Abstract: Pseudorandom functions are vital tools in the construction of cryptographic primitives. Under the hard assumption of SIS (short integer solution), two lattice-based pseudorandom functions are proposed. The first one has parallel structure by the ideal of tree-like pseudorandom synthesizer, and the second one is serial structure whose public key size is reduced. Both constructions have small modulus and provable security. Compared with A Banerjer, C Peikert and A Rosen’s construction (EUROCRYPT 2012), their key sizes are asymptotically smaller, and efficiency are improved by avoiding the “rounding” technology.