Document

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

Current protocols for Multi-Party Computation (MPC) consider the setting where all parties have access to similar resources. For example, all parties have access to channels bounded by the same worst-case delay upper bound Δ, and all channels have the same cost of communication. As a consequence, the overall protocol performance (resp. the communication cost) may be heavily affected by the slowest (resp. the most expensive) channel, even when most channels are fast (resp. cheap). Given the state of affairs, we initiate a systematic study of asymmetric MPC. In asymmetric MPC, the parties are divided into two categories: fast and slow parties, depending on whether they have access to high-end or low-end resources.
We investigate two different models. In the first, we consider asymmetric communication delays: Fast parties are connected via channels with small delay δ among themselves, while channels connected to (at least) one slow party have a large delay Δ ≫ δ. In the second model, we consider asymmetric communication costs: Fast parties benefit from channels with cheap communication, while channels connected to a slow party have an expensive communication. We provide a wide range of positive and negative results exploring the trade-offs between the achievable number of tolerated corruptions t and slow parties s, versus the round complexity and communication cost in each of the models. Among others, we achieve the following results. In the model with asymmetric communication delays, focusing on the information-theoretic (i-t) setting:
- An i-t asymmetric MPC protocol with security with abort as long as t+s < n and t < n/2, in a constant number of slow rounds.
- We show that achieving an i-t asymmetric MPC protocol for t+s = n and with number of slow rounds independent of the circuit size implies an i-t synchronous MPC protocol with round complexity independent of the circuit size, which is a major problem in the field of round-complexity of MPC.
- We identify a new primitive, asymmetric broadcast, that allows to consistently distribute a value among the fast parties, and at a later time the same value to slow parties. We completely characterize the feasibility of asymmetric broadcast by showing that it is possible if and only if 2t + s < n.
- An i-t asymmetric MPC protocol with guaranteed output delivery as long as t+s < n and t < n/2, in a number of slow rounds independent of the circuit size.
In the model with asymmetric communication cost, we achieve an asymmetric MPC protocol for security with abort for t+s < n and t < n/2, based on one-way functions (OWF). The protocol communicates a number of bits over expensive channels that is independent of the circuit size. We conjecture that assuming OWF is needed and further provide a partial result in this direction.

Vipul Goyal, Chen-Da Liu-Zhang, and Rafail Ostrovsky. Asymmetric Multi-Party Computation. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 6:1-6:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goyal_et_al:LIPIcs.ITC.2023.6, author = {Goyal, Vipul and Liu-Zhang, Chen-Da and Ostrovsky, Rafail}, title = {{Asymmetric Multi-Party Computation}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {6:1--6:25}, 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.2023.6}, URN = {urn:nbn:de:0030-drops-183342}, doi = {10.4230/LIPIcs.ITC.2023.6}, annote = {Keywords: multiparty computation, asymmetric, delays, communication} }

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

Most existing work on secure multi-party computation (MPC) ignores a key idiosyncrasy of modern communication networks, that there are a limited number of communication paths between any two nodes, many of which might even be corrupted. The problem becomes particularly acute in the information-theoretic setting, where the lack of trusted setups (and the cryptographic primitives they enable) makes communication over sparse networks more challenging. The work by Garay and Ostrovsky [EUROCRYPT'08] on almost-everywhere MPC (AE-MPC), introduced "best-possible security" properties for MPC over such incomplete networks, where necessarily some of the honest parties may be excluded from the computation.
In this work, we provide a universally composable definition of almost-everywhere security, which allows us to automatically and accurately capture the guarantees of AE-MPC (as well as AE-communication, the analogous "best-possible security" version of secure communication) in the Universal Composability (UC) framework of Canetti. Our results offer the first simulation-based treatment of this important but under-investigated problem, along with the first simulation-based proof of AE-MPC. To achieve that goal, we state and prove a general composition theorem, which makes precise the level or "quality" of AE-security that is obtained when a protocol’s hybrids are replaced with almost-everywhere components.

