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!

# error-correcting codes

# Non-malleable digital Lockers

This is a paper I’m very excited about with Peter Fenteany, a great undergrad at UConn.

**Abstract**: An obfuscated program reveals nothing about its design other than its input/output behavior. A digital locker is an obfuscated program that outputs a stored cryptographic key if and only if a user enters a previously stored password. A digital locker is private if it provides an adversary with no information with high probability. An ideal digital locker would also prevent an adversary from mauling an obfuscation on one password and key into a new program that obfuscates a related password or key. Such a primitive is achievable in the random oracle model. Komargodski and Yogev (Eurocrypt, 2018) constructed a simpler primitive: a non-malleable point function which is a digital locker with no key.

This work describes the first non-malleable digital locker. This construction is built in two main steps:

- Constructing non-malleable digital lockers for short keys. We present one construction for a single bit key and a second for a logarithmic length keys. These constructions can be safely composed with the same input password. This composed construction is non-malleable with respect to the password. Security relies on variants of the strong and power DDH assumptions.
- An extension to polynomial length keys that additionally provides nonmalleability over the stored key. This extension combines the digital locker for short keys and non-malleable codes, and seed- dependent condensers. Our use of seed-dependent condensers require the password distribution to be efficient sampleable. The seed condenser must be public and random but programmability is not required.

Nonmalleability for the password is ensured for functions that can be represented as low degree polynomials. Key nonmalleability is ensured for the class of functions prevented by the non-malleable code.

# 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.

# Reusable Fuzzy Extractors for Low-Entropy Distributions

Ran Canetti, *Benjamin Fuller*, Omer Paneth, Leonid Reyzin, and Adam Smith. Reusable Fuzzy Extractors for Low-Entropy Distributions. Eurocrypt 2016.

Previous titles were “Reusable Fuzzy Extractors via Digital Lockers” and “Key Derivation From Noisy Sources With More Errors Than Entropy.”

### Abstract

Fuzzy extractors (Dodis et al., Eurocrypt 2004) convert repeated noisy readings of a secret into the same uniformly distributed key. To eliminate noise, they require an initial enrollment phase that takes the first noisy reading of the secret and produces a nonsecret helper string to be used in subsequent readings. Reusable fuzzy extractors (Boyen, CCS 2004) remain secure even when this initial enrollment phase is repeated multiple times with noisy versions of the same secret, producing multiple helper strings (for example, when a single person’s biometric is enrolled with multiple unrelated organizations).

We construct the first reusable fuzzy extractor that makes no assumptions about how multiple readings of the source are correlated (the only prior construction assumed a very specific, unrealistic class of correlations). The extractor works for binary strings with Hamming noise; it achieves computational security under assumptions on the security of hash functions or in the random oracle model. It is simple and efficient and tolerates near-linear error rates.

Our reusable extractor is secure for source distributions of linear min-entropy rate. The construction is also secure for sources with much lower entropy rates–lower than those supported by prior (nonreusable) constructions–assuming that the distribution has some additional structure, namely, that random subsequences of the source have sufficient minentropy. We show that such structural assumptions are necessary to support low entropy rates.

We then explore further how different structural properties of a noisy source can be used to construct fuzzy extractors when the error rates are high, providing a computationally secure and an information-theoretically secure construction for large-alphabet sources.

# Strong Key Derivation from Noisy Sources

My Ph.D. has been completed and defended. It is titled Strong Key Derivation from Noisy Sources. It is largely drawn from three papers Computational Fuzzy Extractors, Reusable Fuzzy Extractors for Low-Entropy Distributions, and When are Fuzzy Extractors Possible? Those papers are more up-to-date than my Ph.D. My Ph.D does contain a cohesive summary of the results that can be useful as an introduction.

# Key Derivation from Noisy Sources with More Errors Than Entropy

Our work will appear with about proceedings at Allerton 2014. This work was subsequently published in Reusable Fuzzy Extractors for Low-Entropy Distributions.

# 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.