Lower Bounds for Oblivious Data Structures

TL;DR

This paper establishes fundamental lower bounds of logarithmic time for oblivious data structures like stacks, queues, and search trees, indicating no faster solutions are possible for these problems.

Contribution

It proves $oldsymbol{ ext{Omega}( ext{lg} n)}$ lower bounds for various oblivious data structures, closing the question of whether faster oblivious solutions exist.

Findings

Oblivious stacks, queues, deques, priority queues, and search trees all require at least logarithmic time.

No oblivious data structure for these problems can operate faster than $ ext{O}( ext{lg} n)$ in the worst case.

The results match existing upper bounds, confirming the optimality of current solutions.

Abstract

An oblivious data structure is a data structure where the memory access patterns reveals no information about the operations performed on it. Such data structures were introduced by Wang et al. [ACM SIGSAC'14] and are intended for situations where one wishes to store the data structure at an untrusted server. One way to obtain an oblivious data structure is simply to run a classic data structure on an oblivious RAM (ORAM). Until very recently, this resulted in an overhead of $\omega(\lg n)$ for the most natural setting of parameters. Moreover, a recent lower bound for ORAMs by Larsen and Nielsen [CRYPTO'18] show that they always incur an overhead of at least $\Omega(\lg n)$ if used in a black box manner. To circumvent the $\omega(\lg n)$ overhead, researchers have instead studied classic data structure problems more directly and have obtained efficient solutions for many such problems such as stacks, queues, deques, priority queues and search trees. However, none of these data structures process operations faster than $\Theta(\lg n)$, leaving open the question of whether even faster solutions exist. In this paper, we rule out this possibility by proving $\Omega(\lg n)$ lower bounds for oblivious stacks, queues, deques, priority queues and search trees.