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Cloning Games: A General Framework for Unclonable Primitives

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14085))

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Abstract

The powerful no-cloning principle of quantum mechanics can be leveraged to achieve interesting primitives, referred to as unclonable primitives, that are impossible to achieve classically. In the past few years, we have witnessed a surge of new unclonable primitives. While prior works have mainly focused on establishing feasibility results, another equally important direction, that of understanding the relationship between different unclonable primitives is still in its nascent stages. Moving forward, we need a more systematic study of unclonable primitives.

To this end, we introduce a new framework called cloning games. This framework captures many fundamental unclonable primitives such as quantum money, copy-protection, unclonable encryption, single-decryptor encryption, and many more. By reasoning about different types of cloning games, we obtain many interesting implications to unclonable cryptography, including the following:

  1. 1.

    We obtain the first construction of information-theoretically secure single-decryptor encryption in the one-time setting.

  2. 2.

    We construct unclonable encryption in the quantum random oracle model based on BB84 states, improving upon the previous work, which used coset states. Our work also provides a simpler security proof for the previous work.

  3. 3.

    We construct copy-protection for single-bit point functions in the quantum random oracle model based on BB84 states, improving upon the previous work, which used coset states, and additionally, providing a simpler proof.

  4. 4.

    We establish a relationship between different challenge distributions of copy-protection schemes and single-decryptor encryption schemes.

  5. 5.

    Finally, we present a new construction of one-time encryption with certified deletion.

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Notes

  1. 1.

    An (impatient) reader familiar with the above primitives could skip directly to Sect. 1.1. We still recommend going through the discussion before reading Sect. 1.1.

  2. 2.

    We note that the security in the literature is stated slightly differently. \(\mathcal {A}\) is given encryption of a message \(m_b\), where b is picked uniformly at random and \(\mathcal {B},\mathcal {C}\) are expected to simultaneously guess b. We note that this formulation is identical to the above formulation.

  3. 3.

    The original formulation by [Aar09] is weaker than what is stated here. We follow the game-based definition by [CMP20].

  4. 4.

    We only consider classes of unlearnable functions which are functions that cannot be efficiently learned from its input and output behavior. Copy-protection for learnable functions is impossible.

  5. 5.

    As far as we know, all unclonable primitives can be cast as cloning games by making reasonable minor modifications to the framework.

  6. 6.

    With the exception of copy-protection of point functions [CMP20, AKL+22].

  7. 7.

    In contrast, unclonability allows \(\mathcal {B}\) and \(\mathcal {C}\) to both learn the secret key before passing verification. Note that we use the word “weaker” qualitatively in this sentence, and do not claim that unclonability implies the existence of certified deletion in general.

  8. 8.

    Although this is not true for the initial definition we use to introduce cloning games, deletion games are captured after considering a natural extension of cloning games, where \(\mathcal {B}\) and \(\mathcal {C}\) are not treated symmetrically.

  9. 9.

    Please refer to the definition of unclonability of an unclonable encryption scheme in the introduction.

  10. 10.

    Our theorem is more general than what is stated here; refer to the full version for more details.

  11. 11.

    One example of trivial success probability being large is non-local decision games, where \(\mathcal {B}\) and \(\mathcal {C}\) try to produce binary answers simultaneously.

  12. 12.

    There can be multiple eigenvectors with the same eigenvalues. In the overview, we assume that eigenvalues are unique.

  13. 13.

    There is a one-to-one mapping between \(\{|\phi _p\rangle ,\{|\psi _q\rangle \}\) and the vectors \(\{|\psi ^\mathcal {B}_i\rangle ,\{|\psi ^\mathcal {C}_j\rangle \}\) defined in the Jordan’s lemma.

  14. 14.

    Unlike our result on single-decryptor encryption, which asks for the usual, stronger property of unclonability, here we do not need the simultaneous version of the Goldreich-Levin Lemma because we are in the weaker, certified deletion setting.

  15. 15.

    A simplified version of it without additional properties. The authors show in [BI20] that the construction already satisfies the stronger notion of unclonable indistinguishable security, yet the proof is more involved.

  16. 16.

    This is in order to simplify the notation for the rest of the algorithms. We will sometimes make this inclusion explicit, and other times it is understood implicitly.

  17. 17.

    We assume statistical correctness here.

  18. 18.

    Here we make the natural assumption that correctness and security are defined with respect to the same distribution \(\mathcal {D}'_f\). Intuitively, the scheme should protect against cloning the functionality of the honest evaluator.

  19. 19.

    Note that this captures the average-input correctness as opposed to per-input correctness.

  20. 20.

    Although (2) was defined in a slightly different way in [BL20], the difference is inconsequential, and our version has been used in follow-up works such as [AKL+22]. We also mention that [BL20] considered perfect correctness.

  21. 21.

    We omit the message distribution due to the lack of message.

  22. 22.

    This requirement can be lifted by extending the definition of stateful cloning games and having \(\widetilde{\textsf{GenC}}\) know the random coins of \(\textsf{GenT}'\) (in this case m). We keep the syntax simple for there is no known application to the more general case.

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Acknowledgements

PA and FK are supported by a gift from Cisco.

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Authors and Affiliations

  1. University of California, Santa Barbara, CA, USA

    Prabhanjan Ananth & Fatih Kaleoglu

  2. Simons Institute for the Theory of Computing, Berkeley, CA, USA

    Qipeng Liu

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  1. Prabhanjan Ananth
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Correspondence to Prabhanjan Ananth .

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  1. Rambus Inc., San Jose, CA, USA

    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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Ananth, P., Kaleoglu, F., Liu, Q. (2023). Cloning Games: A General Framework for Unclonable Primitives. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14085. Springer, Cham. https://doi.org/10.1007/978-3-031-38554-4_3

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