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To Label, or Not To Label (in Generic Groups)

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13509))

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Abstract

Generic groups are an important tool for analyzing the feasibility and in-feasibility of group-based cryptosystems. There are two distinct wide-spread versions of generic groups, Shoup’s and Maurer’s, the main difference being whether or not group elements are given explicit labels. The two models are often treated as equivalent. In this work, however, we demonstrate that the models are in fact quite different, and care is needed when stating generic group results:

  • We show that numerous textbook constructions are not captured by Maurer, but are captured by Shoup. In the other direction, any construction captured by Maurer is captured by Shoup.

  • For constructions that exist in both models, we show that security is equivalent for “single stage” games, but Shoup security is strictly stronger than Maurer security for some “multi-stage” games.

  • The existing generic group un-instantiability results do not apply to Maurer. We fill this gap with a new un-instantiability result.

  • We explain how the known black box separations between generic groups and identity-based encryption do not fully apply to Shoup, and resolve this by providing such a separation.

  • We give a new un-instantiability result for the algebraic group model.

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Notes

  1. 1.

    It appears it was never meant to be: Maurer discusses several classes of problems to consider in his model, capturing discrete log, DDH, and more exotic variants. But general cryptosystems are not covered by the classes of problems.

  2. 2.

    Plain PKE and MAC security are single-stage games.

  3. 3.

    Note that many works in the AGM starting from [FKL18] sometimes additionally add a random oracle, and these techniques can be used on the random oracle.

  4. 4.

    g, h could be created by \(\textsf{Gen}\), but we would need to get g, h to the adversary before the first query. We could consider a 1-time signature, where g, h would be included in the public key. Alternatively, we could consider a 2-time MAC, which includes g, h as part of each MAC, giving the adversary g, h in time for the second query.

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Authors and Affiliations

  1. NTT Research, Sunnyvale, USA

    Mark Zhandry

  2. Princeton University, Princeton, USA

    Mark Zhandry

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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Zhandry, M. (2022). To Label, or Not To Label (in Generic Groups). In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13509. Springer, Cham. https://doi.org/10.1007/978-3-031-15982-4_3

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