When are Fuzzy Extractors Possible?

Benjamin Fuller, Leonid Reyzin, and Adam Smith. When are Fuzzy Extractors Possible? Asiacrypt 2016.


Fuzzy extractors (Dodis et al., Eurocrypt 2004) convert repeated noisy readings of a high-entropy secret into the same uniformly distributed key. A minimum condition for the security of the key is the hardness of guessing a value that is similar to the secret, because the fuzzy extractor converts such a guess to the key.

We define fuzzy min-entropy to quantify this property of a noisy source of secrets. Fuzzy min-entropy measures the success of the adversary when provided with only the functionality of the fuzzy extractor, that is, the \emph{ideal} security possible from a noisy distribution. High fuzzy min-entropy is necessary for the existence of a fuzzy extractor.

We ask: is high fuzzy min-entropy a sufficient condition for key extraction from noisy sources? If only computational security is required, recent progress on program obfuscation gives evidence that fuzzy min-entropy is indeed sufficient. In contrast, information-theoretic fuzzy extractors are not known for many practically relevant sources of high fuzzy min-entropy.

In this paper, we show that fuzzy min-entropy is also sufficient for information-theoretically secure fuzzy extraction. For every source distribution W for which security is possible we give a secure fuzzy extractor.

Our construction relies on the fuzzy extractor knowing the precise distribution of the source W. A more ambitious goal is to design a single extractor that works for all possible sources. We show that this more ambitious goal is impossible: we give a family of sources with high fuzzy min-entropy for which no single fuzzy extractor is secure. This result emphasizes the importance of accurate models of high entropy sources.