Condensing and Extracting Against Online Adversaries

2024-11-06

AI Summary

This work studies deterministic randomness compression and extraction for online non-oblivious symbol-fixed (oNOSF) sources—a model where the source comprises a sequence of blocks, some independent with minimal entropy (“good blocks”), while others are adaptively controlled by an online adversary with arbitrary dependence on history (“bad blocks”). Addressing the failure of conventional methods under low entropy and high adversarial power, we: (1) construct the first explicit oNOSF compressor, proving its existence for almost all parameter regimes (e.g., block length ℓ growing and total length n a large constant); (2) introduce the novel notion of “online influence” to quantify the adversary’s dependency strength on future blocks and establish tight lower bounds; and (3) design a new explicit extractor based on leader-election protocols, bypassing limitations of resilient functions and significantly improving efficiency in converting low-entropy oNOSF sources to near-uniform outputs. Our results provide foundational theory and efficient protocols for distributed fault-tolerant computation.

Technology Category

• Search and Optimization: Adversarial Search
• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Data transparency and provenance

Abstract

We investigate the tasks of deterministically condensing and extracting randomness from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective random sources for which extraction is impossible in many parameter regimes [AORSV, EUROCRYPT'20]. A $(g,ell)$-oNOSF source is a sequence of $ell$ blocks where $g$ of the blocks are good (are independent and have some min-entropy), and the remaining bad blocks are controlled by an online adversary - can be arbitrarily correlated with any block that appears before it. The existence of condensers for oNOSF sources was recently studied in [CGR, FOCS'24]. They proved various condensing impossibility results, and showed the existence of condensers when $nggell$. We make significant progress on proving the existence of condensers in almost all parameter regimes, even when $n$ is a large constant and $ell$ is growing. We next construct the first explicit condensers for oNOSF sources, matching the existential results of [CGR, FOCS'24]. We also obtain a much improved construction for transforming low-entropy oNOSF sources into uniform oNOSF sources. We find interesting applications of our results to collective coin flipping and collective sampling, problems that are well-studied in fault-tolerant distributed computing. We use our condensers to provide very simple protocols for these problems. Next, we turn to understanding the possibility of extraction from oNOSF sources. We initiate the study of a new, natural notion of the influence of functions, which we call online influence. We establish tight bounds on the online influence of functions, which imply extraction lower bounds. Lastly, we give explicit extractor constructions for oNOSF sources, using novel connections to leader election protocols. These extractors achieve parameters that go beyond standard resilient functions [AL, Combinatorica'93].

Problem

• Study deterministic randomness extraction from defective online sources.
• Construct explicit condensers for online adversarial sources.
• Explore extraction bounds using online influence of functions.

Innovation

• Explicit condensers for oNOSF sources
• Improved low-entropy to uniform transformation
• Novel extractors using leader election protocols