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53
On Yao’s XOR lemma
 Electronic Colloquium on Computational Complexity
, 1995
"... Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by firs ..."
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Cited by 57 (6 self)
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Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XORLemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XORLemma.
Another proof that BPP ⊆ PH (and more
, 1997
"... Abstract. We provide another proof of the Sipser–Lautemann Theorem by which BPP ⊆ MA ( ⊆ PH). The current proof is based on strong results regarding the amplification of BPP, due to Zuckerman (1996). Given these results, the current proof is even simpler than previous ones. Furthermore, extending th ..."
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Cited by 27 (3 self)
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Abstract. We provide another proof of the Sipser–Lautemann Theorem by which BPP ⊆ MA ( ⊆ PH). The current proof is based on strong results regarding the amplification of BPP, due to Zuckerman (1996). Given these results, the current proof is even simpler than previous ones. Furthermore, extending the proof leads to two results regarding MA: MA ⊆ ZPP N P (which seems to be new), and that twosided error MA equals MA. Finally, we survey the known facts regarding the fragment of the polynomialtime hierarchy that contains MA.
Notes on Levin's Theory of AverageCase Complexity
 Electronic Colloquium on Computational Complexity
, 1997
"... Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author ..."
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Cited by 18 (2 self)
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Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author’s (exposition and exploration) work [4], which was partially reproduded in [1]. An early version of this survey appeared as TR97058 of ECCC. Some of the perspective and conclusions were revised in light of a relatively recent work of Livne [21], but an attempt was made to preserve the spirit of the original survey. The author’s current perspective is better reflected in [7, Sec. 10.2] and [8], which advocate somewhat different definitional choices (e.g., focusing on typical rather than average performace of algorithms). 1
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Cited by 15 (0 self)
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
The Unified Theory of Pseudorandomness
, 2007
"... We survey the close connections between a variety of “pseudorandom objects,” namely pseudorandom generators, expander graphs, listdecodable errorcorrecting codes, randomness extractors, averaging samplers, and hardness amplifiers. ..."
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Cited by 8 (2 self)
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We survey the close connections between a variety of “pseudorandom objects,” namely pseudorandom generators, expander graphs, listdecodable errorcorrecting codes, randomness extractors, averaging samplers, and hardness amplifiers.
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 7 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
Introduction to testing graph properties
 In Property Testing
, 2010
"... Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this ..."
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Cited by 7 (0 self)
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Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this
Are pcps inherent in efficient arguments
 In IEEE Conference on Computational Complexity
"... Abstract. Starting with Kilian (STOC ‘92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collisionresistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithm ..."
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Cited by 7 (0 self)
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Abstract. Starting with Kilian (STOC ‘92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collisionresistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC ‘07) raised the question of whether PCPs are inherent in efficient arguments, and if so, to what extent. We give evidence that they are, by showing how to convert any argument system whose soundness is reducible to the security of some cryptographic primitive into a PCP system whose efficiency is related to that of the argument system and the reduction (under certain complexity assumptions).
To Release Or Not To Release: Evaluating Information Leaks in Aggregate HumanGenome Data
"... Abstract. The rapid progress of human genome studies leads to a strong demand of aggregate human DNA data (e.g, allele frequencies, test statistics, etc.), whose public dissemination, however, has been impeded by privacy concerns. Prior research shows that it is possible to identify the presence of ..."
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Abstract. The rapid progress of human genome studies leads to a strong demand of aggregate human DNA data (e.g, allele frequencies, test statistics, etc.), whose public dissemination, however, has been impeded by privacy concerns. Prior research shows that it is possible to identify the presence of some participants in a study from such data, and in some cases, even fully recover their DNA sequences. A critical issue, therefore, becomes how to evaluate such a risk on individual datasets and determine when they are safe to release. In this paper, we report our research that makes the first attempt to address this issue. We first identified the space of the aggregatedatarelease problem, through examining common types of aggregate data and the typical threats they are facing. Then, we performed an indepth study on different scenarios of attacks on different types of data, which sheds light on several fundamental questions in this problem domain. Particularly, we found that attacks on aggregate data are difficult in general, as the adversary often does not have enough information and needs to solve NPcomplete or NPhard problems. On the other hand, we acknowledge that the attacks can succeed under some circumstances, particularly, when the solution space of the problem is small. Based upon such an understanding, we propose a riskscale system and a methodology to determine when to release an aggregate dataset and when not to. We also used real humangenome data to verify our findings. 1