Password-based authenticated key exchange ($\mathsf {PAKE}$) protocol, a widely used authentication mechanism to realize secure communication, allows protocol participants to establish a high-entropy session key by pre-sharing a low-entropy password. An open challenge in $\mathsf {PAKE}$ is how to design a quantum-resistant round-optimal $\mathsf {PAKE}$. To solve this challenge, lattice-based cryptography is a promising candidate for post-quantum cryptography. In addition, Katz and Vaikuntanathan (ASIACRYPT’09) design the first three-round $\mathsf {PAKE}$ protocol by leveraging the smooth projective hash function ($\mathsf {SPHF}$) over lattices. Subsequently, Zhang and Yu (AISACRYPT’17) optimized Katz-Vaikuntanathan’s approximate $\mathsf {SPHF}$ via a splittable public key encryption. They then constructed a two-round $\mathsf {PAKE}$ by using the simulation-sound non-interactive zero-knowledge (NIZK) proofs, but how to construct a lattice-based simulation-sound NIZK remains an open research question. In other words, how to design a one-round $\mathsf {PAKE}$ via an efficient lattice-based $\mathsf {SPHF}$ still remains a challenge. In this work, we attempt to fill this gap by proposing a lattice-based $\mathsf {SPHF}$ with adaptive smoothness. We then obtain a one-round $\mathsf {PAKE}$ protocol over lattices with rigorous security analysis by integrating the proposed $\mathsf {SPHF}$ into the one-round framework proposed by Katz and Vaikuntananthan (TCC’11). Furthermore, we explore the possibilities of achieving two-round $\mathsf {PAKE}$ and universal composable (UC) security from our $\mathsf {SPHF}$, and show the potential application of our $\mathsf {PAKE}$ in Internet of Things (IoTs) where communication cost is the main consideration.