Verifiable Random Functions (VRFs) and Oblivious Pseudorandom Functions (OPRFs) are essential cryptographic primitives in privacy-preserving applications such as anonymous authentication, private set intersection (PSI), and decentralized identity. Existing constructions, however, rely on number-theoretic assumptions that are vulnerable to quantum attacks. This PhD research project focuses on constructing efficient and practical VRFs and OPRFs from lattice-based assumptions to ensure post-quantum security. A key obstacle in these constructions is the overhead of zero-knowledge proofs (ZKPs), particularly range proofs, which are costly in terms of size and prover complexity. To address this, we investigate probabilistic techniques that relax exact correctness. In particular, we explore approximate range proofs and algebraic transformations, such as using automorphisms in polynomial rings to simulate inner product arguments via polynomial multiplication. These methods enable more efficient and scalable lattice-based constructions of VRFs, including group and context-bound variants, as well as OPRFs. The goal is to make these primitives practical for deployment in post-quantum privacy-preserving systems.