26 Search Results for "Ishai, Yuval"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.16
A Characterization of Optimal-Rate Linear Homomorphic Secret Sharing Schemes, and Applications

Authors: Keller Blackwell and Mary Wootters

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
A Homomorphic Secret Sharing (HSS) scheme is a secret-sharing scheme that shares a secret x among s servers, and additionally allows an output client to reconstruct some function f(x), using information that can be locally computed by each server. A key parameter in HSS schemes is download rate, which quantifies how much information the output client needs to download from each server. Recent work (Fosli, Ishai, Kolobov, and Wootters, ITCS 2022) established a fundamental limitation on the download rate of linear HSS schemes for computing low-degree polynomials, and gave an example of HSS schemes that meet this limit. In this paper, we further explore optimal-rate linear HSS schemes for polynomials. Our main result is a complete characterization of such schemes, in terms of a coding-theoretic notion that we introduce, termed optimal labelweight codes. We use this characterization to answer open questions about the amortization required by HSS schemes that achieve optimal download rate. In more detail, the construction of Fosli et al. required amortization over 𝓁 instances of the problem, and only worked for particular values of 𝓁. We show that - perhaps surprisingly - the set of 𝓁’s for which their construction works is in fact nearly optimal, possibly leaving out only one additional value of 𝓁. We show this by using our coding-theoretic characterization to prove a necessary condition on the 𝓁’s admitting optimal-rate linear HSS schemes. We then provide a slightly improved construction of optimal-rate linear HSS schemes, where the set of allowable 𝓁’s is optimal in even more parameter settings. Moreover, based on a connection to the MDS conjecture, we conjecture that our construction is optimal for all parameter regimes.
Cite as

Keller Blackwell and Mary Wootters. A Characterization of Optimal-Rate Linear Homomorphic Secret Sharing Schemes, and Applications. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blackwell_et_al:LIPIcs.ITCS.2024.16,
  author =	{Blackwell, Keller and Wootters, Mary},
  title =	{{A Characterization of Optimal-Rate Linear Homomorphic Secret Sharing Schemes, and Applications}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{16:1--16:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.16},
  URN =		{urn:nbn:de:0030-drops-195447},
  doi =		{10.4230/LIPIcs.ITCS.2024.16},
  annote =	{Keywords: Error Correcting Codes, Homomorphic Secret Sharing}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.23
Bounded Simultaneous Messages

Authors: Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Sruthi Sekar

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

Abstract
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.
Cite as

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-dev.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}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.56
Robustness for Space-Bounded Statistical Zero Knowledge

Authors: Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively.
Cite as

Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang. Robustness for Space-Bounded Statistical Zero Knowledge. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 56:1-56:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{allender_et_al:LIPIcs.APPROX/RANDOM.2023.56,
  author =	{Allender, Eric and Gray, Jacob and Mutreja, Saachi and Tirumala, Harsha and Wang, Pengxiang},
  title =	{{Robustness for Space-Bounded Statistical Zero Knowledge}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{56:1--56:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.56},
  URN =		{urn:nbn:de:0030-drops-188815},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.56},
  annote =	{Keywords: Interactive Proofs}
}
Document
DOI: 10.4230/LIPIcs.ITC.2023.14
Locally Covert Learning

Authors: Justin Holmgren and Ruta Jawale

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

Abstract
The goal of a covert learning algorithm is to learn a function f by querying it, while ensuring that an adversary, who sees all queries and their responses, is unable to (efficiently) learn any more about f than they could learn from random input-output pairs. We focus on a relaxation that we call local covertness, in which queries are distributed across k servers and we only limit what is learnable by k - 1 colluding servers. For any constant k, we give a locally covert algorithm for efficiently learning any Fourier-sparse function (technically, our notion of learning is improper, agnostic, and with respect to the uniform distribution). Our result holds unconditionally and for computationally unbounded adversaries. Prior to our work, such an algorithm was known only for the special case of O(log n)-juntas, and only with k = 2 servers [Yuval Ishai et al., 2019]. Our main technical observation is that the original Goldreich-Levin algorithm only utilizes i.i.d. pairs of correlated queries, where each half of every pair is uniformly random. We give a simple generalization of this algorithm in which pairs are replaced by k-tuples in which any k - 1 components are jointly uniform. The cost of this generalization is that the number of queries needed grows exponentially with k.
Cite as

