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P=BPP unless E has subexponential circuits: Derandomizing the XOR Lemma (Preliminary Version)
 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 41 (6 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
The Complexity of Hardness Amplification and
, 2006
"... This thesis studies the interplay between randomness and computation. We investigate this interplay from the perspectives of hardness amplification and derandomization. Hardness amplification is the task of taking a function that is hard to compute on some input or on some fraction of inputs, and pr ..."
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This thesis studies the interplay between randomness and computation. We investigate this interplay from the perspectives of hardness amplification and derandomization. Hardness amplification is the task of taking a function that is hard to compute on some input or on some fraction of inputs
Abstract P=BPP unless E has subexponential circuits: Derandomizing the XOR Lemma
"... Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem 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 ins ..."
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Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem 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 instances of a problem for which the analog of the XOR lemma holds. Combining this generator with the results of [25, 6] gives substantially improved results for hardness vs randomness tradeoffs. In particular, we show that if any problem in E = DT IME(2 O(n) ) has circuit complexity 2 Ω(n), then P = BP P. Our generator is a combination of two known ones the random walks on expander graphs of [1, 10, 19] and the nearly disjoint subsets generator of [23, 25]. The quality of the generator is proved via a new proof of the XOR lemma which may be useful for other direct product results.
Using Nondeterminism to Amplify Hardness
, 2004
"... We revisit the problem of hardness amplification in N P, as recently studied by O’Donnell (STOC ‘02). We prove that if N P has a balanced function f such that any circuit of size s(n) fails to compute f on a 1 / poly(n) fraction of inputs, then N P has a function f ′ such that any circuit of size s ..."
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Cited by 34 (6 self)
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We revisit the problem of hardness amplification in N P, as recently studied by O’Donnell (STOC ‘02). We prove that if N P has a balanced function f such that any circuit of size s(n) fails to compute f on a 1 / poly(n) fraction of inputs, then N P has a function f ′ such that any circuit of size
The complexity of constructing pseudorandom generators from hard functions
 Computational Complexity
, 2004
"... Abstract. We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constantdepth circuits. We show that, starting from a function f: {0, 1} l → {0, 1} computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a ..."
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Cited by 44 (9 self)
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computable by constantdepth circuits of size polynomial in n. We also study worstcase hardness amplification, which is the related problem of producing an averagecase hard function starting from a worstcase hard one. In particular, we deduce that there is no blackbox worstcase hardness amplification
On Randomization in Sequential and Distributed Algorithms
 ACM Computing Surveys
, 1994
"... I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, nezther yet bread to the wise, nor yet riches to men of understanding, nor yet favor to men of skdl; but tzme and chance happeneth to them all. Chaos umpue sits, And by decision more embrods the fray ..."
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correctness and its computational complexity. Several related topics of interest are also addressed, including the theory of probabilistic automata, probabilistic analysis of conventional algorithms, deterministic amplification, and derandomization of randomized algorithms. Finally, a comprehensive annotated
Hardness vs. Randomness within Alternating Time
, 2003
"... We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f: {0, 1} l → {0, 1} that is mildly hard on average, i.e. every circuit of size 2 Ω(l) fails to compute f on at least a 1/poly(l) fraction of in ..."
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Cited by 10 (0 self)
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on worstcase hard functions. We also prove a tight lower bound on blackbox worstcase hardness amplification, which is the problem of producing an averagecase hard function starting from a worstcase hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute
Results 1  10
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57