Lower bounds for one-way probabilistic communication complexity and their application to space complexity

Ablayev, Farid

doi:10.1016/0304-3975(95)00157-3

Cited by 76 publications

(103 citation statements)

References 7 publications

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“…The figure shows a frequency distribution of [2,3,9,4,1,3,2,0]. Over these leaves, we impose a binary tree, and for each internal node w in the tree we showf (w).…”

Section: A First Results For Frequent Itemsmentioning

confidence: 99%

Practical verified computation with streaming interactive proofs

Cormode

Mitzenmacher

Thaler

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

154

269

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When delegating computation to a service provider, as in the cloud computing paradigm, we seek some reassurance that the output is correct and complete. Yet recomputing the output as a check is inefficient and expensive, and it may not even be feasible to store all the data locally. We are therefore interested in what can be validated by a streaming (sublinear space) user, who cannot store the full input, or perform the full computation herself. Our aim in this work is to advance a recent line of work on "proof systems" in which the service provider proves the correctness of its output to a user. The goal is to minimize the time and space costs of both parties in generating and checking the proof. Only very recently have there been attempts to implement such proof systems, and thus far these have been quite limited in functionality.Here, our approach is two-fold. First, we describe a carefully chosen instantiation of one of the most efficient general-purpose constructions for arbitrary computations (streaming or otherwise), due to Goldwasser, Kalai, and Rothblum [19]. This requires several new insights and enhancements to move the methodology from a theoretical result to a practical possibility. Our main contribution is in achieving a prover that runs in time O(S(n) log S(n)), where S(n) is the size of an arithmetic circuit computing the function of interest; this compares favorably to the poly(S(n)) runtime for the prover promised in [19]. Our experimental results demonstrate that a practical general-purpose protocol for verifiable computation may be significantly closer to reality than previously realized.Second, we describe a set of techniques that achieve genuine scalability for protocols fine-tuned for specific important problems in streaming and database processing. Focusing in particular on noninteractive protocols for problems ranging from matrix-vector multiplication to bipartite perfect matching, we build on prior work [8,15] to achieve a prover that runs in nearly linear-time, while obtaining optimal tradeoffs between communication cost and the user's working memory. Existing techniques required (substantially) superlinear time for the prover. Finally, we develop improved interactive protocols for specific problems based on a linearization technique originally due to Shen [34]. We argue that even if general-purpose methods improve, fine-tuned protocols will remain valuable in real-world settings for key problems, and hence special attention to specific problems is warranted.

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“…The figure shows a frequency distribution of [2,3,9,4,1,3,2,0]. Over these leaves, we impose a binary tree, and for each internal node w in the tree we showf (w).…”

Section: A First Results For Frequent Itemsmentioning

confidence: 99%

Practical verified computation with streaming interactive proofs

Cormode

Mitzenmacher

Thaler

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

154

269

View full text Add to dashboard Cite

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“…We note that Theorem 6.7 does not hold when ρ = 0, i.e., when the protocol is not allowed to err. Recall (Proposition 3.22) that IC (EQ, 0) = O (1). At the same time, it is not hard to see that if the function EQ is on the space {0, 1} m ×{0, 1} m , then R n ρ (f n ) = Ω(m·n), and thus the right-hand side in Theorem 6.7 is Ω(m).…”

Section: The Nondistributional Casementioning

confidence: 99%

“…For example, Ablayev [1] used information theory to prove a lower bound for Index, 1 Nayak [59] used it in the quantum setting, and Klauck et al [51] used it for pointer jumping.…”

mentioning

confidence: 99%

Interactive Information Complexity

Braverman¹

2017

SIAM Rev.

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Abstract. The primary goal of this paper is to define and study the interactive information complexity of functions. Let f (x, y) be a function, and suppose Alice is given x and Bob is given y. Informally, the interactive information complexity IC(f ) of f is the least amount of information Alice and Bob need to reveal to each other to compute f . Previously, information complexity has been defined with respect to a prior distribution on the input pairs (x, y). Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity IC(f ). In particular, we show that IC(f ) is equal to the amortized (randomized) communication complexity of f . We also show a direct sum theorem for IC(f ) and give the first general connection between information complexity and (nonamortized) communication complexity. This connection implies that a nontrivial exchange of information is required when solving problems that have nontrivial communication complexity. We explore the information complexity of two specific problems: equality and disjointness. We show that only a constant amount of information needs to be exchanged when solving equality with no errors, while solving disjointness with a constant error probability requires the parties to reveal a linear amount of information to each other. This paper originally appeared in [SIAM J. Comput., 44 (2015), pp. 1698-1739]. The present version has been slightly updated with some recent developments.

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“…We now proceed to recall the basic definitions for communication complexity [49,29], specifically one-way communication complexity with one-sided error [1,28,37].…”

Section: Communication Complexitymentioning

confidence: 99%

“…Note that for ABE ciphertext and key sizes, we ignore the contributions from the descriptive values x, y as well as multiplicative factors in the security parameter. 1 We are particularly interested in three regimes of parameters for ( A , B ):…”

Section: Introductionmentioning

confidence: 99%

Communication Complexity of Conditional Disclosure of Secrets and Attribute-Based Encryption

Gay

Kerenidis

Wee

2015

Lecture Notes in Computer Science

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Abstract. We initiate a systematic treatment of the communication complexity of conditional disclosure of secrets (CDS), where two parties want to disclose a secret to a third party if and only if their respective inputs satisfy some predicate. We present a general upper bound and the first non-trivial lower bounds for conditional disclosure of secrets. Moreover, we achieve tight lower bounds for many interesting setting of parameters for CDS with linear reconstruction, the latter being a requirement in the application to attribute-based encryption. In particular, our lower bounds explain the trade-off between ciphertext and secret key sizes of several existing attribute-based encryption schemes based on the dual system methodology.

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Lower bounds for one-way probabilistic communication complexity and their application to space complexity

Cited by 76 publications

References 7 publications

Practical verified computation with streaming interactive proofs

Practical verified computation with streaming interactive proofs

Interactive Information Complexity

Communication Complexity of Conditional Disclosure of Secrets and Attribute-Based Encryption

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