Computational Robust (Fuzzy) Extractors for CRS-dependent Sources with Minimal Min-entropy

Robust (fuzzy) extractors are very useful for, e.g., authenticated exchange from shared weak secret and remote biometric authentication against active adversaries. They enable two parties to extract the same uniform randomness with the ``helper\u27\u27 string. More importantly, they have an authentication mechanism built in that tampering of the ``helper\u27\u27 string will be detected. Unfortunately, as shown by Dodis and Wichs, in the information-theoretic setting, a robust extractor for an $(n,k)$-source requires $k>n/2$, which is in sharp contrast with randomness extractors which only require $k=\omega(\log n)$. Existing work either relies on random oracles or introduces CRS and works only for CRS-independent sources (even in the computational setting). In this work, we give a systematic study of robust (fuzzy) extractors for general CRS-dependent sources. We show in the information-theoretic setting, the same entropy lower bound holds even in the CRS model; we then show we can have robust extractors in the computational setting for general CRS-dependent source that is only with minimal entropy. At the heart of our construction lies a new primitive called $\kappa$-MAC that is unforgeable with a weak key and hides all partial information about the key (both against auxiliary input), by which we can compile any conventional randomness extractor into a robust one. We further augment $\kappa$-MAC to defend against ``key manipulation attacks, which yields a robust fuzzy extractor for CRS-dependent sources