computational entropy

Pseudoentropic Isometries

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I was excited to join the paper Pseudoentropic Isometries: A New framework for fuzzy extractor reusability by Quentin Alamélou, Paul-Edmond Berthier, Chloe Cachet, Stéphane Cauchie, Benjamin Fuller, Philippe Gaborit, and Sailesh Simhadri.  This paper describes how to use the random oracle to build a reusable fuzzy extractor that corrects a linear fraction of errors.  Presented at AsiaCCS 2018.  The abstract is below.

Abstract:

Fuzzy extractors (Dodis et al., Eurocrypt 2004) turn a noisy secret into a stable, uniformly distributed key. Reusable fuzzy extractors remain secure when multiple keys are produced from a single noisy secret (Boyen, CCS 2004). Boyen proved that any information-theoretically secure reusable fuzzy extractor is subject to strong limitations. Simoens et al. (IEEE S&P, 2009) then showed deployed constructions suffer severe security breaks when reused. Canetti et al. (Eurocrypt 2016) proposed using computational security to sidestep this problem. They constructed a computationally secure reusable fuzzy extractor for the Hamming metric that corrects a sublinear fraction of errors.

We introduce a generic approach to constructing reusable fuzzy extractors. We define a new primitive called a reusable pseudoentropic isometry that projects an input metric space to an output metric space. This projection preserves distance and entropy even if the same input is mapped to multiple output metric spaces. A reusable pseudoentropy isometry yields a reusable fuzzy extractor by 1) randomizing the noisy secret using the isometry and 2) applying a traditional fuzzy extractor to derive a secret key.

We propose reusable pseudoentropic isometries for the set difference and Hamming metrics. The set difference construction is built from composable digital lockers (Canetti and Dakdouk, Eurocrypt 2008) yielding the first reusable fuzzy extractor that corrects a linear fraction of errors. For the Hamming metric, we show that the second construction of Canetti et al. (Eurocrypt 2016) can be seen as an instantiation of our framework. In both cases, the pseudoentropic isometry’s reusability requires noisy secrets distributions to have entropy in each symbol of the alphabet.

Lastly, we implement our set difference solution and describe two use cases.

Reusable Fuzzy Extractors for Low-Entropy Distributions

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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.

Computational Fuzzy Extractors

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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.