10 Search Results for "Komargodski, Ilan"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.46
Scalable Distributed Agreement from LWE: Byzantine Agreement, Broadcast, and Leader Election

Authors: Rex Fernando, Yuval Gelles, and Ilan Komargodski

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

Abstract
Distributed agreement is a general name for the task of ensuring consensus among non-faulty nodes in the presence of faulty or malicious behavior. Well-known instances of agreement tasks are Byzantine Agreement, Broadcast, and Committee or Leader Election. Since agreement tasks lie at the heart of many modern distributed applications, there has been an increased interest in designing scalable protocols for these tasks. Specifically, we want protocols where the per-party communication complexity scales sublinearly with the number of parties. With unconditional security, the state of the art protocols have Õ(√ n) per-party communication and Õ(1) rounds, where n stands for the number of parties, tolerating 1/3-ε fraction of corruptions for any ε > 0. There are matching lower bounds showing that these protocols are essentially optimal among a large class of protocols. Recently, Boyle-Cohen-Goel (PODC 2021) relaxed the attacker to be computationally bounded and using strong cryptographic assumptions showed a protocol with Õ(1) per-party communication and rounds (similarly, tolerating 1/3-ε fraction of corruptions). The security of their protocol relies on SNARKs for NP with linear-time extraction, a somewhat strong and non-standard assumption. Their protocols further relies on a public-key infrastructure (PKI) and a common-reference-string (CRS). In this work, we present a new protocol with Õ(1) per-party communication and rounds but relying only on the standard Learning With Errors (LWE) assumption. Our protocol also relies on a PKI and a CRS, and tolerates 1/3-ε fraction of corruptions, similarly to Boyle et al. Technically, we leverage (multi-hop) BARGs for NP directly and in a generic manner which significantly deviate from the framework of Boyle et al.
Cite as

Rex Fernando, Yuval Gelles, and Ilan Komargodski. Scalable Distributed Agreement from LWE: Byzantine Agreement, Broadcast, and Leader Election. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 46:1-46:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fernando_et_al:LIPIcs.ITCS.2024.46,
  author =	{Fernando, Rex and Gelles, Yuval and Komargodski, Ilan},
  title =	{{Scalable Distributed Agreement from LWE: Byzantine Agreement, Broadcast, and Leader Election}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{46:1--46:23},
  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.2024.46},
  URN =		{urn:nbn:de:0030-drops-195744},
  doi =		{10.4230/LIPIcs.ITCS.2024.46},
  annote =	{Keywords: Byzantine agreement, scalable, learning with errors}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2023.42
Brief Announcement: Scalable Agreement Protocols with Optimal Optimistic Efficiency

Authors: Yuval Gelles and Ilan Komargodski

Published in: LIPIcs, Volume 281, 37th International Symposium on Distributed Computing (DISC 2023)

Abstract
Designing efficient distributed protocols for various agreement tasks such as Byzantine Agreement, Broadcast, and Committee Election is a fundamental problem. We are interested in scalable protocols for these tasks, where each (honest) party communicates a number of bits which is sublinear in n, the number of parties. The first major step towards this goal is due to King et al. (SODA 2006) who showed a protocol where each party sends only Õ(1) bits throughout Õ(1) rounds, but guarantees only that 1-o(1) fraction of honest parties end up agreeing on a consistent output, assuming constant < 1/3 fraction of static corruptions. Few years later, King et al. (ICDCN 2011) managed to get a full agreement protocol in the same model but where each party sends Õ(√n) bits throughout Õ(1) rounds. Getting a full agreement protocol with o(√n) communication per party has been a major challenge ever since. In light of this barrier, we propose a new framework for designing efficient agreement protocols. Specifically, we design Õ(1)-round protocols for all of the above tasks (assuming constant < 1/3 fraction of static corruptions) with optimistic and pessimistic guarantees: - Optimistic complexity: In an honest execution, all parties send only Õ(1) bits. - Pessimistic complexity: In any other case, (honest) parties send Õ(√n) bits. Thus, all an adversary can gain from deviating from the honest execution is that honest parties will need to work harder (i.e., transmit more bits) to reach agreement and terminate. Besides the above agreement tasks, we also use our new framework to get a scalable secure multiparty computation (MPC) protocol with optimistic and pessimistic complexities. Technically, we identify a relaxation of Byzantine Agreement (of independent interest) that allows us to fall-back to a pessimistic execution in a coordinated way by all parties. We implement this relaxation with Õ(1) communication bits per party and within Õ(1) rounds.
Cite as

