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Impossibility of efficient information-theoretic fuzzy extraction

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Abstract

Fuzzy extractors convert noisy signals from the physical world into reliable cryptographic keys. Fuzzy min-entropy measures the limit of the length of key that a fuzzy extractor can derive from a distribution (Fuller et al. in IEEE Trans Inf Theory 66(8):5282–5298, 2020). In general, fuzzy min-entropy that is superlogarithmic in the security parameter is required for a noisy distribution to be suitable for key derivation. There is a wide gap between what is possible with respect to computational and information-theoretic adversaries. Under the assumption of general-purpose obfuscation, keys can be securely derived from all distributions with superlogarithmic entropy. Against information-theoretic adversaries, however, it is impossible to build a single fuzzy extractor that works for all distributions (Fuller et al. 2020). A weaker information-theoretic goal is building a fuzzy extractor for each probability distribution. This is the approach taken by Woodage et al. (in: Advances in Cryptology—CRYPTO, Springer, pp 682–710, 2017). Prior approaches use the full description of the probability mass function and are inefficient. We show this is inherent: for a quarter of distributions with fuzzy min-entropy and \(2^k\) points there is no secure fuzzy extractor that uses less \(2^{\Theta (k)}\) bits of information about the distribution. We show an analogous result with stronger parameters for information-theoretic secure sketches. Secure sketches are frequently used to construct fuzzy extractors.

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Notes

  1. Fuzzy extractors were first designed as an information-theoretic primitive because of strong connections to randomness extraction and coding theory. An important application is in quantum key agreement which does not allow computational assumptions. Many computational constructions use an information-theoretic secure sketch [35, 36]. (Exceptions exist such as the universal constructions listed above and constructions for distributions with statistical properties beyond fuzzy min-entropy [1, 2, 9, 16].)

  2. Our result for secure sketches requires them to retain at least 5 bits of min-entropy about the input in comparison with [16] which required the sketch to maintain 3 bits of entropy.

  3. As in Sect. 3.3 the additional constraint that \( \gamma \le \beta - \textsf{log}\!\left( 2e\right) \) imposes an additive \(\textsf{log}\!\left( 2e\right) \) impact on the maximal fuzzy min-entropy that can be supported.

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  1. Department of Computer Science and Engineering, University of Connecticut, 371 Fairfield Way, Storrs, CT, 06269, USA

    Benjamin Fuller

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Correspondence to Benjamin Fuller.

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Communicated by S. D. Galbraith.

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The author thanks the reviewers for their helpful feedback and Luke Demarest and Alexander Russell for their helpful discussions. B.F. is supported by National Science Foundation Grants #2232813 and #2141033 and the Office of Naval Research. This manuscript has no associated data.

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Fuller, B. Impossibility of efficient information-theoretic fuzzy extraction. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01376-z

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