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Optimal Statistical Oblivious RAM and Search Tree in the External-Memory Model

ePrint Report: Optimal Statistical Oblivious RAM and Search Tree in the External-Memory Model Andrew Krapivin, Elaine Shi Oblivious RAM (ORAM) allows a trusted client to access (encrypted) data stored in untrusted memory, such that the access patterns reveal nothing about the data or the queries. It is well-known that any ORAM scheme must suffer from a logarithmic overhead. More precisely stated, each request must incur $\Omega(\log_B N)$ block transfers on average, where $N$ is the size of the logical memory and $B$ is the block size of the oblivious implementation. Moreover, the same lower bound extends to oblivious Search Trees (ST). For upper bounds, however, we have been long stuck with a $(\log_B N)^2$ curse on two fronts: (1) Log-squared curse for statistically secure ORAMs: although optimality is known for computationally secure ORAMs for $B = \Theta(1)$ or $B \geq \log^{1+\epsilon} N$ (where $\epsilon > 0$ is an arbitrarily small constant), progress has been stagnant when it comes to statistical security. (2) Log-squared curse for oblivious search trees: for search trees, we do not know how to overcome the log-squared curse even when allowing computational assumptions. In this paper, we break this long-standing log-squared curse for both oblivious search trees and statistically secure ORAM on broad parameter regimes: (1) Our first result gives an optimal, statistically secure ORAM with $O(\log_B N)$ cost as long as $B \geq \log^{1+\epsilon} N$. (2) Our second result gives an optimal, statistically secure oblivious search tree with $O(\log_B N)$ cost when $B \geq \log^{2+\epsilon} N$, matching the performance of the best known non-oblivious search trees in this parameter region. Notably, this means that for search trees, we can get obliviousness for free! The assumptions on $B$ required to achieve optimality are standard and mild. They not only coincide with the standard wide cache-line assumption prevalent in the classical algorithms literature, but also align well with the hardware-enclave architectures that employ oblivious algorithms in practice, such as the implementations of Signal and Meta.

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