We prove several new results for seedless condensers in the context of three related classes of sources: NOSF sources, SHELA sources as defined by [AORSV, EUROCRYPT'20], and almost CG sources as defined by [DMOZ, STOC'23]. We will think of these sources as a sequence of random variables $\mathbf{X}=\mathbf{X}_1,\dots,\mathbf{X}_\ell$ on $\ell$ symbols where at least $g$ symbols are "good" (i.e., uniformly random), denoted as a $(g,\ell)$-source, and the remaining "bad" $\ell-g$ symbols may adversarially depend on these $g$ good blocks. The difference between each of these sources is realized by restrictions on the power of the adversary, with the adversary in NOSF sources having no restrictions. Prior to our work, the only known seedless condenser upper or lower bound in these settings is due to [DMOZ, STOC'23] which explicitly constructs a seedless condenser for a restricted subset of $(g,\ell)$-almost CG sources. The following are our main results concerning seedless condensers for each of these three sources. 1. When $g\leq \frac{\ell}{2}$, we prove for all three classes of sources that condensing with error 0.99 above rate $\frac{1}{\lfloor \ell/g \rfloor}$ is impossible. 2. We show that condensing from (2, 3) NOSF sources above rate $\frac{2}{3}$ is impossible. 3. Quite surprisingly, we show the existence of excellent condensers for uniform $(2,3)$-SHELA and uniform almost CG sources, thus proving a separation from NOSF sources. Further, we explicitly construct a condenser that outputs $m = \frac{n}{16}$ bits and condenses any uniform $(2,3)$-SHELA source to entropy $m - O(\log(m / \varepsilon))$ (with error $\varepsilon$). Our construction is based on a new type of seeded extractor that we call output-light, which could be of independent interest. In contrast, we show that it is impossible to extract from uniform $(2,3)$-SHELA sources.

翻译：我们针对三类相关信源背景下无种子凝聚器证明了若干新结果：NOSF信源、[AORSV, EUROCRYPT'20]定义的SHELA信源，以及[DMOZ, STOC'23]定义的近似CG信源。我们将这些信源视为由ℓ个符号构成的随机变量序列$\mathbf{X}=\mathbf{X}_1,\dots,\mathbf{X}_\ell$，其中至少g个符号是“好的”（即均匀随机），记为$(g,\ell)$-信源，而剩余$\ell-g$个“坏的”符号可能恶意依赖于这g个好区块。这些信源之间的差异通过对手能力的限制体现，其中NOSF信源中的对手不受任何限制。在本工作之前，这些设定下已知的唯一无种子凝聚器上界或下界来自[DMOZ, STOC'23]，该工作针对$(g,\ell)$-近似CG信源的受限子集显式构造了一个无种子凝聚器。以下是关于这三类信源无种子凝聚器的主要结果： 1. 当$g\leq \frac{\ell}{2}$时，我们证明对于所有三类信源，以高于速率$\frac{1}{\lfloor \ell/g \rfloor}$且误差为0.99的凝聚是不可能的。 2. 我们证明对于(2,3)-NOSF信源，以高于速率$\frac{2}{3}$的凝聚是不可能的。 3. 令人惊讶的是，我们证明均匀(2,3)-SHELA信源与均匀近似CG信源存在优秀的凝聚器，从而证明其与NOSF信源存在分离。进一步，我们显式构造了一个输出$m = \frac{n}{16}$比特的凝聚器，该凝聚器能将任意均匀(2,3)-SHELA信源凝聚至熵$m - O(\log(m / \varepsilon))$（误差为$\varepsilon$）。该构造基于一种称为输出轻量型的新型种子提取器，这种提取器可能具有独立的研究价值。相比之下，我们证明从均匀(2,3)-SHELA信源中提取信息是不可能的。