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Security Definitions for Hash Functions: Combining UCE and Indifferentiability

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Security and Cryptography for Networks (SCN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11035))

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Abstract

Hash functions are one of the most important cryptographic primitives, but their desired security properties have proven to be remarkably hard to formalize. To prove the security of a protocol using a hash function, nowadays often the random oracle model (ROM) is used due to its simplicity and its strong security guarantees. Moreover, hash function constructions are commonly proven to be secure by showing them to be indifferentiable from a random oracle when using an ideal compression function. However, it is well known that no hash function realizes a random oracle and no real compression function realizes an ideal one.

As an alternative to the ROM, Bellare et al. recently proposed the notion of universal computational extractors (UCE). This notion formalizes that a family of functions “behaves like a random oracle” for “real-world” protocols while avoiding the general impossibility results. However, in contrast to the indifferentiability framework, UCE is formalized as a multi-stage game without clear composition guarantees.

As a first contribution, we introduce context-restricted indifferentiability (CRI), a generalization of indifferentiability that allows us to model that the random oracle does not compose generally but can only be used within a well-specified set of protocols run by the honest parties, thereby making the provided composition guarantees explicit. We then show that UCE and its variants can be phrased as a special case of CRI. Moreover, we show how our notion of CRI leads to generalizations of UCE. As a second contribution, we prove that the hash function constructed by Merkle-Damgåd satisfies one of the well-known UCE variants, if we assume that the compression function satisfies one of our generalizations of UCE, basing the overall security on a plausible assumption. This result further validates the Merkle-Damgård construction and shows that UCE-like assumptions can serve both as a valid reference point for modular protocol analyses, as well as for the design of hash functions, linking those two aspects in a framework with explicit composition guarantees.

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Notes

  1. 1.

    The set of distinguishers \(\mathcal {D}\) needs to be closed under emulation of a resource and converter. The set of converters needs to be closed under sequential composition and parallel composition with the identity converter.

  2. 2.

    The choice to consider a private random oracle stems from the fact that in the UCE framework the hash key is just chosen uniformly at random instead of allowing an arbitrary efficient simulator with access to the random oracle to generate this key.

  3. 3.

    We would like to stress that while split-security was originally introduced for the computational setting, it is still a natural class to consider even when combined with a statistical unpredictability notion.

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Authors and Affiliations

  1. Department of Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Daniel Jost & Ueli Maurer

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  1. Daniel Jost
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  2. Ueli Maurer
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Correspondence to Daniel Jost .

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Editors and Affiliations

  1. University of Catania, Catania, Italy

    Dario Catalano

  2. University of Salerno, Fisciano, Italy

    Roberto De Prisco

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Jost, D., Maurer, U. (2018). Security Definitions for Hash Functions: Combining UCE and Indifferentiability. In: Catalano, D., De Prisco, R. (eds) Security and Cryptography for Networks. SCN 2018. Lecture Notes in Computer Science(), vol 11035. Springer, Cham. https://doi.org/10.1007/978-3-319-98113-0_5

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  • DOI: https://doi.org/10.1007/978-3-319-98113-0_5

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