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215
The Complexity of Hardness Amplification and Derandomization
, 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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Cited by 4 (2 self)
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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
On the complexity of hardness amplification
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We study the task of transforming a hard function f, with which any small circuit disagrees on (1 − δ)/2 fraction of the input, into a harder function f ′ , with which any small circuit disagrees on (1 − δ k)/2 fraction of the input, for δ ∈ (0, 1) and k ∈ N. We show that this process can not be car ..."
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Cited by 8 (1 self)
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not be carried out in a blackbox way by a circuit of depth d and size 2 o(k1/d) or by a nondeterministic circuit of size o(k / log k) (and arbitrary depth). In particular, for k = 2 Ω(n) , such hardness amplification can not be done in ATIME(O(1), 2 o(n)). Therefore, hardness amplification in general requires a
Hardness Amplification for Errorless Heuristics
, 2007
"... An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol ⊥, which means “I don’t know. ” A central question in averagecase complexity is whether every distributional decision problem in NP has an errorless heuristic scheme: This is an algorit ..."
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Cited by 3 (0 self)
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: This is an algorithm that, for every δ> 0, runs in time polynomial in the instance size and 1/δ and answers ⊥ only on a δ fraction of instances. We study the question from the standpoint of hardness amplification and show that • If every problem in (NP, U) has errorless heuristic circuits that output the correct
Hardness Amplification within NP
, 2002
"... In this paper we investigate the following question: If NP is slightly hard on average, is it very hard on average? We show the answer is yes; if there is a function in NP which is infinitely often balanced and (11/poly(n))hard for circuits of polynomial size, then there is a function in NP which ..."
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Cited by 48 (1 self)
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In this paper we investigate the following question: If NP is slightly hard on average, is it very hard on average? We show the answer is yes; if there is a function in NP which is infinitely often balanced and (11/poly(n))hard for circuits of polynomial size, then there is a function in NP which
Input Locality and Hardness Amplification
"... We establish new hardness amplification results for oneway functions in which each input bit influences only a small number of output bits (a.k.a. inputlocal functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain the ..."
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Cited by 4 (1 self)
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We establish new hardness amplification results for oneway functions in which each input bit influences only a small number of output bits (a.k.a. inputlocal functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain
Proofs of retrievability via hardness amplification
 In TCC
, 2009
"... Proofs of Retrievability (PoR), introduced by Juels and Kaliski [JK07], allow the client to store a file F on an untrusted server, and later run an efficient audit protocol in which the server proves that it (still) possesses the client’s data. Constructions of PoR schemes attempt to minimize the cl ..."
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Cited by 84 (4 self)
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of Shacham and Waters [SW08]. • Build the first boundeduse scheme with informationtheoretic security. The main insight of our work comes from a simple connection between PoR schemes and the notion of hardness amplification, extensively studied in complexity theory. In particular, our improvements come from
Hardness Amplification in Proof Complexity
, 2009
"... We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as ..."
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Cited by 7 (1 self)
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We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as Th(k) proofs.) As special cases, such systems include: LovászSchrijver systems (LS, LS+), high degree analogues of LovászSchrijver (LS(k), LS+(k)), Cutting Planes and high degree versions of Cutting Planes (CP(k)), as well as SheraliAdams and Lasserre proofs. We introduce two very general families of proof systems, denoted by T cc (k) and R cc (k). The proof lines of T cc (k) are arbitrary Boolean functions, each of which can be evaluated by an efficient kparty randomized communication protocol. T cc (k) proofs are very powerful and include Th(k − 1) proofs as a special case. R cc (k) proofs generalize T cc (k) proofs and require only that each inference be checkable (in a certain weak sense) by an efficient kparty randomized communication protocol. Our main results are the following:
Extractors Using Hardness Amplification
, 2009
"... Zimand [24] presented simple constructions of locally computable strong extractors whose analysis relies on the direct product theorem for oneway functions and on the BlumMicaliYao generator. For Nbit sources of entropy γN, his extractor has seed O(log 2 N)and extracts N γ/3 random bits. We show ..."
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Cited by 4 (0 self)
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Zimand [24] presented simple constructions of locally computable strong extractors whose analysis relies on the direct product theorem for oneway functions and on the BlumMicaliYao generator. For Nbit sources of entropy γN, his extractor has seed O(log 2 N)and extracts N γ/3 random bits. We show that his construction can be analyzed based solely on the direct product theorem for general functions. Using the direct product theorem of Impagliazzo et al. [6], we show that Zimand’s construction can extract ˜ Ωγ(N 1/3) random bits. (As in Zimand’s construction, the seed length is O(log 2 N)bits.) We also show that a simplified construction can be analyzed based solely on the XOR lemma. Using Levin’s proof of the XOR lemma [8], we provide an alternative simpler construction of a locally computable extractor with seed length O(log 2 N) and output length ˜ Ωγ(N 1/3). Finally, we show that the derandomized direct product theorem of Impagliazzo and Wigderson [7] can be used to derive a locally computable extractor construction with O(log N) seed length and ˜ Ω(N 1/5) output length. Zimand describes a construction with O(log N) seed length and O(2 √ log N) output length.
Hardness amplification proofs require majority
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2008
"... Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f i ..."
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Cited by 20 (5 self)
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Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f
Query Complexity in Errorless Hardness Amplification
, 2010
"... An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know ” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know ” on at most a δ fraction ..."
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Cited by 2 (1 self)
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An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know ” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know ” on at most a δ
Results 1  10
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