Verifiable Random Functions (VRFs) are public-key primitives that enable the key owner to generate a pseudorandom output for any given input, along with a proof that allows others to verify the correctness of the output. VRFs are closely related to signature schemes with uniqueness property, meaning that for any given message and public key—even if the key is maliciously generated—there is only one valid signature. There are known approaches for turning unique-signature schemes into VRFs. With the advent of quantum computers, problems such as prime factorization can be solved efficiently. Consequently, it is critical to move toward post-quantum VRF constructions that remain secure. In this thesis, we discuss a novel symmetric-key primitive based one-time signature scheme, named POTS, whose security relies solely on pseudorandom functions (PRFs), a primitive that is secure against quantum adversaries. We prove that POTS is a secure one-time signature scheme and can be extended to a many-time signature scheme with provable security. Compared to other one-time signature schemes, POTS offers faster verification at the cost of a larger public key. Then we construct PVRF, a one-time VRF scheme based on POTS. We introduce a new assumption on the underlying PRF needed by POTS and, based on this assumption, prove PVRF’s security. Finally, we implement POTS and compare our performance results to WOTS+, a standardized one-time signature scheme. This comparison highlights the practical advantages of our approach in terms of verification efficiency.