The known constructions of negligible error (non-malleable) two-source extractors can be broadly classified in three categories: (1) Constructions where one source has min-entropy rate about $1/2$, the other source can have small min-entropy rate, but the extractor doesn't guarantee non-malleability. (2) Constructions where one source is uniform, and the other can have small min-entropy rate, and the extractor guarantees non-malleability when the uniform source is tampered. (3) Constructions where both sources have entropy rate very close to $1$ and the extractor guarantees non-malleability against the tampering of both sources. We introduce a new notion of collision resistant extractors and in using it we obtain a strong two source non-malleable extractor where we require the first source to have $0.8$ entropy rate and the other source can have min-entropy polylogarithmic in the length of the source. We show how the above extractor can be applied to obtain a non-malleable extractor with output rate $\frac 1 2$, which is optimal. We also show how, by using our extractor and extending the known protocol, one can obtain a privacy amplification secure against memory tampering where the size of the secret output is almost optimal.

翻译：已知构造的具有可忽略误差（非 malleable）双源提取器大致可分为三类：（1）一个源的最小熵率约为 $1/2$，另一个源可以具有较小的最小熵率，但提取器不保证非 malleability 的构造。（2）一个源是均匀的，另一个源可以具有较小的最小熵率，并且当均匀源被篡改时提取器保证非 malleability 的构造。（3）两个源的熵率都非常接近 $1$，并且提取器保证对两个源的篡改具有非 malleability 的构造。我们引入了一个新的碰撞抵抗提取器概念，并利用它获得了一个强双源非 malleable 提取器，其中我们要求第一个源具有 $0.8$ 的熵率，另一个源的最小熵率可以是源长度的多对数阶。我们展示了上述提取器如何应用于获得输出率为 $1/2$ 的非 malleable 提取器，该输出率是最优的。我们还展示了如何通过使用我们的提取器并扩展已知协议，获得一种针对内存篡改安全的隐私放大方法，其中秘密输出的大小几乎是最优的。