Justin Holmgren and Ruta Jawale. Locally Covert Learning. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{holmgren_et_al:LIPIcs.ITC.2023.14,
  author =	{Holmgren, Justin and Jawale, Ruta},
  title =	{{Locally Covert Learning}},
  booktitle =	{4th Conference on Information-Theoretic Cryptography (ITC 2023)},
  pages =	{14:1--14:12},
  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.2023.14},
  URN =		{urn:nbn:de:0030-drops-183421},
  doi =		{10.4230/LIPIcs.ITC.2023.14},
  annote =	{Keywords: learning theory, adversarial machine learning, zero knowledge, Fourier analysis of boolean functions, Goldreich-Levin algorithm, Kushilevitz-Mansour algorithm}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.23
On Low-End Obfuscation and Learning

Authors: Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, and Gal Yehuda

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Most recent works on cryptographic obfuscation focus on the high-end regime of obfuscating general circuits while guaranteeing computational indistinguishability between functionally equivalent circuits. Motivated by the goals of simplicity and efficiency, we initiate a systematic study of "low-end" obfuscation, focusing on simpler representation models and information-theoretic notions of security. We obtain the following results. - Positive results via "white-box" learning. We present a general technique for obtaining perfect indistinguishability obfuscation from exact learning algorithms that are given restricted access to the representation of the input function. We demonstrate the usefulness of this approach by obtaining simple obfuscation for decision trees and multilinear read-k arithmetic formulas. - Negative results via PAC learning. A proper obfuscation scheme obfuscates programs from a class C by programs from the same class. Assuming the existence of one-way functions, we show that there is no proper indistinguishability obfuscation scheme for k-CNF formulas for any constant k ≥ 3; in fact, even obfuscating 3-CNF by k-CNF is impossible. This result applies even to computationally secure obfuscation, and makes an unexpected use of PAC learning in the context of negative results for obfuscation. - Separations. We study the relations between different information-theoretic notions of indistinguishability obfuscation, giving cryptographic evidence for separations between them.
Cite as

Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, and Gal Yehuda. On Low-End Obfuscation and Learning. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 23:1-23:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{boyle_et_al:LIPIcs.ITCS.2023.23,
  author =	{Boyle, Elette and Ishai, Yuval and Meyer, Pierre and Robere, Robert and Yehuda, Gal},
  title =	{{On Low-End Obfuscation and Learning}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{23:1--23:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.23},
  URN =		{urn:nbn:de:0030-drops-175265},
  doi =		{10.4230/LIPIcs.ITCS.2023.23},
  annote =	{Keywords: Indistinguishability obfuscation, cryptography, learning}
}
Document
DOI: 10.4230/LIPIcs.ITC.2022.10
Refuting the Dream XOR Lemma via Ideal Obfuscation and Resettable MPC

Authors: Saikrishna Badrinarayanan, Yuval Ishai, Dakshita Khurana, Amit Sahai, and Daniel Wichs

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

Abstract
We provide counterexamples to the "dream" version of Yao’s XOR Lemma. In particular, we put forward explicit candidates for hard predicates, such that the advantage of predicting the XOR of many independent copies does not decrease beyond some fixed negligible function, even as the number of copies gets arbitrarily large. We provide two such constructions: - Our first construction is in the ideal obfuscation model (alternatively, assuming virtual black-box obfuscation for a concrete class of circuits). It develops a general framework that may be of broader interest, and allows us to embed an instance of a resettably-secure multiparty computation protocol into a one-way function. Along the way, we design the first resettably-secure multiparty computation protocol for general functionalities in the plain model with super-polynomial simulation, under standard assumptions. - The second construction relies on public-coin differing-inputs obfuscation (PCdiO) along with a certain form of hash-function security called extended second-preimage resistance (ESPR). It starts with a previously known counterexample to the dream direct-product hardness amplification based on ESPR, and uses PCdiO to upgrade it into a counterexample for the XOR lemma. Prior to our work, even completely heuristic counterexamples of this type were not known.
Cite as

