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We present PanORAMa, the first Oblivious RAM construction that achieves communication overhead O(log N log log N) for database of N blocks and for any block size B = Ω(log N) while requiring client memory of only a constant number of memory blocks. Our scheme can be instantiated in the "balls and bins" model in which Goldreich and Ostrovsky [JACM 96] showed an Ω(log N) lower bound for ORAM communication. Our construction follows the hierarchical approach to ORAM design and relies on two main building blocks of independent interest: a new oblivious hash table construction with improved amortized O(log N + poly(log log λ)) communication overhead for security parameter λ and N = poly(λ), assuming its input is randomly shuffled; and a complementary new oblivious random multi-array shuffle construction, which shuffles N blocks of data with communication O(N log log λ + N log N/log λ) when the …