Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

Ford, Jeff; Gál, Anna

doi:10.1007/11523468_94

Cited by 17 publications

(7 citation statements)

References 22 publications

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“…The relevance of Prop. 1 consists in the fact that matrices M having maximal excess have been exhaustively studied by mathematicians during the last five decades, including a wide range of Hadamard matrices [28,[32][33][34][35][36]43], complex Hadamard matrices [44], Hadamard tensors [45] and orthogonal designs [46]. These results considerably extend the set of Bell inequalities for which the LHV value is known [25], which is not a minor observation taking into account the hardness to calculate the LHV value of a bipartite Bell inequality [47].…”

Section: Excess and Lhv Modelsmentioning

confidence: 99%

Local hidden variable values without optimization procedures

Goyeneche

Bruzda

Turek

et al. 2023

Quantum

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The problem of computing the local hidden variable (LHV) value of a Bell inequality plays a central role in the study of quantum nonlocality. In particular, this problem is the first step towards characterizing the LHV polytope of a given scenario. In this work, we establish a relation between the LHV value of bipartite Bell inequalities and the mathematical notion of excess of a matrix. Inspired by the well developed theory of excess, we derive several results that directly impact the field of quantum nonlocality. We show infinite families of bipartite Bell inequalities for which the LHV value can be computed exactly, without needing to solve any optimization problem, for any number of measurement settings. We also find tight Bell inequalities for a large number of measurement settings.

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Section: Excess and Lhv Modelsmentioning

confidence: 99%

Local hidden variable values without optimization procedures

Goyeneche

Bruzda

Turek

et al. 2023

Quantum

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“…Note that this lemma implies hardness in the almost adaptive one-round SM model (since this model is more restrictive), and hence, together with Theorems 2 and 3, implies security of the Γ pos ,D,t and Γ ka ,D,t protocols. The communication complexity of the GIP function has been studied in multiple papers [5,6,32,23,22], but up to our knowledge, not in the strong randomized settings that we need in this work. Our proof is a rather straightforward adaptation of the techniques from this prior work.…”

Section: Protocols In the Plain Modelmentioning

confidence: 99%

Position-Based Cryptography and Multiparty Communication Complexity

Brody

Dziembowski

Faust

et al. 2017

Theory of Cryptography

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Position based cryptography (PBC), proposed in the seminal work of Chandran, Goyal, Moriarty, and Ostrovsky (SIAM J. Computing, 2014), aims at constructing cryptographic schemes in which the identity of the user is his geographic position. Chandran et al. construct PBC schemes for secure positioning and position-based key agreement in the bounded-storage model (Maurer, J. Cryptology, 1992). Apart from bounded memory, their security proofs need a strong additional restriction on the power of the adversary: he cannot compute joint functions of his inputs. Removing this assumption is left as an open problem.We show that an answer to this question would resolve a long standing open problem in multiparty communication complexity: finding a function that is hard to compute with low communication complexity in the simultaneous message model, but easy to compute in the fully adaptive model.On a more positive side: we also show some implications in the other direction, i.e.: we prove that lower bounds on the communication complexity of certain multiparty problems imply existence of PBC primitives. Using this result we then show two attractive ways to "bypass" our hardness result: the first uses the random oracle model, the second weakens the locality requirement in the bounded-storage model to online computability. The random oracle construction is arguably one of the simplest proposed so far in this area. Our results indicate that constructing improved provably secure protocols for PBC requires a better understanding of multiparty communication complexity. This is yet another example where negative results in one area (in our case: lower bounds in multiparty communication complexity) can be used to construct secure cryptographic schemes.

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“…Lemma 2.2 (Ford & Gál 2005;Raz 2000). Let M ∈ {−1, 1} |X|×|Y | , and let μ be a probability distribution over X ×Y.…”

Section: Discrepancy Under Product Distributionsmentioning

confidence: 99%

“…Ford and Gál used Lemma 2.2 in an elegant way to relate discrepancy to the pairwise correlations of the matrix rows: Lemma 2.3 (Ford & Gál 2005). For every Boolean function f (x, y) and every…”

Section: Discrepancy Under Product Distributionsmentioning

confidence: 99%

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Halfspace Matrices

Sherstov

2008

comput. complex.

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We introduce the notion of a halfspace matrix, which is a sign matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x ∈ {−1, 1} n , and the entries given by A f,x = f (x). We use halfspace matrices to solve the following problems. In communication complexity, we exhibit a Boolean function f with discrepancy Ω(1/n 4 ) under every product distribution but O( √ n/2 n/4 ) under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan (1997).In learning theory, we give a short and simple proof that the statisticalquery (SQ) dimension of halfspaces in n dimensions is less than 2(n + 1) 2 under all distributions. This improves on the n O(1) estimate from the fundamental paper of Blum et al. (1998). We show that estimating the SQ dimension of natural classes of Boolean functions can resolve major open problems in complexity theory, such as separating PSPACE cc and PH cc . Finally, we construct a matrix A ∈ {−1, 1} N ×N log N with dimension complexity log N but margin complexity Ω(N 1/4 / √ log N ). This gap is an exponential improvement over previous work. We prove several other relationships among the complexity measures of sign matrices, complementing work by Linial et al. ( , 2007.

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Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

Cited by 17 publications

References 22 publications

Local hidden variable values without optimization procedures

Local hidden variable values without optimization procedures

Position-Based Cryptography and Multiparty Communication Complexity

Halfspace Matrices

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