Imagine that you walk up to a house, and you knock on the door. The door opens up, but then the floor under your feet opens up and you are dropped into a deep pit. This is what we call a trapdoor. It was very easy for you to fall all the way down into darkness, but climbing back up is a difficult task. The only way to enter the house is to climb up!
When we create a (Common Reference String)CRS based on the discrete logarithm, we say that the only way to exploit this is to break the trapdoor. The trapdoor being the Discrete logarithm problem, which is very easy to compute given that you know the private key, but believed to be hard if you do not know it.
Imagine you walk up to the same house, but instead of knocking on the door, you notice that the door to the garden is open. Moreover, to your surprise, the door to enter the house from the rear is unlocked. You can now enter the house with ease. This is what we call a backdoor. Note, you can have a trapdoor and a backdoor.
Trusted Setup (KoE)
When we create a Structured Reference String(SRS), we usually prefix it’s security with “as long as a single party is honest in the MPC”. If this is not the case, then a backdoor can be exploited which ultimately breaks one of the security assumptions of your system.