Saikrishna Badrinarayanan, Yuval Ishai, Dakshita Khurana, Amit Sahai, and Daniel Wichs. Refuting the Dream XOR Lemma via Ideal Obfuscation and Resettable MPC. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{badrinarayanan_et_al:LIPIcs.ITC.2022.10,
  author =	{Badrinarayanan, Saikrishna and Ishai, Yuval and Khurana, Dakshita and Sahai, Amit and Wichs, Daniel},
  title =	{{Refuting the Dream XOR Lemma via Ideal Obfuscation and Resettable MPC}},
  booktitle =	{3rd Conference on Information-Theoretic Cryptography (ITC 2022)},
  pages =	{10:1--10:21},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-164885},
  doi =		{10.4230/LIPIcs.ITC.2022.10},
  annote =	{Keywords: XOR Lemma, Resettable MPC, Obfuscation}
}
Document
DOI: 10.4230/LIPIcs.ITC.2022.17
Information-Theoretic Distributed Point Functions

Authors: Elette Boyle, Niv Gilboa, Yuval Ishai, and Victor I. Kolobov

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

Abstract
A distributed point function (DPF) (Gilboa-Ishai, Eurocrypt 2014) is a cryptographic primitive that enables compressed additive secret-sharing of a secret weight-1 vector across two or more servers. DPFs support a wide range of cryptographic applications, including efficient private information retrieval, secure aggregation, and more. Up to now, the study of DPFs was restricted to the computational security setting, relying on one-way functions. This assumption is necessary in the case of a dishonest majority. We present the first statistically private 3-server DPF for domain size N with subpolynomial key size N^{o(1)}. We also present a similar perfectly private 4-server DPF. Our constructions offer benefits over their computationally secure counterparts, beyond the superior security guarantee, including better computational complexity and better protocols for distributed key generation, all while having comparable communication complexity for moderate-sized parameters.
Cite as

Elette Boyle, Niv Gilboa, Yuval Ishai, and Victor I. Kolobov. Information-Theoretic Distributed Point Functions. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boyle_et_al:LIPIcs.ITC.2022.17,
  author =	{Boyle, Elette and Gilboa, Niv and Ishai, Yuval and Kolobov, Victor I.},
  title =	{{Information-Theoretic Distributed Point Functions}},
  booktitle =	{3rd Conference on Information-Theoretic Cryptography (ITC 2022)},
  pages =	{17:1--17:14},
  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.2022.17},
  URN =		{urn:nbn:de:0030-drops-164957},
  doi =		{10.4230/LIPIcs.ITC.2022.17},
  annote =	{Keywords: Information-theoretic cryptography, homomorphic secret sharing, private information retrieval, secure multiparty computation}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.26
Bounded Indistinguishability for Simple Sources

Authors: Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Akshayaram Srinivasan

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.27
Locality-Preserving Hashing for Shifts with Connections to Cryptography

Authors: Elette Boyle, Itai Dinur, Niv Gilboa, Yuval Ishai, Nathan Keller, and Ohad Klein

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

Abstract
Can we sense our location in an unfamiliar environment by taking a sublinear-size sample of our surroundings? Can we efficiently encrypt a message that only someone physically close to us can decrypt? To solve this kind of problems, we introduce and study a new type of hash functions for finding shifts in sublinear time. A function h:{0,1}ⁿ → ℤ_n is a (d,δ) locality-preserving hash function for shifts (LPHS) if: (1) h can be computed by (adaptively) querying d bits of its input, and (2) Pr[h(x) ≠ h(x ≪ 1) + 1] ≤ δ, where x is random and ≪ 1 denotes a cyclic shift by one bit to the left. We make the following contributions. - Near-optimal LPHS via Distributed Discrete Log. We establish a general two-way connection between LPHS and algorithms for distributed discrete logarithm in the generic group model. Using such an algorithm of Dinur et al. (Crypto 2018), we get LPHS with near-optimal error of δ = Õ(1/d²). This gives an unusual example for the usefulness of group-based cryptography in a post-quantum world. We extend the positive result to non-cyclic and worst-case variants of LPHS. - Multidimensional LPHS. We obtain positive and negative results for a multidimensional extension of LPHS, making progress towards an optimal 2-dimensional LPHS. - Applications. We demonstrate the usefulness of LPHS by presenting cryptographic and algorithmic applications. In particular, we apply multidimensional LPHS to obtain an efficient "packed" implementation of homomorphic secret sharing and a sublinear-time implementation of location-sensitive encryption whose decryption requires a significantly overlapping view.
Cite as

