发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造显式非延展仿射提取器，实现小误差与线性输出，并导出针对弱只读线性分支程序、局部决策树及分辨率反驳的复杂度下界，改进已知权衡。
AI 中文摘要
我们为每个常数熵率构造了显式非延展仿射提取器，具有线性输出长度和指数级小误差，针对任意固定数量的无不动点仿射篡改。对于每个固定的$0<\eta<1$和$t$，我们还获得了熵阈值$C_{\eta,t}n/\log n$、输出长度$\lfloor n^{1-\eta}\rfloor$以及针对$t$次篡改的误差$2^{-n^{1-\eta}}$。我们的提取器以及Li和Zhong（CCC 2024）的方向仿射提取器，产生了与大小为$2^{\Omega(n)}$的弱只读线性分支程序具有相关性$2^{-\Omega(n)}$的显式布尔函数。对于非遗忘决策树，我们证明了每个固定度数$r\ge2$的查询的线性深度下界。应用Li的sumset提取器（FOCS 2023）给出了增长局部性$\ell\le n^{1-\delta}$（其中$0<\delta<1$固定）下的深度$\Omega_\delta((n/\ell)\log\ell)$。在同一范围内，方向仿射提取器给出了深度为$c(n/\ell)\log\ell/\log\log\ell$的局部树的相关性$2^{-\Omega(n/\sqrt\ell)}$，其中$c>0$是足够小的常数。我们的提取器对Efremenko和Itsykson（STOC 2026）的无损提升进行了去随机化。对于每个固定的$0<\xi<1$，这给出了$N$个变量上的显式多项式大小不可满足CNF，其深度至多$N$的$\mathrm{Res}(\oplus)$反驳要求大小至少$2^{(1-\xi)N}$。另外，奇偶替换给出了$N$个变量上的多项式大小CNF，具有多项式大小的普通分辨率证明，而每个大小为$S$、深度为$d$的$\mathrm{Res}(\oplus)$反驳满足$d\log(2S)=\Omega(N^2)$。这消除了Itsykson、Podolskii和Shekhovtsov（CCC 2026）权衡中的$\log^2 N$损失。
英文摘要
We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<η<1$ and $t$, we also obtain entropy threshold $C_{η,t}n/\log n$, output length $\lfloor n^{1-η}\rfloor$, and error $2^{-n^{1-η}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-Ω(n)}$ against weakly read-once linear branching programs of size $2^{Ω(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $Ω_δ((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-δ}$, where $0<δ<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-Ω(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<ξ<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-ξ)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=Ω(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).
CommentsAbstract shortened due to constraints. 140 pages
