Randomized simultaneous messages: solution of a problem of Yao in communication complexity

Babai, László; Kimmel, Peter G.

doi:10.1109/ccc.1997.612319

Cited by 53 publications

(85 citation statements)

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“…Their result was generalized by Babai and Kimmel [BK97] to the result that the randomized and deterministic complexity can be at most quadratically far apart for any function in this model. Babai and Kimmel attribute a simplified proof of this fact to Bourgain and Wigderson.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, once the key is distributed, it must be stored securely until the inputs are obtained. This is because an adversary who knows the value of the key can easily choose inputs x and y such that x = y but for which the output of the protocol always indicates that x = y. bits suffice if we allow a small error probability (see also [KNR95,NS96,BK97]). Note that in this setting Alice and Bob still have access to random bits, but their random bits may not be correlated.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Fingerprinting

et al. 2001

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Classical fingerprinting associates with each string a shorter string (its fingerprint), such that, with high probability, any two distinct strings can be distinguished by comparing their fingerprints alone. The fingerprints can be exponentially smaller than the original strings if the parties preparing the fingerprints share a random key, but not if they only have access to uncorrelated random sources. In this paper we show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties: we give a scheme where the quantum fingerprints are exponentially shorter than the original strings and we give a test that distinguishes any two unknown quantum fingerprints with high probability. Our scheme implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity. We optimize several aspects of our scheme.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum Fingerprinting

et al. 2001

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“…However, the separation does not hold in the presence of public coins. Buhrman et al [6] were able to solve the equality problem in the SM model with a quantum protocol of complexity O(log n) rather than the Θ( √ n) bits necessary in any bounded-error randomized SM protocol with private coins [17,5]. Again, if we allow the players to share random coins, then equality can be solved classically with O(1) communication.…”

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confidence: 99%

Exponential separation of quantum and classical one-way communication complexity

Bar-Yossef

Jayram

Kerenidis

2004

Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing

101

135

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Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple i, j, b such that the edge (i, j) belongs to the matching M and b = x i ⊕ x j . We prove that the quantum one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω( √ n) bits of communication.No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω( 3 √ n log n) bits of communication.Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P681. Introduction. The investigation of the strength and limitations of quantum computing has become an important field of study in theoretical computer science. The celebrated algorithm of Shor [22] for factoring numbers in polynomial time on a quantum computer gives strong evidence that quantum computers are more powerful than classical ones. The further study of the relationship between quantum and classical computing in models like black-box computation, communication complexity, and interactive proof systems help towards a better understanding of quantum and classical computing.In this paper we answer an open question about the relative power of quantum one-way communication protocols. We describe a problem which can be solved by a quantum one-way communication protocol exponentially faster than any classical one. No asymptotic gap was previously known. We prove a similar result in the model of Simultaneous Messages.Communication complexity, defined by Yao [23] in 1979, is a central model of computation with numerous applications. It has been used for proving lower bounds in many areas including Boolean circuits, time-space tradeoffs, data structures, automata, formulae size, etc. Examples of these applications can be found in the textbook of Kushilevitz and Nisan [15]. A communication complexity problem is defined by three sets X, Y, Z and a relation R ⊆ X ×Y ×Z. Two computationally all-powerful players, Alice and Bob, are given inputs x ∈ X and y ∈ Y , respectively. Neither of the players has any information about the other player's input. Alice and Bob exchange

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“…In both results we use the simultaneous messages (SM) model of communication complexity, suggested by Yao in his seminal paper on communication complexity [Yao79], and especially a lower bound of Babai and Kimmel [BK97] (who generalized the lower bound of Newman and Szegedy [NS96]) for the communication complexity of SM protocols for equality testing. To the best of our knowledge, the only prior application of the SM model outside the field of communication complexity was for a positive result (a PIR construction) in the work of Beimel, Ishai and Kushilevitz [BIK05].…”

Section: Our Resultsmentioning

confidence: 99%

The complexity of online memory checking

Naor

Rothblum

2009

J. ACM

107

104

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We consider the problem of storing a large file on a remote and unreliable server. To verify that the file has not been corrupted, a user could store a small private (randomized) "fingerprint" on his own computer. This is the setting for the well-studied authentication problem in cryptography, and the required fingerprint size is well understood. We study the problem of sub-linear authentication: suppose the user would like to encode and store the file in a way that allows him to verify that it has not been corrupted, but without reading the entire file. If the user only wants to read q bits of the file, how large does the size s of the private fingerprint need to be? We define this problem formally, and show a tight lower bound on the relationship between s and q when the adversary is not computationally bounded, namely: s × q = Ω(n), where n is the file size. This is an easier case of the online memory checking problem, introduced by Blum et al. in 1991, and hence the same (tight) lower bound applies also to that problem.It was previously shown that when the adversary is computationally bounded, under the assumption that one-way functions exist, it is possible to construct much better online memory checkers. T he same is also true for sub-linear authentication schemes. We show that the existence of one-way functions is also a necessary condition: even slightly breaking the s × q = Ω(n) lower bound in a computational setting implies the existence of one-way functions.

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Randomized simultaneous messages: solution of a problem of Yao in communication complexity

Cited by 53 publications

References 5 publications

Quantum Fingerprinting

Quantum Fingerprinting

Exponential separation of quantum and classical one-way communication complexity

The complexity of online memory checking

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