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**Published in:** LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

A secret-sharing scheme enables a dealer, holding a secret string, to distribute shares to parties such that only pre-defined authorized subsets of parties can reconstruct the secret. The collection of authorized sets is called an access structure. There is a huge gap between the best known upper bounds on the share size of a secret-sharing scheme realizing an arbitrary access structure and the best known lower bounds on the size of these shares. For an arbitrary n-party access structure, the best known upper bound on the share size is 2^{O(n)}. On the other hand, the best known lower bound on the total share size is much smaller, i.e., Ω(n²/log(n)) [Csirmaz, Studia Sci. Math. Hungar.]. This lower bound was proved more than 25 years ago and no major progress has been made since.
In this paper, we study secret-sharing schemes for k-hypergraphs, i.e., for access structures where all minimal authorized sets are of size exactly k (however, unauthorized sets can be larger). We consider the case where k is small, i.e., constant or at most log(n). The trivial upper bound for these access structures is O(n⋅ binom(n-1,k-1)) and this can be slightly improved. If there were efficient secret-sharing schemes for such k-hypergraphs (e.g., 2-hypergraphs or 3-hypergraphs), then we would be able to construct secret-sharing schemes for arbitrary access structures that are better than the best known schemes. Thus, understanding the share size required for k-hypergraphs is important. Prior to our work, the best known lower bound for these access structures was Ω(n log(n)), which holds already for graphs (i.e., 2-hypergraphs).
We improve this lower bound, proving a lower bound of Ω(n^{2-1/(k-1)}/k) on the total share size for some explicit k-hypergraphs, where 3 ≤ k ≤ log(n). For example, for 3-hypergraphs we prove a lower bound of Ω(n^{3/2}). For log(n)-hypergraphs, we prove a lower bound of Ω(n²/log(n)), i.e., we show that the lower bound of Csirmaz holds already when all minimal authorized sets are of size log(n). Our proof is simple and shows that the lower bound of Csirmaz holds for a simple variant of the access structure considered by Csirmaz. Using our results, we prove a near quadratic separation between the required share size for realizing an explicit access structure and the monotone circuit size describing the access structure, i.e., the share size in Ω(n²/log(n)) and the monotone circuit size is O(nlog(n)) (where the circuit has depth 3).

Amos Beimel. Lower Bounds for Secret-Sharing Schemes for k-Hypergraphs. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{beimel:LIPIcs.ITC.2023.16, author = {Beimel, Amos}, title = {{Lower Bounds for Secret-Sharing Schemes for k-Hypergraphs}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {16:1--16:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-271-6}, ISSN = {1868-8969}, year = {2023}, volume = {267}, editor = {Chung, Kai-Min}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.16}, URN = {urn:nbn:de:0030-drops-183440}, doi = {10.4230/LIPIcs.ITC.2023.16}, annote = {Keywords: Secret Sharing, Share Size, Lower Bounds, Monotone Circuits} }

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

A secret-sharing scheme allows to distribute a secret s among n parties such that only some predefined "authorized" sets of parties can reconstruct the secret s, and all other "unauthorized" sets learn nothing about s. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size 2^{n-o(n)} and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to 2^{0.994n+o(n)}, and this was further improved by several follow-ups accumulating in an upper bound of 1.5^{n+o(n)} (Applebaum and Nir, CRYPTO 2021). Following these advances, it is natural to ask whether these new approaches can lead to a truly sub-exponential upper-bound of 2^{n^{1-ε}} for some constant ε > 0, or even all the way down to polynomial upper-bounds.
In this paper, we relate this question to the complexity of computing monotone Boolean functions by monotone real circuits (MRCs) - a computational model that was introduced by Pudlák (J. Symb. Log., 1997) in the context of proof complexity. We introduce a new notion of "separable" MRCs that lies between monotone real circuits and monotone real formulas (MRFs). As our main results, we show that recent constructions of general secret-sharing schemes implicitly give rise to separable MRCs for general monotone functions of similar complexity, and that some monotone functions (in monotone NP) cannot be computed by sub-exponential size separable MRCs. Interestingly, it seems that proving similar lower-bounds for general MRCs is beyond the reach of current techniques.
We use this connection to obtain lower-bounds against a natural family of secret-sharing schemes, as well as new non-trivial upper-bounds for MRCs. Specifically, we conclude that recent approaches for secret-sharing schemes cannot achieve sub-exponential share size and that every monotone function can be realized by an MRC (or even MRF) of complexity 1.5^{n+o(n)}. To the best of our knowledge, this is the first improvement over the trivial 2^{n-o(n)} upper-bound. Along the way, we show that the recent constructions of general secret-sharing schemes implicitly give rise to Boolean formulas over slice functions and prove that such formulas can be simulated by separable MRCs of similar size. On a conceptual level, our paper continues the rich line of study that relates the share size of secret-sharing schemes to monotone complexity measures.

