In a recent work, Gryaznov, Pudl\'{a}k, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of $\mathsf{ROBP}$s where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs ($\mathsf{SROLBP}$s). Their main result gives explicit constructions of directional affine extractors for entropy $k > 2n/3$, which implies average-case complexity $2^{n/3-o(n)}$ against $\mathsf{SROLBP}$s with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (ECCC' 22) improves the hardness to $2^{n-o(n)}$ at the price of increasing the correlation to polynomially large. In this paper we show: An explicit construction of directional affine extractors with $k=o(n)$ and exponentially small error, which gives average-case complexity $2^{n-o(n)}$ against $\mathsf{SROLBP}$s with exponentially small correlation, thus answering the two open questions raised in previous works. An explicit function in $\mathsf{AC}^0$ that gives average-case complexity $2^{(1-\delta)n}$ against $\mathsf{ROBP}$s with negligible correlation, for any constant $\delta>0$. Previously, no such average-case hardness is known, and the best size lower bound for any function in $\mathsf{AC}^0$ against $\mathsf{ROBP}$s is $2^{\Omega(n)}$. One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in $\mathsf{F}_2^n$.

翻译：在近期工作中，Gryaznov、Pudlák 与 Talebanford（CCC' 22）引入了被称为定向仿射提取器的仿射提取器强化版本，并结合了每个节点可进行线性查询的 $\mathsf{ROBP}$ 推广模型，证明了前者可对特定类型的后者（即强只读一次线性分支程序 $\mathsf{SROLBP}$）产生强下界。他们的主要成果给出了熵值 $k > 2n/3$ 的显式定向仿射提取器构造，这意味着针对具有指数级小相关性的 $\mathsf{SROLBP}$ 可实现 $2^{n/3-o(n)}$ 的平均情况复杂度。Chattopadhyay 与 Liao 的后续研究（ECCC' 22）将硬度提升至 $2^{n-o(n)}$，但代价是将相关性增大至多项式量级。本文中我们证明：针对 $k=o(n)$ 且具有指数级小误差的显式定向仿射提取器构造，可实现对具有指数级小相关性的 $\mathsf{SROLBP}$ 达到 $2^{n-o(n)}$ 的平均情况复杂度，从而解决了先前工作中提出的两个开放性问题。我们进一步构造了 $\mathsf{AC}^0$ 中的显式函数，该函数对具有可忽略相关性的 $\mathsf{ROBP}$ 可实现 $2^{(1-\delta)n}$ 的平均情况复杂度（其中 $\delta>0$ 为任意常数）。此前尚未有已知的平均情况硬度结果，且针对 $\mathsf{ROBP}$ 的 $\mathsf{AC}^0$ 函数最佳规模下界仅为 $2^{\Omega(n)}$。我们构造的核心要素之一是基于维度扩展器的新型仿射源线性某处冷凝器。该冷凝器还使得具有可忽略误差的显式仿射提取器的熵需求得到无条件改进。我们进一步证明，在 $\mathsf{F}_2^n$ 的多项式 Freiman-Ruzsa 定理条件下，该冷凝器同样适用于一般弱随机源。