Yuval Gelles and Ilan Komargodski. Brief Announcement: Scalable Agreement Protocols with Optimal Optimistic Efficiency. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 42:1-42:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gelles_et_al:LIPIcs.DISC.2023.42,
  author =	{Gelles, Yuval and Komargodski, Ilan},
  title =	{{Brief Announcement: Scalable Agreement Protocols with Optimal Optimistic Efficiency}},
  booktitle =	{37th International Symposium on Distributed Computing (DISC 2023)},
  pages =	{42:1--42:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-301-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{281},
  editor =	{Oshman, Rotem},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2023.42},
  URN =		{urn:nbn:de:0030-drops-191684},
  doi =		{10.4230/LIPIcs.DISC.2023.42},
  annote =	{Keywords: Byzantine Agreement, Consensus, Optimistic-Pessimistic, Secure Multi-Party Computation}
}
Document
DOI: 10.4230/LIPIcs.ITC.2023.2
A Lower Bound on the Share Size in Evolving Secret Sharing

Authors: Noam Mazor

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

Abstract
Secret sharing schemes allow sharing a secret between a set of parties in a way that ensures that only authorized subsets of the parties learn the secret. Evolving secret sharing schemes (Komargodski, Naor, and Yogev [TCC '16]) allow achieving this end in a scenario where the parties arrive in an online fashion, and there is no a-priory bound on the number of parties. An important complexity measure of a secret sharing scheme is the share size, which is the maximum number of bits that a party may receive as a share. While there has been a significant progress in recent years, the best constructions for both secret sharing and evolving secret sharing schemes have a share size that is exponential in the number of parties. On the other hand, the best lower bound, by Csirmaz [Eurocrypt '95], is sub-linear. In this work, we give a tight lower bound on the share size of evolving secret sharing schemes. Specifically, we show that the sub-linear lower bound of Csirmaz implies an exponential lower bound on evolving secret sharing.
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Noam Mazor. A Lower Bound on the Share Size in Evolving Secret Sharing. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 2:1-2:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mazor:LIPIcs.ITC.2023.2,
  author =	{Mazor, Noam},
  title =	{{A Lower Bound on the Share Size in Evolving Secret Sharing}},
  booktitle =	{4th Conference on Information-Theoretic Cryptography (ITC 2023)},
  pages =	{2:1--2:9},
  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.2023.2},
  URN =		{urn:nbn:de:0030-drops-183300},
  doi =		{10.4230/LIPIcs.ITC.2023.2},
  annote =	{Keywords: Secret sharing, Evolving secret sharing}
}
Document
DOI: 10.4230/LIPIcs.ITC.2023.4
The Cost of Statistical Security in Proofs for Repeated Squaring

Authors: Cody Freitag and Ilan Komargodski

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

Abstract
In recent years, the number of applications of the repeated squaring assumption has been growing rapidly. The assumption states that, given a group element x, an integer T, and an RSA modulus N, it is hard to compute x^2^T mod N - or even decide whether y?=x^2^T mod N - in parallel time less than the trivial approach of simply computing T squares. This rise has been driven by efficient proof systems for repeated squaring, opening the door to more efficient constructions of verifiable delay functions, various secure computation primitives, and proof systems for more general languages. In this work, we study the complexity of statistically sound proofs for the repeated squaring relation. Technically, we consider proofs where the prover sends at most k ≥ 0 elements and the (probabilistic) verifier performs generic group operations over the group ℤ_N^⋆. As our main contribution, we show that for any (one-round) proof with a randomized verifier (i.e., an MA proof) the verifier either runs in parallel time Ω(T/(k+1)) with high probability, or is able to factor N given the proof provided by the prover. This shows that either the prover essentially sends p,q such that N = p⋅ q (which is infeasible or undesirable in most applications), or a variant of Pietrzak’s proof of repeated squaring (ITCS 2019) has optimal verifier complexity O(T/(k+1)). In particular, it is impossible to obtain a statistically sound one-round proof of repeated squaring with efficiency on par with the computationally-sound protocol of Wesolowski (EUROCRYPT 2019), with a generic group verifier. We further extend our one-round lower bound to a natural class of recursive interactive proofs for repeated squaring. For r-round recursive proofs where the prover is allowed to send k group elements per round, we show that the verifier either runs in parallel time Ω(T/(k+1)^r) with high probability, or is able to factor N given the proof transcript.
Cite as