Nishanth Chandran, Pouyan Forghani, Juan Garay, Rafail Ostrovsky, Rutvik Patel, and Vassilis Zikas. Universally Composable Almost-Everywhere Secure Computation. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 14:1-14:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chandran_et_al:LIPIcs.ITC.2022.14, author = {Chandran, Nishanth and Forghani, Pouyan and Garay, Juan and Ostrovsky, Rafail and Patel, Rutvik and Zikas, Vassilis}, title = {{Universally Composable Almost-Everywhere Secure Computation}}, booktitle = {3rd Conference on Information-Theoretic Cryptography (ITC 2022)}, pages = {14:1--14:25}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2022.14}, URN = {urn:nbn:de:0030-drops-164929}, doi = {10.4230/LIPIcs.ITC.2022.14}, annote = {Keywords: Secure multi-party computation, universal composability, almost-everywhere secure computation, sparse graphs, secure message transmission} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ε > 0, for any instance with n input points and for any positive integer n' ≤ n, we compute in polynomial time a clustering of at least (1-ε) n' points of cost at most a constant factor greater than the optimal cost of clustering n' points. The approximation guarantee grows with 1/(ε). Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).

Sandip Banerjee, Rafail Ostrovsky, and Yuval Rabani. Min-Sum Clustering (With Outliers). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 16:1-16:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{banerjee_et_al:LIPIcs.APPROX/RANDOM.2021.16, author = {Banerjee, Sandip and Ostrovsky, Rafail and Rabani, Yuval}, title = {{Min-Sum Clustering (With Outliers)}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.16}, URN = {urn:nbn:de:0030-drops-147093}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.16}, annote = {Keywords: Clustering, approximation algorithms, primal-dual} }

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

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.

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.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} }

Document

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

Data-oblivious algorithms are a key component of many secure computation protocols.
In this work, we show that advances in secure multiparty shuffling algorithms can be used to increase the efficiency of several key cryptographic tools.
The key observation is that many secure computation protocols rely heavily on secure shuffles. The best data-oblivious shuffling algorithms require O(n log n), operations, but in the two-party or multiparty setting, secure shuffling can be achieved with only O(n) communication.
Leveraging the efficiency of secure multiparty shuffling, we give novel, information-theoretic algorithms that improve the efficiency of securely sorting sparse lists, secure stable compaction, and securely merging two sorted lists.
Securely sorting private lists is a key component of many larger secure computation protocols. The best data-oblivious sorting algorithms for sorting a list of n elements require O(n log n) comparisons. Using black-box access to a linear-communication secure shuffle, we give a secure algorithm for sorting a list of length n with t ≪ n nonzero elements with communication O(t log² n + n), which beats the best oblivious algorithms when the number of nonzero elements, t, satisfies t < n/log² n.
Secure compaction is the problem of removing dummy elements from a list, and is essentially equivalent to sorting on 1-bit keys. The best oblivious compaction algorithms run in O(n)-time, but they are unstable, i.e., the order of the remaining elements is not preserved. Using black-box access to a linear-communication secure shuffle, we give an information-theoretic stable compaction algorithm with only O(n) communication.
Our main result is a novel secure merge protocol. The best previous algorithms for securely merging two sorted lists into a sorted whole required O(n log n) secure operations. Using black-box access to an O(n)-communication secure shuffle, we give the first multi-party secure merge algorithm that requires only O(n log log n) communication. Our algorithm takes as input n secret-shared values, and outputs a secret-sharing of the sorted list.
All our algorithms are generic, i.e., they can be implemented using generic secure computations techniques and make black-box access to a secure shuffle. Our techniques extend naturally to the multiparty situation (with a constant number of parties) as well as to handle malicious adversaries without changing the asymptotic efficiency.
These algorithm have applications to securely computing database joins and order statistics on private data as well as multiparty Oblivious RAM protocols.