Elette Boyle, Itai Dinur, Niv Gilboa, Yuval Ishai, Nathan Keller, and Ohad Klein. Locality-Preserving Hashing for Shifts with Connections to Cryptography. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 27:1-27:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boyle_et_al:LIPIcs.ITCS.2022.27,
  author =	{Boyle, Elette and Dinur, Itai and Gilboa, Niv and Ishai, Yuval and Keller, Nathan and Klein, Ohad},
  title =	{{Locality-Preserving Hashing for Shifts with Connections to Cryptography}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{27:1--27:24},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.27},
  URN =		{urn:nbn:de:0030-drops-156231},
  doi =		{10.4230/LIPIcs.ITCS.2022.27},
  annote =	{Keywords: Sublinear algorithms, metric embeddings, shift finding, discrete logarithm, homomorphic secret sharing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.30
PCPs and Instance Compression from a Cryptographic Lens

Authors: Liron Bronfman and Ron D. Rothblum

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

Abstract
Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary’s power in other information theoretic proof-systems and show how to use this assumption to bypass impossibility results. We first consider the question of constructing succinct PCPs. These are PCPs whose length is polynomial only in the length of the original NP witness (in contrast to standard PCPs whose length is proportional to the non-deterministic verification time). Unfortunately, succinct PCPs are known to be impossible to construct under standard complexity assumptions. Assuming the sub-exponential hardness of the learning with errors (LWE) problem, we construct succinct probabilistically checkable arguments or PCAs (Kalai and Raz 2009), which are PCPs in which soundness is guaranteed against efficiently generated false proofs. Our PCA construction is for every NP relation that can be verified by a small-depth circuit (e.g., SAT, clique, TSP, etc.) and in contrast to prior work is publicly verifiable and has constant query complexity. Curiously, we also show, as a proof-of-concept, that such publicly-verifiable PCAs can be used to derive hardness of approximation results. Second, we consider the notion of Instance Compression (Harnik and Naor, 2006). An instance compression scheme lets one compress, for example, a CNF formula φ on m variables and n ≫ m clauses to a new formula φ' with only poly(m) clauses, so that φ is satisfiable if and only if φ' is satisfiable. Instance compression has been shown to be closely related to succinct PCPs and is similarly highly unlikely to exist. We introduce a computational analog of instance compression in which we require that if φ is unsatisfiable then φ' is effectively unsatisfiable, in the sense that it is computationally infeasible to find a satisfying assignment for φ' (although such an assignment may exist). Assuming the same sub-exponential LWE assumption, we construct such computational instance compression schemes for every bounded-depth NP relation. As an application, this lets one compress k formulas ϕ₁,… ,ϕ_k into a single short formula ϕ that is effectively satisfiable if and only if at least one of the original formulas was satisfiable.
Cite as

Liron Bronfman and Ron D. Rothblum. PCPs and Instance Compression from a Cryptographic Lens. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bronfman_et_al:LIPIcs.ITCS.2022.30,
  author =	{Bronfman, Liron and Rothblum, Ron D.},
  title =	{{PCPs and Instance Compression from a Cryptographic Lens}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{30:1--30:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.30},
  URN =		{urn:nbn:de:0030-drops-156269},
  doi =		{10.4230/LIPIcs.ITCS.2022.30},
  annote =	{Keywords: PCP, Succinct Arguments, Instance Compression}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.67
Small Circuits Imply Efficient Arthur-Merlin Protocols

Authors: Michael Ezra and Ron D. Rothblum

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

Abstract
The inner product function ⟨ x,y ⟩ = ∑_i x_i y_i mod 2 can be easily computed by a (linear-size) AC⁰(⊕) circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on the bottom most layer (closest to the input)? Namely, can the inner product function be computed by an AC⁰ circuit composed with a single layer of parity gates? This seemingly simple question is an important open question at the frontier of circuit lower bound research. In this work, we focus on a minimalistic version of the above question. Namely, whether the inner product function cannot be approximated by a small DNF augmented with a single layer of parity gates. Our main result shows that the existence of such a circuit would have unexpected implications for interactive proofs, or more specifically, for interactive variants of the Data Streaming and Communication Complexity models. In particular, we show that the existence of such a small (i.e., polynomial-size) circuit yields: 1) An O(d)-message protocol in the Arthur-Merlin Data Streaming model for every n-variate, degree d polynomial (over GF(2)), using only Õ(d) ⋅log(n) communication and space complexity. In particular, this gives an AM[2] Data Streaming protocol for a variant of the well-studied triangle counting problem, with poly-logarithmic communication and space complexities. 2) A 2-message communication complexity protocol for any sparse (or low degree) polynomial, and for any function computable by an AC⁰(⊕) circuit. Specifically, for the latter, we obtain a protocol with communication complexity that is poly-logarithmic in the size of the AC⁰(⊕) circuit.
Cite as