Cody Freitag and Ilan Komargodski. The Cost of Statistical Security in Proofs for Repeated Squaring. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{freitag_et_al:LIPIcs.ITC.2023.4,
  author =	{Freitag, Cody and Komargodski, Ilan},
  title =	{{The Cost of Statistical Security in Proofs for Repeated Squaring}},
  booktitle =	{4th Conference on Information-Theoretic Cryptography (ITC 2023)},
  pages =	{4:1--4:23},
  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.2023.4},
  URN =		{urn:nbn:de:0030-drops-183326},
  doi =		{10.4230/LIPIcs.ITC.2023.4},
  annote =	{Keywords: Cryptographic Proofs, Repeated Squaring, Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.68
Differentially Oblivious Turing Machines

Authors: Ilan Komargodski and Elaine Shi

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Oblivious RAM (ORAM) is a machinery that protects any RAM from leaking information about its secret input by observing only the access pattern. It is known that every ORAM must incur a logarithmic overhead compared to the non-oblivious RAM. In fact, even the seemingly weaker notion of differential obliviousness, which intuitively "protects" a single access by guaranteeing that the observed access pattern for every two "neighboring" logical access sequences satisfy (ε,δ)-differential privacy, is subject to a logarithmic lower bound. In this work, we show that any Turing machine computation can be generically compiled into a differentially oblivious one with only doubly logarithmic overhead. More precisely, given a Turing machine that makes N transitions, the compiled Turing machine makes O(N ⋅ log log N) transitions in total and the physical head movements sequence satisfies (ε,δ)-differential privacy (for a constant ε and a negligible δ). We additionally show that Ω(log log N) overhead is necessary in a natural range of parameters (and in the balls and bins model). As a corollary, we show that there exist natural data structures such as stack and queues (supporting online operations) on N elements for which there is a differentially oblivious implementation on a Turing machine incurring amortized O(log log N) overhead per operation, while it is known that any oblivious implementation must consume Ω(log N) operations unconditionally even on a RAM. Therefore, we obtain the first unconditional separation between obliviousness and differential obliviousness in the most natural setting of parameters where ε is a constant and δ is negligible. Before this work, such a separation was only known in the balls and bins model. Note that the lower bound applies in the RAM model while our upper bound is in the Turing machine model, making our separation stronger.
Cite as

Ilan Komargodski and Elaine Shi. Differentially Oblivious Turing Machines. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 68:1-68:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{komargodski_et_al:LIPIcs.ITCS.2021.68,
  author =	{Komargodski, Ilan and Shi, Elaine},
  title =	{{Differentially Oblivious Turing Machines}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{68:1--68:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.68},
  URN =		{urn:nbn:de:0030-drops-136071},
  doi =		{10.4230/LIPIcs.ITCS.2021.68},
  annote =	{Keywords: Differential privacy, Turing machines, obliviousness}
}
Document
DOI: 10.4230/LIPIcs.ITC.2020.11
Oblivious Parallel Tight Compaction

Authors: Gilad Asharov, Ilan Komargodski, Wei-Kai Lin, Enoch Peserico, and Elaine Shi