Brett Hemenway Falk and Rafail Ostrovsky. Secure Merge with O(n log log n) Secure Operations. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 7:1-7:29, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{falk_et_al:LIPIcs.ITC.2021.7, author = {Falk, Brett Hemenway and Ostrovsky, Rafail}, title = {{Secure Merge with O(n log log n) Secure Operations}}, booktitle = {2nd Conference on Information-Theoretic Cryptography (ITC 2021)}, pages = {7:1--7:29}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.7}, URN = {urn:nbn:de:0030-drops-143265}, doi = {10.4230/LIPIcs.ITC.2021.7}, annote = {Keywords: Secure computation, Data-oblivious algorithms, Sorting, Merging, Shuffling, Compaction} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Osborne's iteration is a method for balancing n x n matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over R^n, but it is normally used in the L_2 norm. The choice of norm not only affects the desired balance condition, but also defines the iterated balancing step itself.
In this paper we focus on Osborne's iteration in any L_p norm, where p < infty. We design a specific implementation of Osborne's iteration in any L_p norm that converges to a strictly epsilon-balanced matrix in O~(epsilon^{-2}n^{9} K) iterations, where K measures, roughly, the number of bits required to represent the entries of the input matrix.
This is the first result that proves a variant of Osborne's iteration in the L_2 norm (or any L_p norm, p < infty) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in L_p norms. Previously, Schulman and Sinclair (STOC 2015) showed strict balancing of another variant of Osborne's iteration in the L_infty norm. Their result does not imply any bounds on strict balancing in other norms.

Rafail Ostrovsky, Yuval Rabani, and Arman Yousefi. Strictly Balancing Matrices in Polynomial Time Using Osborne's Iteration. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 93:1-93:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ostrovsky_et_al:LIPIcs.ICALP.2018.93, author = {Ostrovsky, Rafail and Rabani, Yuval and Yousefi, Arman}, title = {{Strictly Balancing Matrices in Polynomial Time Using Osborne's Iteration}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {93:1--93:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.93}, URN = {urn:nbn:de:0030-drops-90976}, doi = {10.4230/LIPIcs.ICALP.2018.93}, annote = {Keywords: Numerical Linear Algebra, Optimization} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Protecting software from malware injection is one of the biggest challenges of modern computer science. Despite intensive efforts by the scientific and engineering community, the number of successful attacks continues to increase.
This work sets first footsteps towards a provably secure investigation of malware detection. We provide a formal model and cryptographic security definitions of attestation for systems with dynamic memory, and suggest novel provably secure attestation schemes. The key idea underlying our schemes is to use the very insertion of the malware itself to allow for the systems to detect it. This is, in our opinion, close in spirit to the quantum Observer Effect. The attackers, no matter how clever, no matter when they insert their malware, change the state of the system they are attacking. This fundamental idea can be a game changer. And our system does not rely on heuristics; instead, our scheme enjoys the unique property that it is proved secure in a formal and precise mathematical sense and with minimal and realistic CPU modification achieves strong provable security guarantees. We envision such systems with a formal mathematical security treatment as a venue for new directions in software protection.

Richard J. Lipton, Rafail Ostrovsky, and Vassilis Zikas. Provably Secure Virus Detection: Using The Observer Effect Against Malware. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lipton_et_al:LIPIcs.ICALP.2016.32, author = {Lipton, Richard J. and Ostrovsky, Rafail and Zikas, Vassilis}, title = {{Provably Secure Virus Detection: Using The Observer Effect Against Malware}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.32}, URN = {urn:nbn:de:0030-drops-63113}, doi = {10.4230/LIPIcs.ICALP.2016.32}, annote = {Keywords: Cryptography, Software Attestation, Provable Security} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol.
In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and tolerates noise rates of up to 1/18 - epsilon. To this end we develop a novel primitive we name edit distance tree code. The edit distance tree code is designed to replace the Hamming distance constraints in Schulman's tree codes (STOC 93), with a stronger edit distance requirement. However, the straightforward generalization of tree codes to edit distance does not seem to yield a primitive that suffices for communication in the presence of synchronization problems. Giving the "right" definition of edit distance tree codes is a main conceptual contribution of this work.