Benny Applebaum, Amos Beimel, Oded Nir, Naty Peter, and Toniann Pitassi. Secret Sharing, Slice Formulas, and Monotone Real Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{applebaum_et_al:LIPIcs.ITCS.2022.8, author = {Applebaum, Benny and Beimel, Amos and Nir, Oded and Peter, Naty and Pitassi, Toniann}, title = {{Secret Sharing, Slice Formulas, and Monotone Real Circuits}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {8:1--8:23}, 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.2022.8}, URN = {urn:nbn:de:0030-drops-156046}, doi = {10.4230/LIPIcs.ITCS.2022.8}, annote = {Keywords: Secret Sharing Schemes, Monotone Real Circuits} }

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**Published in:** LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

Motivated by the desire to bridge the utility gap between local and trusted curator models of differential privacy for practical applications, we initiate the theoretical study of a hybrid model introduced by "Blender" [Avent et al., USENIX Security '17], in which differentially private protocols of n agents that work in the local-model are assisted by a differentially private curator that has access to the data of m additional users. We focus on the regime where m ≪ n and study the new capabilities of this (m,n)-hybrid model. We show that, despite the fact that the hybrid model adds no significant new capabilities for the basic task of simple hypothesis-testing, there are many other tasks (under a wide range of parameters) that can be solved in the hybrid model yet cannot be solved either by the curator or by the local-users separately. Moreover, we exhibit additional tasks where at least one round of interaction between the curator and the local-users is necessary - namely, no hybrid model protocol without such interaction can solve these tasks. Taken together, our results show that the combination of the local model with a small curator can become part of a promising toolkit for designing and implementing differential privacy.

Amos Beimel, Aleksandra Korolova, Kobbi Nissim, Or Sheffet, and Uri Stemmer. The Power of Synergy in Differential Privacy: Combining a Small Curator with Local Randomizers. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 14:1-14:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{beimel_et_al:LIPIcs.ITC.2020.14, author = {Beimel, Amos and Korolova, Aleksandra and Nissim, Kobbi and Sheffet, Or and Stemmer, Uri}, title = {{The Power of Synergy in Differential Privacy: Combining a Small Curator with Local Randomizers}}, booktitle = {1st Conference on Information-Theoretic Cryptography (ITC 2020)}, pages = {14:1--14:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-151-1}, ISSN = {1868-8969}, year = {2020}, volume = {163}, editor = {Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.14}, URN = {urn:nbn:de:0030-drops-121195}, doi = {10.4230/LIPIcs.ITC.2020.14}, annote = {Keywords: differential privacy, hybrid model, private learning, local model} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In a recent paper, Chan et al. [SODA '19] proposed a relaxation of the notion of (full) memory obliviousness, which was introduced by Goldreich and Ostrovsky [J. ACM '96] and extensively researched by cryptographers. The new notion, differential obliviousness, requires that any two neighboring inputs exhibit similar memory access patterns, where the similarity requirement is that of differential privacy. Chan et al. demonstrated that differential obliviousness allows achieving improved efficiency for several algorithmic tasks, including sorting, merging of sorted lists, and range query data structures.
In this work, we continue the exploration of differential obliviousness, focusing on algorithms that do not necessarily examine all their input. This choice is motivated by the fact that the existence of logarithmic overhead ORAM protocols implies that differential obliviousness can yield at most a logarithmic improvement in efficiency for computations that need to examine all their input. In particular, we explore property testing, where we show that differential obliviousness yields an almost linear improvement in overhead in the dense graph model, and at most quadratic improvement in the bounded degree model. We also explore tasks where a non-oblivious algorithm would need to explore different portions of the input, where the latter would depend on the input itself, and where we show that such a behavior can be maintained under differential obliviousness, but not under full obliviousness. Our examples suggest that there would be benefits in further exploring which class of computational tasks are amenable to differential obliviousness.

Amos Beimel, Kobbi Nissim, and Mohammad Zaheri. Exploring Differential Obliviousness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{beimel_et_al:LIPIcs.APPROX-RANDOM.2019.65, author = {Beimel, Amos and Nissim, Kobbi and Zaheri, Mohammad}, title = {{Exploring Differential Obliviousness}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {65:1--65:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.65}, URN = {urn:nbn:de:0030-drops-112803}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.65}, annote = {Keywords: Differential Obliviousness, Differential Privacy, Oblivious RAM, Graph Property Testing} }

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