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Beyond Uber: Instantiating Generic Groups via PGGs

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Theory of Cryptography (TCC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13749))

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Abstract

The generic-group model (GGM) has been very successful in making the analyses of many cryptographic assumptions and protocols tractable. It is, however, well known that the GGM is “uninstantiable,” i.e., there are protocols secure in the GGM that are insecure when using any real-world group. This motivates the study of standard-model notions formalizing that a real-world group in some sense “looks generic.”

We introduce a standard-model definition called pseudo-generic group (PGG), where we require exponentiations with base an (initially) unknown group generator to result in random-looking group elements. In essence, our framework delicately lifts the influential notion of Universal Computational Extractors of Bellare, Hoang, and Keelveedhi (BHK, CRYPTO 2013) to a setting where the underlying ideal reference object is a generic group. The definition we obtain simultaneously generalizes the Uber assumption family, as group exponents no longer need to be polynomially induced. At the core of our definitional contribution is a new notion of algebraic unpredictability, which reinterprets the standard Schwartz–Zippel lemma as a restriction on sources. We prove the soundness of our definition in the GGM with auxiliary-input (AI-GGM).

Our remaining results focus on applications of PGGs. We first show that PGGs are indeed a generalization of Uber. We then present a number of applications in settings where exponents are not polynomially induced. In particular we prove that simple variants of ElGamal meet several advanced security goals previously achieved only by complex and inefficient schemes. We also show that PGGs imply UCEs for split sources, which in turn are sufficient in several applications. As corollaries of our AI-GGM feasibility, we obtain the security of all these applications in the presence of preprocessing attacks.

Some of our implications utilize a novel type of hash function, which we call linear-dependence destroyers (LDDs) and use to convert standard into algebraic unpredictability. We give an LDD for low-degree sources, and establish their plausibility for all sources by showing, via a compression argument, that random functions meet this definition.

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Notes

  1. 1.

    An alternative formulation of the GGM is given by Maurer [45]; we follow Shoup’s presentation in this paper. Relations and comparisons between different flavors of the GGM are discussed in the recent work [56].

  2. 2.

    Similar work on the algebraic-group model (discussed later) was first carried out in simple groups [37] and later in bilinear ones [5].

  3. 3.

    Recently, Bartusek, Ma, and Zhandry (BMZ) [4] studied the “fixed-generator” and “random-generator” settings in group-based assumptions. We necessarily work in the latter since, as we shall see, otherwise attacks arise.

  4. 4.

    The idea is to have assumptions and models organized into a hierarchy, where higher levels justify lower ones and, conversely, proving a scheme secure at some level shows that it meets higher ones as well. This allows us to identify precisely how strong an assumption is needed for a given application. Moreover, proving security of a scheme at a lower level typically gives more insight into its inner workings.

  5. 5.

    We note that our understanding of the role played by the AGM in assessing the hardness of group-related assumptions is evolving in light of recent works [43, 56].

  6. 6.

    Note that \(\sigma \) can be lazily sampled, so that the game runs in polynomial time.

  7. 7.

    For instance, one could allow for more expressive forms of post-processing. However, we have not yet been able to find applications of this wider class of sources.

  8. 8.

    On the other hand, it is unclear how to rule this attack out using an extended notion of algebraic unpredictability that takes operation queries into account.

  9. 9.

    In particular, this model allows a restricted class of sources that leak arbitrary information (without any unpredictability requirements), as long as the sampling of the exponents is unpredictable (e.g., random, as is the case for the DLP).

  10. 10.

    Distinguish \((g, g^x, g^y, g^{xy})\) from \((g, g^x, g^y, g^z)\) for a random generator g, unpredictable x, and random y and z.

  11. 11.

    Accordingly, we also impose a statistical notion of unpredictability on sources by giving predictors access to the full table.

  12. 12.

    Interestingly, this simplification provides another avenue to circumvent iO-based attacks that exploit repetitions in \(\textbf{x}\).

  13. 13.

    We note, however, that in iterative constructions of block-ciphers from hash functions [12], or indeed in domain extenders for hash functions [52], adaptive calls to the hash function seem to be necessary.

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Acknowledgments

We thank Sogol Mazaheri for collaborating in the early stages of this work. We also thank anonymous reviewers who helped improve the presentation of our results. Pooya Farshim was supported in part by EPSRC grant EP/V034065/1. Patrick Harasser was funded by the Deutsche Forschungsgemeinschaft (DFG) – SFB 1119 – 236615297. Adam O’Neill is supported in part by a gift from Cisco.

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Authors and Affiliations

  1. IRIF, CNRS, Paris, France

    Balthazar Bauer

  2. IOHK and Durham University, Durham, UK

    Pooya Farshim

  3. Technische Universität Darmstadt, Darmstadt, Germany

    Patrick Harasser

  4. Manning College of Information and Computer Sciences, University of Massachusetts Amherst, Amherst, USA

    Adam O’Neill

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    Eike Kiltz

  2. Massachusetts Institute of Technology, Cambridge, MA, USA

    Vinod Vaikuntanathan

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Bauer, B., Farshim, P., Harasser, P., O’Neill, A. (2022). Beyond Uber: Instantiating Generic Groups via PGGs. In: Kiltz, E., Vaikuntanathan, V. (eds) Theory of Cryptography. TCC 2022. Lecture Notes in Computer Science, vol 13749. Springer, Cham. https://doi.org/10.1007/978-3-031-22368-6_8

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