OptORAMa: Optimal Oblivious RAM

Oblivious RAM (ORAM), first introduced in the ground-breaking work of Goldreich and Ostrovsky (STOC '87 and J. ACM '96) is a technique for provably obfuscating programs' access patterns, such that the access patterns leak no information about the programs' secret inputs. To compile a general program to an oblivious counterpart, it is well-known that Omega(log N) amortized blowup is necessary, where N is the size of the logical memory. This was shown in Goldreich and Ostrovksy's original ORAM work for statistical security and in a somewhat restricted model (the so called balls-and-bins model), and recently by Larsen and Nielsen (CRYPTO '18) for computational security. A long standing open question is whether there exists an optimal ORAM construction that matches the aforementioned logarithmic lower bounds (without making large memory word assumptions, and assuming a constant number of CPU registers). In this paper, we resolve this problem and present the first secure ORAM with O(log N) amortized blowup, assuming one-way functions. Our result is inspired by and non-trivially improves on the recent beautiful work of Patel et al. (FOCS '18) who gave a construction with O(log N . log log N) amortized blowup, assuming one-way functions. One of our building blocks of independent interest is a linear-time deterministic oblivious algorithm for tight compaction: Given an array of n elements where some elements are marked, we permute the elements in the array so that all marked elements end up in the front of the array. Our O(n) algorithm improves the previously best known deterministic or randomized algorithms whose running time is O(n . log n) or O(n . log log n), respectively.