Physical Unclonable Functions (PUFs) have gained widespread attention for their secure key storage, authentication, and anti-counterfeiting applications. While traditional PUFs based on Complementary Metal-Oxide-Semiconductor (CMOS) have been extensively studied, the emergence of memristors offers new opportunities due to their inherent device variations and distinctive resistive switching behaviors. This study explores the construction of reliable PUFs using self-rectifying analog BiFeO3 (BFO) memristors. We assess the raw bit error rate (rBER) of the BFO-based PUF under varying voltage challenges and classify the switching behavior into stochastic, transition, and deterministic regions. As the primary objective of this study, we identify the sources of stochastic behavior in the three distinct regions while investigating the physical switching mechanism in BFO cells. Additionally, we propose a key storage method based on memristor variability, including an error correction scheme to enhance the reliability of PUF. This research contributes to a comprehensive understanding of PUF reliability and the underlying sources of intrinsic stochastic behavior in memristive technology.