Physical unclonable functions (PUFs) exploit the intrinsic complexity and irreproducibility of physical systems to generate secret information. The advantage is that PUFs have the potential to provide fundamentally higher security than traditional cryptographic methods by preventing the cloning of devices and the extraction of secret keys. Most PUF designs focus on exploiting process variations in Complementary Metal Oxide Semiconductor (CMOS) technology. In recent years, progress in nanoelectronic devices such as memristors has demonstrated the prevalence of process variations in scaling electronics down to the nano region. In this paper, we exploit the extremely large information density available in nanocrossbar architectures and the significant resistance variations of memristors to develop an on-chip memristive device based strong PUF (mrSPUF). Our novel architecture demonstrates desirable characteristics of PUFs, including uniqueness, reliability, and large number of challenge-response pairs (CRPs) and desirable characteristics of strong PUFs. More significantly, in contrast to most existing PUFs, our PUF can act as a reconfigurable PUF (rPUF) without additional hardware and is of benefit to applications needing revocation or update of secure key information.