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A Physical Unclonable Function (PUF) is a promising solution to many security issues due its ability to generate a die unique identifier that can resist cloning attempts as well as physical tampering. However, the efficiency of a PUF depends on its implementation cost, its reliability, its resiliency to attacks, and the amount of entropy in it. PUF entropy is used to construct crypto graphic keys, chip identifiers, or challenge-response pairs (CRPs) in a chip authentication mechanism. The amount of entropy in a PUF is limited by the circuit resources available to build a PUF. As a result, generating longer keys or larger sets of CRPs may increase PUF circuit cost. We address this limitation in a PUF by proposing an identity-mapping function that expands the set of CRPs of a ring-oscillator PUF (RO-PUF) with low area cost. The CRPs generated through this function exhibit strong PUF qualities in terms of uniqueness and reliability. To introduce the identity-mapping function, we formulate a novel PUF system model that uncouples PUF measurement from PUF identifier formation. We show the enhanced CRP generation capability of the new function using a statistical hypothesis test. An implementation of our technique on a low-cost FPGA platform shows at least 2 times savings in area compared to the traditional RO-PUF. The proposed technique is validated using a population of 125 chips, and its reliability over varying environmental conditions is shown.