Non-malleable extractors are generalizations and strengthening of standard randomness extractors, that are resilient to adversarial tampering. Such extractors have wide applications in cryptography and explicit construction of extractors. In the well-studied models of two-source and affine non-malleable extractors, the previous best constructions only work for entropy rate $>2/3$ and $1-\gamma$ respectively by Li (FOCS' 23). We present explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard ones for small entropy. Our main results include two-source and affine non-malleable extractors (over $\mathsf{F}_2$) for sources on $n$ bits with min-entropy $k \ge \log^C n$ and polynomially small error, matching the parameters of standard extractors by Chattopadhyay and Zuckerman (STOC' 16, Annals of Mathematics' 19) and Li (FOCS' 16), as well as those with min-entropy $k = O(\log n)$ and constant error, matching the parameters of standard extractors by Li (FOCS' 23). Our constructions significantly improve previous results, and the parameters (entropy requirement and error) are the best possible without first improving the constructions of standard extractors. In addition, our improved affine non-malleable extractors give strong lower bounds for a certain kind of read-once linear branching programs, recently introduced by Gryaznov, Pudl\'{a}k, and Talebanfard (CCC' 22) as a generalization of several well-studied computational models. These bounds match the previously best-known average-case hardness results given by Chattopadhyay and Liao (CCC' 23) and Li (FOCS' 23), where the branching program size lower bounds are close to optimal, but the explicit functions we use here are different.\ Our results also suggest a possible deeper connection between non-malleable extractors and standard ones.

翻译：非弹性提取器是对标准随机性提取器的推广与强化，能够抵御敌手篡改攻击。此类提取器在密码学及提取器的显式构造中具有广泛应用。在已被充分研究的双源与仿射非弹性提取器模型中，之前的最佳构造分别仅适用于熵率>2/3和1-γ的情形（Li, FOCS' 23）。我们提出了双源与仿射非弹性提取器的显式构造，其性能与小熵源的标准提取器当前最优构造相匹配。主要成果包括：在n比特源上达到最小熵k≥log^C n且误差为多项式级小的双源与仿射非弹性提取器（基于域F_2），参数与Chattopadhyay和Zuckerman（STOC' 16, Annals of Mathematics' 19）及Li（FOCS' 16）的标准提取器相当；以及最小熵k=O(log n)且误差为常数级的构造，参数与Li（FOCS' 23）的标准提取器相当。我们的构造显著优于先前结果，其参数（熵需求与误差）在不优先改进标准提取器构造的前提下已达到最优。此外，改进的仿射非弹性提取器为一类由Gryaznov、Pudlák和Talebanfard（CCC' 22）近期提出的只读线性分支程序（该程序可视为多种经典计算模型的推广）提供了强下界。这些下界与Chattopadhyay和Liao（CCC' 23）及Li（FOCS' 23）先前给出的最优平均情形困难性结果相匹配——其中分支程序规模下界接近最优，但我们此处所使用的显式函数不同。我们的结果还暗示了非弹性提取器与标准提取器之间可能存在更深刻的内在联系。