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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

We consider the following question of bounded simultaneous messages (BSM) protocols: Can computationally unbounded Alice and Bob evaluate a function f(x,y) of their inputs by sending polynomial-size messages to a computationally bounded Carol? The special case where f is the mod-2 inner-product function and Carol is bounded to AC⁰ has been studied in previous works. The general question can be broadly motivated by applications in which distributed computation is more costly than local computation.
In this work, we initiate a more systematic study of the BSM model, with different functions f and computational bounds on Carol. In particular, we give evidence against the existence of BSM protocols with polynomial-size Carol for naturally distributed variants of NP-complete languages.

Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Sruthi Sekar. Bounded Simultaneous Messages. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bogdanov_et_al:LIPIcs.FSTTCS.2023.23, author = {Bogdanov, Andrej and Dinesh, Krishnamoorthy and Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Sekar, Sruthi}, title = {{Bounded Simultaneous Messages}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.23}, URN = {urn:nbn:de:0030-drops-193961}, doi = {10.4230/LIPIcs.FSTTCS.2023.23}, annote = {Keywords: Simultaneous Messages, Instance Hiding, Algebraic degree, Preprocessing, Lower Bounds} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A pair of sources X, Y over {0,1}ⁿ are k-indistinguishable if their projections to any k coordinates are identically distributed. Can some AC^0 function distinguish between two such sources when k is big, say k = n^{0.1}? Braverman’s theorem (Commun. ACM 2011) implies a negative answer when X is uniform, whereas Bogdanov et al. (Crypto 2016) observe that this is not the case in general.
We initiate a systematic study of this question for natural classes of low-complexity sources, including ones that arise in cryptographic applications, obtaining positive results, negative results, and barriers. In particular:
- There exist Ω(√n)-indistinguishable X, Y, samplable by degree-O(log n) polynomial maps (over F₂) and by poly(n)-size decision trees, that are Ω(1)-distinguishable by OR.
- There exists a function f such that all f(d, ε)-indistinguishable X, Y that are samplable by degree-d polynomial maps are ε-indistinguishable by OR for all sufficiently large n. Moreover, f(1, ε) = ⌈log(1/ε)⌉ + 1 and f(2, ε) = O(log^{10}(1/ε)).
- Extending (weaker versions of) the above negative results to AC^0 distinguishers would require settling a conjecture of Servedio and Viola (ECCC 2012). Concretely, if every pair of n^{0.9}-indistinguishable X, Y that are samplable by linear maps is ε-indistinguishable by AC^0 circuits, then the binary inner product function can have at most an ε-correlation with AC^0 ◦ ⊕ circuits.
Finally, we motivate the question and our results by presenting applications of positive results to low-complexity secret sharing and applications of negative results to leakage-resilient cryptography.

Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Akshayaram Srinivasan. Bounded Indistinguishability for Simple Sources. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2022.26, author = {Bogdanov, Andrej and Dinesh, Krishnamoorthy and Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Srinivasan, Akshayaram}, title = {{Bounded Indistinguishability for Simple Sources}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.26}, URN = {urn:nbn:de:0030-drops-156223}, doi = {10.4230/LIPIcs.ITCS.2022.26}, annote = {Keywords: Pseudorandomness, bounded indistinguishability, complexity of sampling, constant-depth circuits, secret sharing, leakage-resilient cryptography} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

It is a classical result that the inner product function cannot be computed by an AC⁰ circuit [Merrick L. Furst et al., 1981; Miklós Ajtai, 1983; Johan Håstad, 1986]. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output n + n/(log^{ω(1)} n) bits. Our methods extend to many other functions, including pseudorandom functions, and imply a (weak but nontrivial) limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the main conjecture with the question of learning AC⁰ under simple input distributions.

Yuval Filmus, Yuval Ishai, Avi Kaplan, and Guy Kindler. Limits of Preprocessing. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 17:1-17:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filmus_et_al:LIPIcs.CCC.2020.17, author = {Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Kindler, Guy}, title = {{Limits of Preprocessing}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {17:1--17:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.17}, URN = {urn:nbn:de:0030-drops-125697}, doi = {10.4230/LIPIcs.CCC.2020.17}, annote = {Keywords: circuit, communication complexity, IPPP, preprocessing, PRF, simultaneous messages} }