Michael Ezra and Ron D. Rothblum. Small Circuits Imply Efficient Arthur-Merlin Protocols. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ezra_et_al:LIPIcs.ITCS.2022.67,
  author =	{Ezra, Michael and Rothblum, Ron D.},
  title =	{{Small Circuits Imply Efficient Arthur-Merlin Protocols}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{67:1--67:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.67},
  URN =		{urn:nbn:de:0030-drops-156635},
  doi =		{10.4230/LIPIcs.ITCS.2022.67},
  annote =	{Keywords: Circuits Complexity, Circuit Lower Bounds, Communication Complexity, Data Streaming, Arthur-Merlin games, Interactive Proofs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.71
On the Download Rate of Homomorphic Secret Sharing

Authors: Ingerid Fosli, Yuval Ishai, Victor I. Kolobov, and Mary Wootters

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

Abstract
A homomorphic secret sharing (HSS) scheme is a secret sharing scheme that supports evaluating functions on shared secrets by means of a local mapping from input shares to output shares. We initiate the study of the download rate of HSS, namely, the achievable ratio between the length of the output shares and the output length when amortized over 𝓁 function evaluations. We obtain the following results. - In the case of linear information-theoretic HSS schemes for degree-d multivariate polynomials, we characterize the optimal download rate in terms of the optimal minimal distance of a linear code with related parameters. We further show that for sufficiently large 𝓁 (polynomial in all problem parameters), the optimal rate can be realized using Shamir’s scheme, even with secrets over 𝔽₂. - We present a general rate-amplification technique for HSS that improves the download rate at the cost of requiring more shares. As a corollary, we get high-rate variants of computationally secure HSS schemes and efficient private information retrieval protocols from the literature. - We show that, in some cases, one can beat the best download rate of linear HSS by allowing nonlinear output reconstruction and 2^{-Ω(𝓁)} error probability.
Cite as

Ingerid Fosli, Yuval Ishai, Victor I. Kolobov, and Mary Wootters. On the Download Rate of Homomorphic Secret Sharing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 71:1-71:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fosli_et_al:LIPIcs.ITCS.2022.71,
  author =	{Fosli, Ingerid and Ishai, Yuval and Kolobov, Victor I. and Wootters, Mary},
  title =	{{On the Download Rate of Homomorphic Secret Sharing}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{71:1--71:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.71},
  URN =		{urn:nbn:de:0030-drops-156675},
  doi =		{10.4230/LIPIcs.ITCS.2022.71},
  annote =	{Keywords: Information-theoretic cryptography, homomorphic secret sharing, private information retrieval, secure multiparty computation, regenerating codes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.117
Low-Bandwidth Recovery of Linear Functions of Reed-Solomon-Encoded Data

Authors: Noah Shutty and Mary Wootters

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

Abstract
We study the problem of efficiently computing on encoded data. More specifically, we study the question of low-bandwidth computation of functions F:F^k → F of some data 𝐱 ∈ F^k, given access to an encoding 𝐜 ∈ Fⁿ of 𝐱 under an error correcting code. In our model - relevant in distributed storage, distributed computation and secret sharing - each symbol of 𝐜 is held by a different party, and we aim to minimize the total amount of information downloaded from each party in order to compute F(𝐱). Special cases of this problem have arisen in several domains, and we believe that it is fruitful to study this problem in generality. Our main result is a low-bandwidth scheme to compute linear functions for Reed-Solomon codes, even in the presence of erasures. More precisely, let ε > 0 and let 𝒞: F^k → Fⁿ be a full-length Reed-Solomon code of rate 1 - ε over a field F with constant characteristic. For any γ ∈ [0, ε), our scheme can compute any linear function F(𝐱) given access to any (1 - γ)-fraction of the symbols of 𝒞(𝐱), with download bandwidth O(n/(ε - γ)) bits. In contrast, the naive scheme that involves reconstructing the data 𝐱 and then computing F(𝐱) uses Θ(n log n) bits. Our scheme has applications in distributed storage, coded computation, and homomorphic secret sharing.
Cite as

