We investigate the task of deterministically condensing randomness from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model for which extraction is impossible [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks where at least $g$ of the blocks are good (independent and have some min-entropy) and the remaining bad blocks are controlled by an online adversary where each bad block can be arbitrarily correlated with any block that appears before it. The existence of condensers was studied in [CGR, FOCS'24]. They proved condensing impossibility results for various values of $g, \ell$ and showed the existence of condensers matching the impossibility results in the case when $n$ is extremely large compared to $\ell$. In this work, we make significant progress on proving the existence of condensers with strong parameters in almost all parameter regimes, even when $n$ is a large enough constant and $\ell$ is growing. This almost resolves the question of the existence of condensers for oNOSF sources, except when $n$ is a small constant. We construct the first explicit condensers for oNOSF sources, achieve parameters that match the existential results of [CGR, FOCS'24], and obtain an improved construction for transforming low-entropy oNOSF sources into uniform ones. We find applications of our results to collective coin flipping and sampling, well-studied problems in fault-tolerant distributed computing. We use our condensers to provide simple protocols for these problems. To understand the case of small $n$, we focus on $n=1$ which corresponds to online non-oblivious bit-fixing (oNOBF) sources. We initiate a study of a new, natural notion of influence of Boolean functions which we call online influence. We establish tight bounds on the total online influence of Boolean functions, implying extraction lower bounds.

翻译：我们研究了确定性凝聚在线非遗忘符号固定（oNOSF）源中随机性的任务，这是一种提取不可能的自然模型[AORSV, EUROCRYPT'20]。一个$(g,\ell)$-oNOSF源是一个包含$\ell$个块的序列，其中至少$g$个块是好的（独立且具有最小熵），而其余坏块由在线对抗者控制，每个坏块可以与出现在其之前的任何块任意相关。凝聚器的存在性在[CGR, FOCS'24]中得到了研究。他们证明了对于不同的$g, \ell$值，凝聚是不可能的，并在$n$相对于$\ell$极大的情况下展示了与不可能性结果匹配的凝聚器的存在性。在本工作中，我们在几乎所有参数范围内，即使当$n$是一个足够大的常数且$\ell$增长时，在证明具有强参数的凝聚器存在性方面取得了显著进展。这几乎完全解决了oNOSF源凝聚器存在性的问题，除了当$n$是一个小常数的情况。我们为oNOSF源构造了第一个显式凝聚器，实现了与[CGR, FOCS'24]的存在性结果匹配的参数，并获得了将低熵oNOSF源转化为均匀源的改进构造。我们将我们的结果应用于集体抛币和采样，这些是容错分布式计算中深入研究的问题。我们使用我们的凝聚器为这些问题提供了简单的协议。为了理解小$n$的情况，我们专注于$n=1$，这对应于在线非遗忘比特固定（oNOBF）源。我们启动了对布尔函数一种新的、自然的影响概念的研究，我们称之为在线影响。我们建立了布尔函数总在线影响的紧致界限，这暗示了提取下界。