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

Abstract
In tight compaction one is given an array of balls some of which are marked 0 and the rest are marked 1. The output of the procedure is an array that contains all of the original balls except that now the 0-balls appear before the 1-balls. In other words, tight compaction is equivalent to sorting the array according to 1-bit keys (not necessarily maintaining order within same-key balls). Tight compaction is not only an important algorithmic task by itself, but its oblivious version has also played a key role in recent constructions of oblivious RAM compilers. We present an oblivious deterministic algorithm for tight compaction such that for input arrays of n balls requires O(n) total work and O(log n) depth. Our algorithm is in the Exclusive-Read-Exclusive-Write Parallel-RAM model (i.e., EREW PRAM, the most restrictive PRAM model), and importantly we achieve asymptotical optimality in both total work and depth. To the best of our knowledge no earlier work, even when allowing randomization, can achieve optimality in both total work and depth.
Cite as

Gilad Asharov, Ilan Komargodski, Wei-Kai Lin, Enoch Peserico, and Elaine Shi. Oblivious Parallel Tight Compaction. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{asharov_et_al:LIPIcs.ITC.2020.11,
  author =	{Asharov, Gilad and Komargodski, Ilan and Lin, Wei-Kai and Peserico, Enoch and Shi, Elaine},
  title =	{{Oblivious Parallel Tight Compaction}},
  booktitle =	{1st Conference on Information-Theoretic Cryptography (ITC 2020)},
  pages =	{11:1--11:23},
  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.2020.11},
  URN =		{urn:nbn:de:0030-drops-121164},
  doi =		{10.4230/LIPIcs.ITC.2020.11},
  annote =	{Keywords: Oblivious tight compaction, parallel oblivious RAM, EREW PRAM}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.56
Instance Complexity and Unlabeled Certificates in the Decision Tree Model

Authors: Tomer Grossman, Ilan Komargodski, and Moni Naor

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Instance complexity is a measure of goodness of an algorithm in which the performance of one algorithm is compared to others per input. This is in sharp contrast to worst-case and average-case complexity measures, where the performance is compared either on the worst input or on an average one, respectively. We initiate the systematic study of instance complexity and optimality in the query model (a.k.a. the decision tree model). In this model, instance optimality of an algorithm for computing a function is the requirement that the complexity of an algorithm on any input is at most a constant factor larger than the complexity of the best correct algorithm. That is we compare the decision tree to one that receives a certificate and its complexity is measured only if the certificate is correct (but correctness should hold on any input). We study both deterministic and randomized decision trees and provide various characterizations and barriers for more general results. We introduce a new measure of complexity called unlabeled-certificate complexity, appropriate for graph properties and other functions with symmetries, where only information about the structure of the graph is known to the competing algorithm. More precisely, the certificate is some permutation of the input (rather than the input itself) and the correctness should be maintained even if the certificate is wrong. First we show that such an unlabeled certificate is sometimes very helpful in the worst-case. We then study instance optimality with respect to this measure of complexity, where an algorithm is said to be instance optimal if for every input it performs roughly as well as the best algorithm that is given an unlabeled certificate (but is correct on every input). We show that instance optimality depends on the group of permutations in consideration. Our proofs rely on techniques from hypothesis testing and analysis of random graphs.
Cite as

Tomer Grossman, Ilan Komargodski, and Moni Naor. Instance Complexity and Unlabeled Certificates in the Decision Tree Model. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 56:1-56:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{grossman_et_al:LIPIcs.ITCS.2020.56,
  author =	{Grossman, Tomer and Komargodski, Ilan and Naor, Moni},
  title =	{{Instance Complexity and Unlabeled Certificates in the Decision Tree Model}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{56:1--56:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.56},
  URN =		{urn:nbn:de:0030-drops-117418},
  doi =		{10.4230/LIPIcs.ITCS.2020.56},
  annote =	{Keywords: decision tree complexity, instance complexity, instance optimality, query complexity, unlabeled certificates}
}
Document
DOI: 10.4230/LIPIcs.DISC.2018.34
A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols

Authors: Yael Tauman Kalai, Ilan Komargodski, and Ran Raz

Published in: LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Abstract
In 1985, Ben-Or and Linial (Advances in Computing Research '89) introduced the collective coin-flipping problem, where n parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party in the course of the protocol as a function of the messages seen so far. They showed that the majority protocol, in which each player sends a random bit and the output is the majority value, tolerates O(sqrt n) adaptive corruptions. They conjectured that this is optimal for such adversaries. We prove that the majority protocol is optimal (up to a poly-logarithmic factor) among all protocols in which each party sends a single, possibly long, message. Previously, such a lower bound was known for protocols in which parties are allowed to send only a single bit (Lichtenstein, Linial, and Saks, Combinatorica '89), or for symmetric protocols (Goldwasser, Kalai, and Park, ICALP '15).
Cite as

