Physical unclonable functions (PUF) are cryptographic primitives employed to generate true and intrinsic randomness which is critical for cryptographic and secure applications. Thus, the PUF output (response) has properties that can be utilized in building a true random number generator (TRNG) for security applications. The most popular PUF architectures are transistor-based and they focus on exploiting the uncontrollable process variations in conventional CMOS fabrication technology. Recent development in emerging technology such as memristor-based models provides an opportunity to achieve a robust and lightweight PUF architecture. Memristor-based PUF has proven to be more resilient to attacks such as hardware reverse engineering attacks. In this paper, we design a lightweight and low-cost memristor PUF and verify it against cryptographic randomness tests achieving a unique, reliable, irreversible random sequence output. The current research demonstrates the architecture of a low-cost, high endurance Cu/HfO₂/ p^{+ +}Si memristor-based PUF (MR-PUF) which is compatible with advanced CMOS technologies. This paper explores the 15 NIST cryptographic randomness tests that have been applied to our Cu/HfO₂/ p^{+ +}Si MR-PUF. Moreover, security properties such as uniformity, uniqueness, and repeatability of our MR-PUF have been tested in this paper and validated. Additionally, this paper explores the applicability of our MR-PUF on block ciphers to improve the randomness achieved within the encryption process. Our MR-PUF has been used on block ciphers to construct a TRNG cipher block that successfully passed the NIST tests. Additionally, this paper investigated MR-PUF within a new authenticated key exchange and mutual authentication protocol between the head-end system (HES) and smart meters (SM)s in an advanced metering infrastructure (AMI) for smartgrids. The authenticated key exchange protocol utilized within the AMI was verified in this paper to meet the essential security when it comes to randomness by successfully passing the NIST tests without a post-processing algorithm.