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：近二十多年来的一系列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：某个核心对象的渐近最优构造将导致其他所有对象也获得渐近最优解。然而，尽管付出了巨大努力，先前的研究虽能接近目标，却始终缺少实现真正渐近最优构造的最后一步。本文填补了这一关键缺失环节，从而同时实现了各类被长期研究提取器及其应用的显式、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延展提取器，进而得到最小熵为$O(\log n)$的渐近最优双源提取器、$N$顶点上$K=\log^{O(1)} N$的$K$-Ramsey图的显式构造，以及面向主动攻击者的真正最优隐私放大协议；针对某些线性最小熵且误差指数级小的双源非延展提取器与仿射非延展提取器，进而首次给出面向2分裂状态篡改与仿射篡改、恒定速率且误差\textit{指数级}小的非延展码的显式构造；针对仿射源、和集源、交错源与小空间源（空间$s$）的显式提取器，可实现$O(\log n)$或$2s+O(\log n)$的渐近最优最小熵；一个需要大小为$2^{n-O(\log n)}$的强线性一次性分支程序的显式函数，该结果在$O(\cdot)$常数意义下达到最优。此前，即便对于标准一次性分支程序，已知显式函数的大小下界最佳结果仅为$2^{n-O(\log^2 n)}$。