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Universal Reductions: Reductions Relative to Stateful Oracles

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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

We define a framework for analyzing the security of cryptographic protocols that makes minimal assumptions about what a “realistic model of computation is”. In particular, whereas classical models assume that the attacker is a (perhaps non-uniform) probabilistic polynomial-time algorithm, and more recent definitional approaches also consider quantum polynomial-time algorithms, we consider an approach that is more agnostic to what computational model is physically realizable.

Our notion of universal reductions models attackers as PPT algorithms having access to some arbitrary unbounded stateful Nature that cannot be rewound or restarted when queried multiple times. We also consider a more relaxed notion of universal reductions w.r.t. time-evolving, k-window, Natures that makes restrictions on Nature—roughly speaking, Nature’s behavior may depend on number of messages it has received and the content of the last \(k(\lambda )\)-messages (but not on “older” messages).

We present both impossibility results and general feasibility results for our notions, indicating to what extent the extended Church-Turing hypotheses are needed for a well-founded theory of Cryptography.

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Notes

  1. 1.

    Without getting too deep into Philosophy, it seems reasonable to argue that the Extended Church-Turing Hypothesis does not pass Popper’s falsifiability test [Pop05], as we do not have “shared ways of systematically determining” whether a probabilistic Turing machine cannot perform some task (as testified by the fact that the \(\textsf{P}\) v.s. \(\textsf{NP}\) problem is still open). As such, the statement of the hypothesis is no different from the classic example of “All men are mortal”, which according to Popper’s theory is not scientific as we do not have systematic procedures for deducing whether a person is immortal. This is in contrast to assumptions such as “Factoring products of random 1000-bit primes is hard for all physically realizable computation devices”, as we do have a systematic way of determining whether some such device manages to complete the task—simply run it.

  2. 2.

    For concreteness, and to simplify notation, we will model attackers as Turing machines so technically we are still relying on the (more reasonable) non-extended Church-Turing hypothesis. But we highlight that nothing in our treatment requires doing so and none of our results would change if we instead allowed any, even non-computable, attackers. See Sect. 3 for more discussion.

  3. 3.

    This model clearly oversimplifies as, say, \(n^{100}\) computation is not actually feasible. But we start off with a standard asymptotic treatment to get a model that is easy to work with. In practice, a more concrete treatment is desirable, but we leave this for future work.

  4. 4.

    We refer to such reductions as “universal” because they are agnostic to the computational resources of an attacker (and thus can be “universally” applied, independent of the attacker’s computational power). Additionally, on a technical level, and as we discuss in more detail shortly, considering security relative to a stateful entity is related to how security is defined in the framework for Universal Composability of Canetti [Can01].

  5. 5.

    Nevertheless, we note that all our results also hold if restricting the length of \(\rho \) to be polynomial.

  6. 6.

    We emphasize that Theorem 4 is ruling out also so-called “parameter-aware” black-box reductions [BBF13], where the reduction may depend on the success probability a of the attacker; note that Yao’s original reduction is parameter dependent—more specifically, the number of repetitions is required to be superlinear in the adversary’s success probability, and as shown in [LTW05] a dependency on the attackers success probability is inherent for black-box reductions. Theorem 4 rules out also such parameter-aware universal reductions and indeed rules out universal reductions that increase the success probability of the adversary even if assuming that the original attackers success probability is, say, \(\frac{1}{2}\).

  7. 7.

    There is a small subtlety here. Robustness is defined with respect to all previous transcripts, even exponentially long ones, so naively implementing this approach will not work since eventually we can include all possible strings y in the transcripts. Rather, the way we formalize this argument is to consider a \(\textsf{Nat}\) that only has “polynomial memory” and checks for repeated strings y in the most recent part of the transcript it is fed.

  8. 8.

    Again, we highlight that Theorem 5 rules out also “parameter-aware” reductions that depend on the success probability of the attacker—in fact, it rules out also reductions that only work if the underlying attacker’s success probability is 0.99. (As noted in [BBF13], Goldreich-Levin’s standard reduction is parameter-aware, and this is inherent as shown in [LTW05].).

  9. 9.

    In the definition of robust winning above, we require that the augmented adversary win a security game for every prefix \({\rho }\) that \(\textsf{Nat}\) may have previously seen, even those containing exponentially many messages. A natural alternative is to consider a notion of robust winning that considers only those prefixes with \(\textrm{poly}(\lambda )\) many messages; indeed our impossibilities and feasibilities can both be made to work in that setting, but at the expense of definitional complexity.

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Acknowledgements.

This work is supported in part by NSF CNS-2149305, NSF Award SATC-1704788, NSF Award RI-1703846, CNS-2128519, AFOSR Award FA9550-18-1-0267, and a JP Morgan Faculty Award. This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Cody Freitag’s work was done partially during an internship at NTT Research, and he is also supported in part by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2139899. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF, the United States Government, or DARPA.

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  1. Cornell Tech, New York, USA

    Benjamin Chan, Cody Freitag & Rafael Pass

  2. Tel-Aviv University, Tel Aviv-Yafo, Israel

    Rafael Pass

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Correspondence to Benjamin Chan .

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  1. Ruhr University Bochum, Bochum, Germany

    Eike Kiltz

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

    Vinod Vaikuntanathan

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Chan, B., Freitag, C., Pass, R. (2022). Universal Reductions: Reductions Relative to Stateful Oracles. 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_6

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