Abstract: In this paper, we propose new classes of trapdoor functions to solve the bounded distance decoding problem in lattices. Specifically, we construct lattices based on properties of polynomials for which the bounded distance decoding problem is hard to solve unless some trapdoor information is revealed. We thoroughly analyze the security of our proposed functions using state-of-the-art attacks and results on lattice reductions. Finally, we describe how our functions can be used to design quantum-safe encryption schemes with reasonable public key sizes. Our encryption schemes are efficient with respect to key generation, encryption and decryption.