Lowen and I just received word of acceptance of our recent work to ISIT. This papers asks whether you can build a universal fuzzy extractor for all high fuzzy min-entropy distributions. That is, can we have one construction that always just works. Unfortunately, the answer is negative. It is possible to artificially construct families of distributions that are impossible to simultaneously secure. This paper shares a lot of techniques with prior work of myself, Reyzin, and Smith. Excited to talk about these techniques more with the information theory community!

# secure sketches

# Presentation at Asiacrypt 2016

I just presented our paper “When are Fuzzy Extractors Possible?” with Leonid Reyzin and Adam Smith at Asiacrypt 2016. The talk video is available here: https://youtu.be/eiKqok3pNIs?t=13906 and the slides are here: fuzzy-extractors-when-possible-asiacrypt

# When are Fuzzy Extractors Possible?

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

### Abstract

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.

# Computational Fuzzy Extractors

*Benjamin Fuller*, Xianrui Meng, and Leonid Reyzin. Computational Fuzzy Extractors. Asiacrypt 2013.

### Abstract

Fuzzy extractors derive strong keys from noisy sources. Their security is defined information- theoretically, which limits the length of the derived key, sometimes making it too short to be useful. We ask whether it is possible to obtain longer keys by considering computational security, and show the following.

-Negative Result: Noise tolerance in fuzzy extractors is usually achieved using an information reconciliation component called a “secure sketch.” The security of this component, which directly affects the length of the resulting key, is subject to lower bounds from coding theory. We show that, even when defined computationally, secure sketches are still subject to lower bounds from coding theory. Specifically, we consider two computational relaxations of the information-theoretic security requirement of secure sketches, using conditional HILL entropy and unpredictability entropy. For both cases we show that computational secure sketches cannot outperform the best information-theoretic secure sketches in the case of high-entropy Hamming metric sources.

-Positive Result: We show that the negative result can be overcome by analyzing computational fuzzy extractors directly. Namely, we show how to build a computational fuzzy extractor whose output key length equals the entropy of the source (this is impossible in the information-theoretic setting). Our construction is based on the hardness of the Learning with Errors (LWE) problem, and is secure when the noisy source is uniform or symbol-fixing (that is, each dimension is either uniform or fixed). As part of the security proof, we show a result of independent interest, namely that the decision version of LWE is secure even when a small number of dimensions has no error.