Condensing and Extracting Against Online Adversaries

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 $n\gg\ell$. 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].