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

The bottleneck-complexity (BC) of secure multiparty computation (MPC) protocols is a measure of the maximum number of bits which are sent and received by any party in protocol. As the name suggests, the goal of studying BC-efficient protocols is to increase overall efficiency by making sure that the workload in the protocol is somehow "amortized" by the protocol participants.
Orlandi et al. [Orlandi et al., 2022] initiated the study of BC-efficient protocols from simple assumptions in the correlated randomness model and for semi-honest adversaries. In this work, we extend the study of [Orlandi et al., 2022] in two primary directions: (a) to a larger and more general class of functions and (b) to the information-theoretic setting.
In particular, we offer semi-honest secure protocols for the useful function classes of abelian programs, "read-k" non-abelian programs, and "read-k" generalized formulas.
Our constructions use a novel abstraction, called incremental function secret-sharing (IFSS), that can be instantiated with unconditional security or from one-way functions (with different efficiency trade-offs).

Hannah Keller, Claudio Orlandi, Anat Paskin-Cherniavsky, and Divya Ravi. MPC with Low Bottleneck-Complexity: Information-Theoretic Security and More. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 11:1-11:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{keller_et_al:LIPIcs.ITC.2023.11, author = {Keller, Hannah and Orlandi, Claudio and Paskin-Cherniavsky, Anat and Ravi, Divya}, title = {{MPC with Low Bottleneck-Complexity: Information-Theoretic Security and More}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {11:1--11:22}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.11}, URN = {urn:nbn:de:0030-drops-183391}, doi = {10.4230/LIPIcs.ITC.2023.11}, annote = {Keywords: Secure Multiparty Computation, Bottleneck Complexity, Information-theoretic} }

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**Published in:** LIPIcs, Volume 230, 3rd Conference on Information-Theoretic Cryptography (ITC 2022)

Innovative side-channel attacks have repeatedly exposed the secrets of cryptosystems. Benhamouda, Degwekar, Ishai, and Rabin (CRYPTO-2018) introduced local leakage resilience of secret-sharing schemes to study some of these vulnerabilities. In this framework, the objective is to characterize the unintended information revelation about the secret by obtaining independent leakage from each secret share. This work accurately quantifies the vulnerability of the additive secret-sharing scheme to local leakage attacks and its consequences for other secret-sharing schemes.
Consider the additive secret-sharing scheme over a prime field among k parties, where the secret shares are stored in their natural binary representation, requiring λ bits - the security parameter. We prove that the reconstruction threshold k = ω(log λ) is necessary to protect against local physical-bit probing attacks, improving the previous ω(log λ/log log λ) lower bound. This result is a consequence of accurately determining the distinguishing advantage of the "parity-of-parity" physical-bit local leakage attack proposed by Maji, Nguyen, Paskin-Cherniavsky, Suad, and Wang (EUROCRYPT-2021). Our lower bound is optimal because the additive secret-sharing scheme is perfectly secure against any (k-1)-bit (global) leakage and (statistically) secure against (arbitrary) one-bit local leakage attacks when k = ω(log λ).
Any physical-bit local leakage attack extends to (1) physical-bit local leakage attacks on the Shamir secret-sharing scheme with adversarially-chosen evaluation places, and (2) local leakage attacks on the Massey secret-sharing scheme corresponding to any linear code. In particular, for Shamir’s secret-sharing scheme, the reconstruction threshold k = ω(log λ) is necessary when the number of parties is n = O(λ log λ). Our analysis of the "parity-of-parity" attack’s distinguishing advantage establishes it as the best-known local leakage attack in these scenarios.
Our work employs Fourier-analytic techniques to analyze the "parity-of-parity" attack on the additive secret-sharing scheme. We accurately estimate an exponential sum that captures the vulnerability of this secret-sharing scheme to the parity-of-parity attack, a quantity that is also closely related to the "discrepancy" of the Irwin-Hall probability distribution.

