Fast White-Box Adversarial Streaming Without a Random Oracle

TL;DR

This paper introduces a fast, space-efficient white-box adversarial streaming algorithm for sparse recovery and related tasks, eliminating the need for a random oracle and achieving near-optimal performance with low update times.

Contribution

It presents the first near-optimal white-box adversarial streaming algorithm for sparse recovery that does not rely on a random oracle and has polylogarithmic per-item processing time.

Findings

Achieves near-optimal space complexity in white-box adversarial streams.

Eliminates the need for a random oracle in streaming algorithms.

Provides solutions for sparse recovery, distinct element estimation, and low-rank approximation.

Abstract

Recently, the question of adversarially robust streaming, where the stream is allowed to depend on the randomness of the streaming algorithm, has gained a lot of attention. In this work, we consider a strong white-box adversarial model (Ajtai et al. PODS 2022), in which the adversary has access to all past random coins and the parameters used by the streaming algorithm. We focus on the sparse recovery problem and extend our result to other tasks such as distinct element estimation and low-rank approximation of matrices and tensors. The main drawback of previous work is that it requires a random oracle, which is especially problematic in the streaming model since the amount of randomness is counted in the space complexity of a streaming algorithm. Also, the previous work suffers from large update time. We construct a near-optimal solution for the sparse recovery problem in white-box adversarial streams, based on the subexponentially secure Learning with Errors assumption. Importantly, our solution does not require a random oracle and has a polylogarithmic per item processing time. We also give results in a related white-box adversarially robust distributed model. Our constructions are based on homomorphic encryption schemes satisfying very mild structural properties that are currently satisfied by most known schemes.