Condensing and Extracting Against Online Adversaries

TL;DR

This work advances the theory of randomness condensation and extraction under online adversarial constraints by constructing explicit condensers for oNOSF sources in regimes where n is polylogarithmic in ℓ and g≥0.51ℓ, and by developing a low-entropy-to-uniform transformation that strengthens condensers across broader parameter ranges. It introduces online influence, a new analytic tool, to derive extraction lower bounds and ties extractors to leader-election protocols to yield explicit oNOSF/oNOBF extractors. The results yield practical protocols for collective coin flipping and collective sampling, and establish existential and explicit condensers for all ℓ and n, including constant block lengths, with applications to blockchains and fault-tolerant distributed computing. The paper also highlights open questions about constant-gap condensers, seed-length optimizers, and exact extraction thresholds in online settings, guiding future work in resilient randomness processing under online adversaries.

Abstract

We study the tasks of deterministically condensing and extracting from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective randomness where extraction is impossible in many parameter regimes [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks where at least $g$ blocks are good (independent, with min-entropy) and the remaining bad blocks are controlled by an online adversary and can be arbitrarily correlated with prior blocks. Previously, [CGR, FOCS'24] proved impossibility of condensing beyond rate $1/2$ when $g\le 0.5 \ell$ and showed existence of condensers for when $g \ge 0.51\ell$ and $n$ is exponential in $\ell$. In this work, not only do we construct the first explicit condensers matching the existential results of [CGR, FOCS'24], but we make a doubly exponential improvement by handling the case when $g\ge 0.51\ell$ and $n$ is only polylogarithmic in $\ell$. We also obtain a much improved explicit construction for transforming low-entropy oNOSF sources into uniform oNOSF sources. Next, we essentially resolve the question of the existence of condensers for oNOSF sources by showing the existence of condensers even when $n$ is a large enough constant and $\ell$ is growing (provided $g \ge 0.51\ell$). We apply our condensers to collective coin flipping and collective sampling, widely studied problems in fault-tolerant distributed computing, and provide very simple protocols for them. Finally, we study the possibility of extraction from oNOSF sources. For lower bounds, we introduce the notion of online influence - extending the notion of influence of boolean functions - and establish tight bounds that imply extraction lower bounds. We also construct explicit extractors via leader election protocols that beat standard resilient functions [AL, Combinatorica'93].

Theorems & Definitions (158)

• Definition 1.1
• Definition 1.2
• Theorem 1: \ref{['thm overview:condensing 51 with exp']} restated
• Theorem 2: Informal version of \ref{['thm overview: 51 percent good n polylog ell']}
• Corollary 1.3: \ref{['cor:low entropy oNOSF explicit condenser']}, simplified
• Theorem 1.4: Informal version of \ref{['lem:existential low-entropy transformation']}
• Theorem 3: Informal version of \ref{['theorem: simplified large n existential uni onosf condenser']}
• Theorem 4
• Definition 1.5: Online influence
• Theorem 1.6: Informal version of \ref{['cor:impossible_nobf_ext']}