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The increased memory demands of workloads is putting high pressure on Last Level Caches (LLCs). Unfortunately, there is limited opportunity to increase the capacity of LLCs due to the area and power requirements of the underlying SRAM technology. Interestingly, emerging Non-Volatile Memory (NVM) technologies promise a feasible alternative to SRAM for LLCs due to their higher area density. However, NVMs have substantially higher read and write latencies, which offset their area density benefit. Although researchers have proposed methods to tolerate NVM's increased write latency, little emphasis has been placed on reducing the critical NVM read latency.To address this problem, this paper proposes Cloak. Cloak exploits data reuse in the LLC at the page level, to hide NVM read latency. Specifically, on certain L1 TLB misses to a page, Cloak transfers LLC-resident data belonging to the page from the LLC NVM array to a set of small SRAM Page Buffers that will service subsequent requests to this page. Further, to enable the high-bandwidth, low-latency transfer of lines of a page to the page buffers, Cloak uses an LLC layout that accelerates the discovery of LLC-resident cache lines from the page. We evaluate Cloak with full-system simulations of a 4-core processor across 14 workloads. We find that, on average, Cloak outperforms an SRAM LLC by 23.8% and an NVM-only LLC by 8.9%-in both cases, with negligible additional area. Further, Cloak's ED 2 is 39.9% and 17.5% lower, respectively, than these designs.