Noah Shutty and Mary Wootters. Low-Bandwidth Recovery of Linear Functions of Reed-Solomon-Encoded Data. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 117:1-117:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{shutty_et_al:LIPIcs.ITCS.2022.117,
  author =	{Shutty, Noah and Wootters, Mary},
  title =	{{Low-Bandwidth Recovery of Linear Functions of Reed-Solomon-Encoded Data}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{117:1--117:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.117},
  URN =		{urn:nbn:de:0030-drops-157130},
  doi =		{10.4230/LIPIcs.ITCS.2022.117},
  annote =	{Keywords: Reed-Solomon Codes, Regenerating Codes, Coded Computation}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.1
Group Structure in Correlations and Its Applications in Cryptography

Authors: Guru-Vamsi Policharla, Manoj Prabhakaran, Rajeev Raghunath, and Parjanya Vyas

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
Correlated random variables are a key tool in cryptographic applications like secure multi-party computation. We investigate the power of a class of correlations that we term group correlations: A group correlation is a uniform distribution over pairs (x,y) ∈ G² such that x+y ∈ S, where G is a (possibly non-abelian) group and S is a subset of G. We also introduce bi-affine correlation{s}, and show how they relate to group correlations. We present several structural results, new protocols and applications of these correlations. The new applications include a completeness result for black box group computation, perfectly secure protocols for evaluating a broad class of black box "mixed-groups" circuits with bi-affine homomorphisms, and new information-theoretic results. Finally, we uncover a striking structure underlying OLE: In particular, we show that OLE over 𝔽_{2ⁿ}, is isomorphic to a group correlation over ℤ_4^n.
Cite as

Guru-Vamsi Policharla, Manoj Prabhakaran, Rajeev Raghunath, and Parjanya Vyas. Group Structure in Correlations and Its Applications in Cryptography. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{policharla_et_al:LIPIcs.ITC.2021.1,
  author =	{Policharla, Guru-Vamsi and Prabhakaran, Manoj and Raghunath, Rajeev and Vyas, Parjanya},
  title =	{{Group Structure in Correlations and Its Applications in Cryptography}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.1},
  URN =		{urn:nbn:de:0030-drops-143208},
  doi =		{10.4230/LIPIcs.ITC.2021.1},
  annote =	{Keywords: Group correlations, bi-affine correlations, secure computation}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.5
Line-Point Zero Knowledge and Its Applications

Authors: Samuel Dittmer, Yuval Ishai, and Rafail Ostrovsky

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
We introduce and study a simple kind of proof system called line-point zero knowledge (LPZK). In an LPZK proof, the prover encodes the witness as an affine line 𝐯(t) : = at + 𝐛 in a vector space 𝔽ⁿ, and the verifier queries the line at a single random point t = α. LPZK is motivated by recent practical protocols for vector oblivious linear evaluation (VOLE), which can be used to compile LPZK proof systems into lightweight designated-verifier NIZK protocols. We construct LPZK systems for proving satisfiability of arithmetic circuits with attractive efficiency features. These give rise to designated-verifier NIZK protocols that require only 2-5 times the computation of evaluating the circuit in the clear (following an input-independent preprocessing phase), and where the prover communicates roughly 2 field elements per multiplication gate, or roughly 1 element in the random oracle model with a modestly higher computation cost. On the theoretical side, our LPZK systems give rise to the first linear interactive proofs (Bitansky et al., TCC 2013) that are zero knowledge against a malicious verifier. We then apply LPZK towards simplifying and improving recent constructions of reusable non-interactive secure computation (NISC) from VOLE (Chase et al., Crypto 2019). As an application, we give concretely efficient and reusable NISC protocols over VOLE for bounded inner product, where the sender’s input vector should have a bounded L₂-norm.
Cite as

Samuel Dittmer, Yuval Ishai, and Rafail Ostrovsky. Line-Point Zero Knowledge and Its Applications. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dittmer_et_al:LIPIcs.ITC.2021.5,
  author =	{Dittmer, Samuel and Ishai, Yuval and Ostrovsky, Rafail},
  title =	{{Line-Point Zero Knowledge and Its Applications}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.5},
  URN =		{urn:nbn:de:0030-drops-143249},
  doi =		{10.4230/LIPIcs.ITC.2021.5},
  annote =	{Keywords: Zero-knowledge proofs, NIZK, correlated randomness, vector oblivious linear evaluation, non-interactive secure computation}
}
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