Mark Braverman, Ran Gelles, Jieming Mao, and Rafail Ostrovsky. Coding for Interactive Communication Correcting Insertions and Deletions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 61:1-61:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{braverman_et_al:LIPIcs.ICALP.2016.61, author = {Braverman, Mark and Gelles, Ran and Mao, Jieming and Ostrovsky, Rafail}, title = {{Coding for Interactive Communication Correcting Insertions and Deletions}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {61:1--61:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.61}, URN = {urn:nbn:de:0030-drops-61981}, doi = {10.4230/LIPIcs.ICALP.2016.61}, annote = {Keywords: Interactive communication, coding, edit distance} }

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

Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model.

Vladimir Braverman, Rafail Ostrovsky, and Alan Roytman. Zero-One Laws for Sliding Windows and Universal Sketches. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 573-590, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2015.573, author = {Braverman, Vladimir and Ostrovsky, Rafail and Roytman, Alan}, title = {{Zero-One Laws for Sliding Windows and Universal Sketches}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {573--590}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.573}, URN = {urn:nbn:de:0030-drops-53248}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.573}, annote = {Keywords: Streaming Algorithms, Universality, Sliding Windows} }

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

A quantile summary is a data structure that approximates to epsilon-relative error the order statistics of a much larger underlying dataset.
In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses O((1/epsilon) log(1/epsilon)) words of memory. This improves upon the previous best upper bound of O((1/epsilon) (log(1/epsilon))^(3/2)) by Agarwal et al. (PODS 2012). Further, by a lower bound of Hung and Ting (FAW 2010) no deterministic summary for the comparison model can outperform our randomized summary in terms of space complexity. Lastly, our summary has the nice property that O((1/epsilon) log(1/epsilon)) words suffice to ensure that the success probability is 1 - exp(-poly(1/epsilon)).

David Felber and Rafail Ostrovsky. A Randomized Online Quantile Summary in O(1/epsilon * log(1/epsilon)) Words. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 775-785, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{felber_et_al:LIPIcs.APPROX-RANDOM.2015.775, author = {Felber, David and Ostrovsky, Rafail}, title = {{A Randomized Online Quantile Summary in O(1/epsilon * log(1/epsilon)) Words}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {775--785}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.775}, URN = {urn:nbn:de:0030-drops-53357}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.775}, annote = {Keywords: order statistics, data stream, streaming algorithm} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that $4$-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of $4$-wise independent functions on $[n]$.
Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1\pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, and Rafail Ostrovsky. AMS Without 4-Wise Independence on Product Domains. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 119-130, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{braverman_et_al:LIPIcs.STACS.2010.2449, author = {Braverman, Vladimir and Chung, Kai-Min and Liu, Zhenming and Mitzenmacher, Michael and Ostrovsky, Rafail}, title = {{AMS Without 4-Wise Independence on Product Domains}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {119--130}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2449}, URN = {urn:nbn:de:0030-drops-24496}, doi = {10.4230/LIPIcs.STACS.2010.2449}, annote = {Keywords: Data Streams, Randomized Algorithms, Streaming Algorithms, Independence, Sketches} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5411, Anonymous Communication and its Applications (2006)

From 09.10.05 to 14.10.05, the Dagstuhl Seminar 05411 ``Anonymous Communication and its Applications'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Shlomi Dolev, Andreas Pfitzmann, and Rafail Ostrovsky. 05411 Abstracts Collection – Anonymous Communication and its Applications. In Anonymous Communication and its Applications. Dagstuhl Seminar Proceedings, Volume 5411, pp. 1-15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)

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@InProceedings{dolev_et_al:DagSemProc.05411.1, author = {Dolev, Shlomi and Pfitzmann, Andreas and Ostrovsky, Rafail}, title = {{05411 Abstracts Collection – Anonymous Communication and its Applications}}, booktitle = {Anonymous Communication and its Applications}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5411}, editor = {Shlomi Dolev and Rafail Ostrovsky and Andreas Pfitzmann}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05411.1}, URN = {urn:nbn:de:0030-drops-7950}, doi = {10.4230/DagSemProc.05411.1}, annote = {Keywords: Anonymous Communication, Cryptography, Privacy, Security, Anonymity} }

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