Pseudorandom generators without the XOR lemma

Sudan, Madhu; Trevisan, Luca; Vadhan, Salil

doi:10.1109/ccc.1999.766253

Cited by 111 publications

(162 citation statements)

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“…Worst-case to average-case reductions have been shown to exist for complete languages in high complexity classes such as EXP, PSPACE, and P (where hardness is measured both against uniform and nonuniform classes) Babai et al (1993); Impagliazzo & Wigderson (1997); Sudan et al (2001); Trevisan & Vadhan (2006). In contrast, no worst-case to average-case reductions are known for NP-complete problems.…”

Section: Worst-case To Average-case Reductions In Npmentioning

confidence: 99%

“…Here, in contrast to the problem of worst-case to average-case reductions there are many positive results for average classes. It is known that with respect to the uniform distribution (and by Impagliazzo & Levin (1990), it is enough to consider this distribution) hardness amplification can be done for various complexity classes and models of computations and in particular, can be done for functions in EXP and in NP (with various parameters) where the hardness is measured both against uniform and non-uniform classes Babai et al (1993); Healy et al (2006); Impagliazzo & Wigderson (1997);O'Donnell (2004); Sudan et al (2001); Trevisan ( , 2005; Trevisan & Vadhan (2006); Yao (1982).…”

Section: Hardness Amplificationmentioning

confidence: 99%

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If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

2007

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We prove that if NP ⊆ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution. This is the first worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean that there exists one fixed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way functions and cryptography (even if not a sufficient assumption). Nevertheless, we do show that there is a fixed distribution on instances of NPcomplete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an efficient (probabilistic) algorithm that fails to solve SAT in the worst case, we can efficiently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at least one of them.

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Section: Worst-case To Average-case Reductions In Npmentioning

confidence: 99%

Section: Hardness Amplificationmentioning

confidence: 99%

If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

2007

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“…An important question is: given a δ-hard function for size s, can we transform it into a harder function with hardness δ > δ for size about s? This is known as the task of hardness amplification, and it plays a crucial role in the study of derandomization, in which one would like to obtain an extremely hard function, with hardness (1 − ε)/2, so that it looks like a random function and can be used to construct pseudorandom generators, see Babai et al (1993), Impagliazzo (1995), Impagliazzo & Wigderson (1997), Sudan et al (2001), Yao (1982).…”

Section: Introductionmentioning

confidence: 99%

“…It can be used to provide an alternative proof of Yao's celebrated XOR Lemma, see Impagliazzo (1995), or to construct a pseudo-random generator directly from a mildly hard function, bypassing the XOR lemma, see Sudan et al (2001). Recently, it has become a key ingredient in the study of hardness amplification for functions in NP, see Healy et al (2006), O'Donnell (2004), Trevisan (2003Trevisan ( , 2005.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Hard-Core Set Proofs

Tsai

2011

comput. complex.

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We study a fundamental result of Impagliazzo (FOCS'95) known as the hard-core set lemma. Consider any function f : {0, 1} n → {0, 1} which is "mildly hard", in the sense that any circuit of size s must disagree with f on at least a δ fraction of inputs. Then, the hardcore set lemma says that f must have a hard-core set H of density δ on which it is "extremely hard", in the sense that any circuit of size s = O(s/( 1 ε 2 log( 1 εδ ))) must disagree with f on at least (1−ε)/2 fraction of inputs from H. There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low-level complexity class. We introduce two models of hard-core set proofs, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models. First, we show that using any strongly black-box proof, one can only prove the hardness of a hard-core set for smaller circuits of size at most s = O(s/( 1 ε 2 log 1 δ )). Next, we show that any weakly black-box proof must be inherently non-uniform-to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with Ω( 1 ε log |G|) bits of advice. Finally, we show that weakly black-box proofs in general cannot be realized in a low-level complexity class such as AC 0 [p]-the assumption that f is hard for AC 0 [p] is not sufficient to guarantee the existence of a hard-core set.

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“…This result has been subsequently simplified and extended to get derandomization results for a range of parameters (Impagliazzo et al 1999(Impagliazzo et al , 2000Shaltiel & Umans 2001;Sudan et al 2001;Umans 2003).…”

Section: Introductionmentioning

confidence: 99%

Derandomizing Arthur–Merlin Games using Hitting Sets

Miltersen

Vinodchandran

2005

comput. complex.

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We prove that AM (and hence Graph Nonisomorphism) is in NP if for some ǫ > 0, some language in NE ∩ coNE requires nondeterministic circuits of size 2 ǫn . This improves results of Arvind and Köbler and of Klivans and van Melkebeek who proved the same conclusion, but under stronger hardness assumptions. The previous results on derandomizing AM were based on pseudorandom generators. In contrast, our approach is based on a strengthening of Andreev, Clementi and Rolim's hitting set approach to derandomization. As a spin-off, we show that this approach is strong enough to give an easy proof of the following implication: for some ǫ > 0, if there is a language in E which requires nondeterministic circuits of size 2 ǫn , then P = BPP. This differs from Impagliazzo and Wigderson's theorem "only" by replacing deterministic circuits with nondeterministic ones.

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Pseudorandom generators without the XOR lemma

Cited by 111 publications

References 4 publications

If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

Complexity of Hard-Core Set Proofs

Derandomizing Arthur–Merlin Games using Hitting Sets

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