A Classification of Lattice-based Trapdoor Functions

A trapdoor function is a one-way function with trapdoor, which is indispensable for getting a preimage of the function. In lattice-based cryptography, trapdoor function plays an important role in constructing the secure cryptographic schemes like identity-based encryption, homomorphic encryption, or homomorphic signature. There are three categories of trapdoor functions as standard trapdoor, lossy trapdoor, and homomorphic trapdoor functions. Lossy trapdoor function is a trapdoor function that behaves in two ways as a standard trapdoor function or by losing information from the input. Homomorphic trapdoor function can evaluate the computation of trapdoor function results. In this paper, we survey all the public literature on lattices studying lattice-based trapdoor functions and their preimage sampling algorithms to the best of our knowledge. Then, we classify these trapdoor functions into three categories depending on their cryptographic features and suggest their feasibility to design cryptographic primitives.