Leakage-Resilient Extractors against Number-on-Forehead Protocols

2025-06-14

Symposium on the Theory of Computing

AI Summary

This work addresses robust uniform random string generation in multi-source randomness extraction when only a minimal number of high-entropy sources (“good sources”) are available. Prior state-of-the-art explicit extractors required at least $K = N^{0.01}$ or $sqrt{N}$ good sources, whereas this paper achieves the information-theoretic lower bound by constructing the first explicit, low-error extractor requiring only $K = 3$ good sources. Methodologically, we introduce leakage-resilient extractors (LREs) resistant to Number-On-Forehead (NOF) communication protocols, integrating techniques from multiparty communication complexity and non-malleable extraction, and complete the construction via a novel average-case lower-bound analysis. Our results resolve several long-standing open problems in theoretical computer science and significantly enhance the practical feasibility of cryptographic and distributed computing protocols operating under unreliable or adversarially corrupted entropy sources.

Technology Category

• Natural Language Processing: Information Extraction
• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Given a sequence of N independent sources X1,X2,…,XN∼{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 ≥ N1/2 good sources of polylogarithmic min-entropy. In a follow-up, Chattopadhyay and Goodman improved the number of good sources required to just K ≥ N0.01 (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.

Problem

• Extracting random bits from minimal good sources
• Constructing leakage-resilient extractors for NOF protocols
• Improving robustness in multiparty communication complexity

Innovation

• Leakage-resilient extractors against NOF protocols
• Near-optimal explicit LRE construction
• Connection between communication complexity and extractors