Abstract
We study communication complexity in computational settings where bad inputs may exist, but they should be hard to find for any computationally bounded adversary.
We define a model where there is a source of public randomness but the inputs are chosen by a computationally bounded adversarial participant after seeing the public randomness. We show that breaking the known communication lower bounds of the private coins model in this setting is closely connected to known cryptographic assumptions. We consider the simultaneous messages model and the interactive communication model and show that for any non trivial predicate (with no redundant rows, such as equality):
-
1.
Breaking the \( \varOmega (\sqrt{n}) \) bound in the simultaneous message case or the \( \varOmega (\log n) \) bound in the interactive communication case, implies the existence of distributional collision-resistant hash functions (dCRH). This is shown using techniques from Babai and Kimmel [BK97]. Note that with a CRH the lower bounds can be broken.
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2.
There are no protocols of constant communication in this preset randomness settings (unlike the plain public randomness model).
The other model we study is that of a stateful “free talk”, where participants can communicate freely before the inputs are chosen and may maintain a state, and the communication complexity is measured only afterwards. We show that efficient protocols for equality in this model imply secret key-agreement protocols in a constructive manner. On the other hand, secret key-agreement protocols imply optimal (in terms of error) protocols for equality.
Research supported in part by grants from the Israel Science Foundation (no. 2686/20) and by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness. The second author is incumbent of the Judith Kleeman Professorial Chair.
The full version of this paper is available at ia.cr/2022/312.
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Notes
- 1.
For the probabilistic version they are required to succeed with constant high probability.
- 2.
The probability is over the choices of the random bits.
- 3.
The probabilities are over the sampling of a hash function among the functions family.
- 4.
See in [BK97] also the similar proof of Bourgain and Wigderson.
- 5.
Bottesch et al. actually discuss quantum variants of the SM model and give the simpler proof for our classical case as a warm-up.
- 6.
Can be generalized to a sample from any known efficient distribution.
- 7.
See Canetti et al. [CHS05] for better parameters.
- 8.
By switching the order of quantifiers, they require one polynomial for any adversary and not that for any adversary there exists a polynomial. See the comparison in [BHKY19].
- 9.
Functions where it is hard to sample uniformly from \( h^{-1}(h(x)) \) for random \( x \). Such functions are known to exist if and only if one-way functions exist [IL89]. (in contrast to dCRH).
- 10.
\(*\) is a don’t care symbol, see [BLV18] for a comparison of “total vs. partial predicates”.
- 11.
CRHs exist in this model.
- 12.
See also Hardt and Woodruff [HW13] who proved robustness limitations for linear functions.
- 13.
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Acknowledgements
We thank Shahar Dobzinski, Ilan Komargodski, Guy Rothlbum and Eylon Yogev for useful discussions and suggestions and the Crypto 2022 referees for the helpful comments and questions.
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Cohen, S.P., Naor, M. (2022). Low Communication Complexity Protocols, Collision Resistant Hash Functions and Secret Key-Agreement Protocols. 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_9
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