**OptORAMa: Optimal Oblivious RAM**

*Gilad Asharov and Ilan Komargodski and Wei-Kai Lin and Kartik Nayak and Enoch Peserico and Elaine Shi*

**Abstract: **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 $O(\log N\cdot \log\log N)$ 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 $O(n \cdot\log n)$ or $O(n \cdot\log \log n)$, respectively.

**Category / Keywords: **foundations / Oblivious RAM

**Date: **received 21 Sep 2018, last revised 11 Dec 2018

**Contact author: **asharov at cornell edu; komargodski@cornell edu; wklin@cs cornell edu; nkartik@vmware com; enoch@dei unipd it; runting@gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20181211:205753 (All versions of this report)

**Short URL: **ia.cr/2018/892

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