Leakage-Resilient Extractors against Number-on-Forehead Protocols

Abstract

Given a sequence of $N$ independent sources $\mathbf{X}_1,\mathbf{X}_2,\dots,\mathbf{X}_N\sim\{0,1\}^n$, how many of them must be good (i.e., contain some min-entropy) in order to extract a uniformly random string? This question was first raised by Chattopadhyay, Goodman, Goyal and Li (STOC '20), motivated by applications in cryptography, distributed computing, and the unreliable nature of real-world sources of randomness. In their paper, they showed how to construct explicit low-error extractors for just $K \geq N^{1/2}$ good sources of polylogarithmic min-entropy. In a follow-up, Chattopadhyay and Goodman improved the number of good sources required to just $K \geq N^{0.01}$ (FOCS '21). In this paper, we finally achieve $K=3$. Our key ingredient is a near-optimal explicit construction of a new pseudorandom primitive, called a leakage-resilient extractor (LRE) against number-on-forehead (NOF) protocols. Our LRE can be viewed as a significantly more robust version of Li's low-error three-source extractor (FOCS '15), and resolves an open question put forth by Kumar, Meka, and Sahai (FOCS '19) and Chattopadhyay, Goodman, Goyal, Kumar, Li, Meka, and Zuckerman (FOCS '20). Our LRE construction is based on a simple new connection we discover between multiparty communication complexity and non-malleable extractors, which shows that such extractors exhibit strong average-case lower bounds against NOF protocols.