Yael Tauman Kalai, Ilan Komargodski, and Ran Raz. A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{taumankalai_et_al:LIPIcs.DISC.2018.34,
  author =	{Tauman Kalai, Yael and Komargodski, Ilan and Raz, Ran},
  title =	{{A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols}},
  booktitle =	{32nd International Symposium on Distributed Computing (DISC 2018)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-092-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{121},
  editor =	{Schmid, Ulrich and Widder, Josef},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.34},
  URN =		{urn:nbn:de:0030-drops-98230},
  doi =		{10.4230/LIPIcs.DISC.2018.34},
  annote =	{Keywords: Coin flipping, adaptive corruptions, byzantine faults, lower bound}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.850
Negation-Limited Formulas

Authors: Siyao Guo and Ilan Komargodski

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
Cite as

Siyao Guo and Ilan Komargodski. Negation-Limited Formulas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 850-866, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{guo_et_al:LIPIcs.APPROX-RANDOM.2015.850,
  author =	{Guo, Siyao and Komargodski, Ilan},
  title =	{{Negation-Limited Formulas}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{850--866},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  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.2015.850},
  URN =		{urn:nbn:de:0030-drops-53400},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.850},
  annote =	{Keywords: Negation complexity, De Morgan formulas, Shrinkage}
}
Document
DOI: 10.4230/LIPIcs.CCC.2015.1
Strong Locally Testable Codes with Relaxed Local Decoders

Authors: Oded Goldreich, Tom Gur, and Ilan Komargodski

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Abstract
Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester; that is, a tester that makes only a constant number of queries and reject non-codewords with probability that depends solely on their distance from the code. Locally decodable codes (LDCs) are complimentary to LTCs. While the latter allow for highly efficient rejection of strings that are far from being codewords, LDCs allow for highly efficient recovery of individual bits of the information that is encoded in strings that are close to being codewords. While there are known constructions of strong-LTCs with nearly-linear length, the existence of a constant-query LDC with polynomial length is a major open problem. In an attempt to bypass this barrier, Ben-Sasson et al. (SICOMP 2006) introduced a natural relaxation of local decodability, called relaxed-LDCs. This notion requires local recovery of nearly all individual information-bits, yet allows for recovery-failure (but not error) on the rest. Ben-Sasson et al. constructed a constant-query relaxed-LDC with nearly-linear length (i.e., length k^(1 + alpha) for an arbitrarily small constant alpha>0, where k is the dimension of the code). This work focuses on obtaining strong testability and relaxed decodability simultaneously. We construct a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs. This improves upon the previously known constructions, which obtain either weak LTCs or require polynomial length. Our construction heavily relies on tensor codes and PCPs. In particular, we provide strong canonical PCPs of proximity for membership in any linear code with constant rate and relative distance. Loosely speaking, these are PCPs of proximity wherein the verifier is proximity oblivious (similarly to strong-LTCs and every valid statement has a unique canonical proof. Furthermore, the verifier is required to reject non-canonical proofs (even for valid statements). As an application, we improve the best known separation result between the complexity of decision and verification in the setting of property testing.
Cite as

Oded Goldreich, Tom Gur, and Ilan Komargodski. Strong Locally Testable Codes with Relaxed Local Decoders. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 1-41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goldreich_et_al:LIPIcs.CCC.2015.1,
  author =	{Goldreich, Oded and Gur, Tom and Komargodski, Ilan},
  title =	{{Strong Locally Testable Codes with Relaxed Local Decoders}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{1--41},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.1},
  URN =		{urn:nbn:de:0030-drops-50507},
  doi =		{10.4230/LIPIcs.CCC.2015.1},
  annote =	{Keywords: Locally Testable Codes, Locally Decodable Codes, PCPs of Proximity}
}
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