to compute in one direction but hard to invert; the difference is the existence of a special value that allows the function to be easily inverted by any party that has knowledge of this special value.
In this thesis we have contributions in three different facets of backdoors in cryptography, that we describe next.