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With rapid advancements in electronic gadgets, the security and privacy aspects of these devices are significant. For the design of secure systems, physical unclonable function (PUF) and true random number generator (TRNG) are critical hardware security primitives for security applications. This paper proposes novel implementations of PUF and TRNGs on the RRAM crossbar structure. Firstly, two techniques to implement the TRNG in the RRAM crossbar are presented based on write-back and 50% switching probability pulse. The randomness of the proposed TRNGs is evaluated using the NIST test suite. Next, an architecture to implement the PUF in the RRAM crossbar is presented. The initial entropy source for the PUF is used from TRNGs, and challenge-response pairs (CRPs) are collected. The proposed PUF exploits the device variations and sneak-path current to produce unique CRPs. We demonstrate, through extensive experiments, reliability of 100%, uniqueness of 47.78%, uniformity of 49.79%, and bit-aliasing of 48.57% without any post-processing techniques. Finally, the design is compared with the literature to evaluate its implementation efficiency, which is clearly found to be superior to the state-of-the-art.