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Pseudorandom generators without the XOR Lemma (Extended Abstract)
, 1998
"... Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a series of works showing connections between the existence of har ..."
Abstract

Cited by 137 (23 self)
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of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of "hardness amplification" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed
Three xorlemmas  an exposition
 Electronic Colloquium on Computational Complexity (ECCC
, 1995
"... Abstract. We provide an exposition of three lemmas that relate general properties of distributions over bit strings to the exclusiveor (xor) of values of certain bit locations. The first XORLemma, commonly attributed to Umesh Vazirani (1986), relates the statistical distance of a distribution from ..."
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Cited by 18 (1 self)
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from the uniform distribution over bit strings to the maximum bias of the xor of certain bit positions. The second XORLemma, due to Umesh and Vijay Vazirani (19th STOC, 1987), is a computational analogue of the first. It relates the pseudorandomness of a distribution to the difficulty of predicting
ListDecoding Using The XOR Lemma
"... We show that Yao's XOR Lemma, and its essentially equivalent rephrasing as a Direct Product Lemma, can be reinterpreted as a way of obtaining errorcorrecting codes with good listdecoding algorithms from errorcorrecting codes having weak uniquedecoding algorithms. To get codes with good rat ..."
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Cited by 39 (4 self)
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We show that Yao's XOR Lemma, and its essentially equivalent rephrasing as a Direct Product Lemma, can be reinterpreted as a way of obtaining errorcorrecting codes with good listdecoding algorithms from errorcorrecting codes having weak uniquedecoding algorithms. To get codes with good
P=BPP unless E has subexponential circuits: Derandomizing the XOR Lemma
 IN PROCEEDINGS OF THE 29TH STOC
, 1996
"... Yao showed that the XOR of independent random instances of a somewhat hard Boolean function becomes almost completely unpredictable. In this paper we show that, in nonuniform settings, total independence is not necessary for this result to hold. We give a pseudorandom generator which produces n ..."
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Cited by 38 (5 self)
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n instances of the function for which the analog of the XOR lemma holds. This is the first derandomization of a "direct product" result. Our generator is a combination of two known ones  the random walks on expander graphs of [1, 9, 19] and the nearly disjoint subsets generator of [23
Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols
 In Proceedings of the 22nd Annual Conference on Computational Complexity. IEEE
, 2007
"... This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF (2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely rela ..."
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Cited by 23 (6 self)
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related to property testers of this proximity). The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most ɛ ≤ 1/2 with any of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically
On the Derandomization of Constant Depth Circuits
"... Abstract. Nisan [18] and Nisan and Wigderson [19] have constructed a pseudorandom generator which fools any family of polynomialsize constant depth circuits. At the core of their construction is the result ..."
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Abstract. Nisan [18] and Nisan and Wigderson [19] have constructed a pseudorandom generator which fools any family of polynomialsize constant depth circuits. At the core of their construction is the result
Derandomization, Hashing and Expanders
"... Regarding complexity of computation, randomness is a significant resource beside time and space. Particularly from a theoretical viewpoint, it is a fundamental question whether availability of random numbers gives any additional power. Most of randomized algorithms are analyzed under the assumption ..."
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, there are occasional problems with use of weak pseudorandom generators. Further, randomized algorithms are not suited for applications where reliability is a key concern. Derandomization is the process of minimizing the use of random bits, either to small amounts or removing them altogether. We may identify two lines
On Derandomizing Algorithms that Err Extremely Rarely
, 2014
"... Does derandomization of probabilistic algorithms become easier when the number of “bad” random inputs is extremely small? In relation to the above question, we put forward the following quantified derandomization challenge: For a class of circuits C (e.g., P/poly or AC0,AC0[2]) and a bounding functi ..."
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Does derandomization of probabilistic algorithms become easier when the number of “bad” random inputs is extremely small? In relation to the above question, we put forward the following quantified derandomization challenge: For a class of circuits C (e.g., P/poly or AC0,AC0[2]) and a bounding
Derandomized parallel repetition theorems for free games
, 2010
"... Raz’s parallel repetition theorem [Raz98] together with improvements of Holenstein [Hol07] shows that for any twoprover oneround game with value at most 1−ɛ (for ɛ ≤ 1/2), the value of the game repeated n times in parallel on independent inputs is at most (1 − ɛ) Ω ( ɛ2n ℓ) where ℓ is the answer l ..."
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Cited by 2 (0 self)
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(m). Our technique uses strong extractors to “derandomize ” a lemma of [Raz98], and can be also used to derandomize a parallel repetition theorem of Parnafes, Raz and Wigderson [PRW97] for communication games in the special case that the game is free.
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