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Searching for ELFs in the Cryptographic Forest

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14371))

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Abstract

Extremely Lossy Functions (ELFs) are families of functions that, depending on the choice during key generation, either operate in injective mode or instead have only a polynomial image size. The choice of the mode is indistinguishable to an outsider. ELFs were introduced by Zhandry (Crypto 2016) and have been shown to be very useful in replacing random oracles in a number of applications.

One open question is to determine the minimal assumption needed to instantiate ELFs. While all constructions of ELFs depend on some form of exponentially-secure public-key primitive, it was conjectured that exponentially-secure secret-key primitives, such as one-way functions, hash functions or one-way product functions, might be sufficient to build ELFs. In this work we answer this conjecture mostly negative: We show that no primitive, which can be derived from a random oracle (which includes all secret-key primitives mentioned above), is enough to construct even moderately lossy functions in a black-box manner. However, we also show that (extremely) lossy functions themselves do not imply public-key cryptography, leaving open the option to build ELFs from some intermediate primitive between the classical categories of secret-key and public-key cryptography. (The full version can be found at https://eprint.iacr.org/2023/1403.)

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Notes

  1. 1.

    ELFs can be distinguished efficiently using a super-logarithmic amount of non-determinism. It is consistent with our knowledge, however, that NP with an super-logarithmic amount of non-determinism is solvable in polynomial time while polynomially-secure cryptographic primitives exist. Any construction of ELFs from polynomially-secure cryptographic primitives would therefore change our understanding of NP-hardness.

  2. 2.

    For moderately lossy function we could actually use \(\lambda /4\) but for compatibility to the extremely lossy case it is convenient to use \(\lambda /5\) already here.

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Acknowledgments

We thank the anonymous reviewers for valuable comments.

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1119 - 236615297 and by the German Federal Ministry of Education and Research and the Hessian Ministry of Higher Education, Research, Science and the Arts within their joint support of the National Research Center for Applied Cybersecurity ATHENE.

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  1. Cryptoplexity, Technische Universität Darmstadt, Darmstadt, Germany

    Marc Fischlin & Felix Rohrbach

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  1. Marc Fischlin
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Correspondence to Marc Fischlin .

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    Guy Rothblum

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    Hoeteck Wee

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Fischlin, M., Rohrbach, F. (2023). Searching for ELFs in the Cryptographic Forest. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14371. Springer, Cham. https://doi.org/10.1007/978-3-031-48621-0_8

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