A long line of work in the past two decades or so established close connections between several different pseudorandom objects and applications. These connections essentially show that an asymptotically optimal construction of one central object will lead to asymptotically optimal solutions to all the others. However, despite considerable effort, previous works can get close but still lack one final step to achieve truly asymptotically optimal constructions. In this paper we provide the last missing link, thus simultaneously achieving explicit, asymptotically optimal constructions and solutions for various well studied extractors and applications, that have been the subjects of long lines of research. Our results include: Asymptotically optimal seeded non-malleable extractors, which in turn give two source extractors for asymptotically optimal min-entropy of $O(\log n)$, explicit constructions of $K$-Ramsey graphs on $N$ vertices with $K=\log^{O(1)} N$, and truly optimal privacy amplification protocols with an active adversary. Two source non-malleable extractors and affine non-malleable extractors for some linear min-entropy with exponentially small error, which in turn give the first explicit construction of non-malleable codes against $2$-split state tampering and affine tampering with constant rate and \emph{exponentially} small error. Explicit extractors for affine sources, sumset sources, interleaved sources, and small space sources that achieve asymptotically optimal min-entropy of $O(\log n)$ or $2s+O(\log n)$ (for space $s$ sources). An explicit function that requires strongly linear read once branching programs of size $2^{n-O(\log n)}$, which is optimal up to the constant in $O(\cdot)$. Previously, even for standard read once branching programs, the best known size lower bound for an explicit function is $2^{n-O(\log^2 n)}$.

翻译：近二十年来的大量研究工作建立了多种伪随机对象与应用之间的紧密联系。这些联系本质上表明，对一个核心对象的渐近最优构造将导致其他所有对象的渐近最优解。然而，尽管付出了巨大努力，以往的研究虽已接近目标，却始终缺少实现真正渐近最优构造的关键一步。本文填补了最后的缺失环节，从而同步实现了多个长期研究课题中明确定义的渐近最优构造与解决方案。我们的成果包括：渐近最优的带种子非可延展提取器，进而得到针对渐近最优最小熵$O(\log n)$的双源提取器；在$N$个顶点上显式构造$K=\log^{O(1)} N$的$K$-拉姆齐图；以及针对主动攻击者的真正最优隐私放大协议。针对线性最小熵且误差指数级小的双源非可延展提取器与仿射非可延展提取器，进而首次显式构造出针对$2$-分裂态篡改与仿射篡改、码率恒定且误差\emph{指数级}小的非可延展码。针对仿射源、和集源、交错源和小空间源（空间$s$源）的显式提取器，实现了渐近最优的最小熵$O(\log n)$或$2s+O(\log n)$。一个显式函数所需强线性一次读取分支程序的规模达到$2^{n-O(\log n)}$，这是对$O(\cdot)$中常数意义下的最优结果。此前，即使是针对标准一次读取分支程序，已知显式函数大小的最佳下界仅为$2^{n-O(\log^2 n)}$。