The Black-Box Query Complexity of Polynomial Summation

Juma, Ali; Kabanets, Valentine; Rackoff, Charles; Shpilka, Amir

doi:10.1007/s00037-009-0263-7

Cited by 9 publications

(15 citation statements)

References 7 publications

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“…Recently (and independently of us), Juma, Kabanets, Rackoff and Shpilka[14] studied an algebraic query complexity model closely related to ours, and proved lower bounds in this model. In our terminology, they "almost" constructed an oracle A, and a multiquadratic extension A of A, such that #P A ⊂ FP A /poly 3.…”

mentioning

confidence: 73%

“…The goal will be to show that querying points outside the Boolean cube is useless if one wants to gain information about values on the Boolean cube. In full generality, this is of course false (as witnessed by interactive proofs and PCPs on the one hand, and by the result of Juma et al [14] on the other). To make our adversary arguments work, it will be crucial to give ourselves sufficient freedom, by using polynomials of multidegree 2 rather than multilinear polynomials.…”

Section: Lower Bounds By Direct Constructionmentioning

confidence: 99%

“…Again, the results of Juma et al [14] imply that multidegree 2 is essential here, since for multilinear polynomials it is possible to solve the OR problem with only one query (over fields of characteristic greater than 2).…”

Section: Deterministic Lower Boundsmentioning

confidence: 99%

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Algebrization

Aaronson

Wigderson

2008

Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing

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Any proof of P = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory.In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring.We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization-both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MA EXP ⊂ P/poly -do indeed algebrize. Second, we show that almost all of the major open problemsincluding P versus NP, P versus RP, and NEXP versus P/polywill require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP.Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck). * Extended abstract. For the full version, please go to www.scottaaronson.com/papers.

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mentioning

confidence: 73%

Section: Lower Bounds By Direct Constructionmentioning

confidence: 99%

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Algebrization

Aaronson

Wigderson

2008

Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing

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“…However, giving B additional structure calls into question the hiding property of the scheme. Indeed, surprisingly, a result of Juma et al [JKRS09] shows that this new scheme is in fact not hiding (in fields of odd characteristic): it holds that B(2 −1 , . .…”

Section: Algebraic Commitments From Algebraic Query Complexity Lower Boundsmentioning

confidence: 98%

“…However, we need d to be small to achieve soundness. Fortunately, a result of [JKRS09] shows that d = 2 suffices: given a random multiquadratic extension B of B, one needs 2 k queries to B to determine β∈{0,1} k B( β). 12 Committing to a polynomial.…”

Section: Algebraic Commitments From Algebraic Query Complexity Lower Boundsmentioning

confidence: 99%

Spatial Isolation Implies Zero Knowledge Even in a Quantum World

Chiesa

Forbes

Gur

et al. 2022

J. ACM

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Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP , as long as one is willing to make a suitable physical assumption : if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting. In this work, we study the following question: Does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * . Our proof consists of constructing a zero knowledge interactive probabilistically checkable proof with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.

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