Hemanta K. Maji, Hai H. Nguyen, Anat Paskin-Cherniavsky, Tom Suad, Mingyuan Wang, Xiuyu Ye, and Albert Yu. Tight Estimate of the Local Leakage Resilience of the Additive Secret-Sharing Scheme & Its Consequences. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{maji_et_al:LIPIcs.ITC.2022.16, author = {Maji, Hemanta K. and Nguyen, Hai H. and Paskin-Cherniavsky, Anat and Suad, Tom and Wang, Mingyuan and Ye, Xiuyu and Yu, Albert}, title = {{Tight Estimate of the Local Leakage Resilience of the Additive Secret-Sharing Scheme \& Its Consequences}}, booktitle = {3rd Conference on Information-Theoretic Cryptography (ITC 2022)}, pages = {16:1--16:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-238-9}, ISSN = {1868-8969}, year = {2022}, volume = {230}, editor = {Dachman-Soled, Dana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2022.16}, URN = {urn:nbn:de:0030-drops-164943}, doi = {10.4230/LIPIcs.ITC.2022.16}, annote = {Keywords: leakage resilience, additive secret-sharing, Shamir’s secret-sharing, physical-bit probing leakage attacks, Fourier analysis} }

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

Nearly all secret sharing schemes studied so far are linear or multi-linear schemes. Although these schemes allow to implement any monotone access structure, the share complexity, SC, may be suboptimal - there are access structures for which the gap between the best known lower bounds and best known multi-linear schemes is exponential.
There is growing evidence in the literature, that non-linear schemes can improve share complexity for some access structures, with the work of Beimel and Ishai (CCC '01) being among the first to demonstrate it. This motivates further study of non linear schemes.
We initiate a systematic study of polynomial secret sharing schemes (PSSS), where shares are (multi-variate) polynomials of secret and randomness vectors ~s,~r respectively over some finite field 𝔽_q. Our main hope is that the algebraic structure of polynomials would help obtain better lower bounds than those known for the general secret sharing. Some of the initial results we prove in this work are as follows.
On share complexity of polynomial schemes. First we study degree (at most) 1 in randomness variables ~r (where the degree of secret variables is unlimited). We have shown that for a large subclass of these schemes, there exist equivalent multi-linear schemes with O(n) share complexity overhead. Namely, PSSS where every polynomial misses monomials of exact degree c≥ 2 in ~s and 0 in ~r, and PSSS where all polynomials miss monomials of exact degree ≥ 1 in ~s and 1 in ~r. This translates the known lower bound of Ω(n^{log(n)}) for multi linear schemes onto a class of schemes strictly larger than multi linear schemes, to contrast with the best Ω(n²/log(n)) bound known for general schemes, with no progress since 94'. An observation in the positive direction we make refers to the share complexity (per bit) of multi linear schemes (polynomial schemes of total degree 1). We observe that the scheme by Liu et. al obtaining share complexity O(2^{0.994n}) can be transformed into a multi-linear scheme with similar share complexity per bit, for sufficiently long secrets. For the next natural degree to consider, 2 in ~r, we have shown that PSSS where all share polynomials are of exact degree 2 in ~r (without exact degree 1 in ~r monomials) where 𝔽_q has odd characteristic, can implement only trivial access structures where the minterms consist of single parties.
Obtaining improved lower bounds for degree-2 in ~r PSSS, and even arbitrary degree-1 in ~r PSSS is left as an interesting open question.
On the randomness complexity of polynomial schemes. We prove that for every degree-2 polynomial secret sharing scheme, there exists an equivalent degree-2 scheme with identical share complexity with randomness complexity, RC, bounded by 2^{poly(SC)}. For general PSSS, we obtain a similar bound on RC (preserving SC and 𝔽_q but not degree). So far, bounds on randomness complexity were known only for multi linear schemes, demonstrating that RC ≤ SC is always achievable. Our bounds are not nearly as practical as those for multi-linear schemes, and should be viewed as a proof of concept. If a much better bound for some degree bound d=O(1) is obtained, it would lead directly to super-polynomial counting-based lower bounds for degree-d PSSS over constant-sized fields. Another application of low (say, polynomial) randomness complexity is transforming polynomial schemes with polynomial-sized (in n) algebraic formulas C(~s,~r) for each share, into a degree-3 scheme with only polynomial blowup in share complexity, using standard randomizing polynomials constructions.

Anat Paskin-Cherniavsky and Radune Artiom. On Polynomial Secret Sharing Schemes. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{paskincherniavsky_et_al:LIPIcs.ITC.2020.12, author = {Paskin-Cherniavsky, Anat and Artiom, Radune}, title = {{On Polynomial Secret Sharing Schemes}}, booktitle = {1st Conference on Information-Theoretic Cryptography (ITC 2020)}, pages = {12:1--12:21}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.12}, URN = {urn:nbn:de:0030-drops-121174}, doi = {10.4230/LIPIcs.ITC.2020.12}, annote = {Keywords: Secret sharing, polynomial, lower bounds, linear program} }

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