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56
PseudoRandom Generation from OneWay Functions
 PROC. 20TH STOC
, 1988
"... Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show howto construct a pseudorandom generator from any oneway function. Since it is easy to construct a oneway function from a pseudorandom generator, this result shows that there is a pseudorandom gene ..."
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Cited by 725 (21 self)
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Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show howto construct a pseudorandom generator from any oneway function. Since it is easy to construct a oneway function from a pseudorandom generator, this result shows that there is a pseudorandom generator iff there is a oneway function.
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... , Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [2 ..."
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Cited by 182 (4 self)
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, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...
Simulating BPP Using a General Weak Random Source
 ALGORITHMICA
, 1996
"... We show how to simulate BPP and approximation algorithms in polynomial time using the output from a ffisource. A ffisource is a weak random source that is asked only once for R bits, and must output an Rbit string according to some distribution that places probability no more than 2 \GammaffiR on ..."
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Cited by 106 (19 self)
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We show how to simulate BPP and approximation algorithms in polynomial time using the output from a ffisource. A ffisource is a weak random source that is asked only once for R bits, and must output an Rbit string according to some distribution that places probability no more than 2 \GammaffiR on any particular string. We also give an application to the unapproximability of Max Clique.
Dispersers, Deterministic Amplification, and Weak Random Sources.
, 1989
"... We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of r ..."
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Cited by 93 (11 self)
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We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of random bits and are robust with respect to the quality of the random bits. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2 \Gamman ), with only (3 + ffl)n random bits, as opposed to standard amplification techniques that require \Omega\Gamma n 2 ) random bits for the same task. This result is nearly optimal, since the informati...
Extracting Randomness: A Survey and New Constructions
, 1999
"... this paper we do two things. First, we survey extractors and dispersers: what they are, how they can be designed, and some of their applications. The work described in the survey is due to a long list of research papers by various authors##most notably by David Zuckerman. Then, we present a new tool ..."
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Cited by 90 (5 self)
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this paper we do two things. First, we survey extractors and dispersers: what they are, how they can be designed, and some of their applications. The work described in the survey is due to a long list of research papers by various authors##most notably by David Zuckerman. Then, we present a new tool for constructing explicit extractors and give two new constructions that greatly improve upon previous results. The new tool we devise, a merger," is a function that accepts d strings, one of which is uniformly distributed and outputs a single string that is guaranteed to be uniformly distributed. We show how to build good explicit mergers, and how mergers can be used to build better extractors. Using this, we present two new constructions. The first construction succeeds in extracting all of the randomness from any somewhat random source. This improves upon previous extractors that extract only some of the randomness from somewhat random sources with enough" randomness. The amount of truly random bits used by this extractor, however, is not optimal. The second extractor we build extracts only some of the randomness and works only for sources with enough randomness, but uses a nearoptimal amount of truly random bits. Extractors and dispersers have many applications in removing randomness" in various settings and in making randomized constructions explicit. We survey some of these applications and note whenever our new constructions yield better results, e.g., plugging our new extractors into a previous construction we achieve the first explicit Nsuperconcentrators of linear size and polyloglog(N) depth. ] 1999 Academic Press CONTENTS 1.
Computing with Very Weak Random Sources
, 1994
"... For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6source wath minentropy R‘. Such a weak random source is asked once for R bits; it outputs an Rbit string such that any string has probability at most 2Rc. If 6> 1 l/(k + l), our BPP simulations tak ..."
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Cited by 73 (7 self)
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For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6source wath minentropy R‘. Such a weak random source is asked once for R bits; it outputs an Rbit string such that any string has probability at most 2Rc. If 6> 1 l/(k + l), our BPP simulations take time no(‘og(k)n) (log(k) is the logarithm iterated k times). We also gave a polynomialtime BPP simulation using ChorGoldreich sources of minentropy Ro(’), which is optimal. We present applications to timespace tradeoffs, expander constructions, and the hardness of approximation. Also of interest is our randomnessefficient Leflover Hash Lemma, found independently by Goldreich & Wigderson.
Extracting randomness from samplable distributions
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, ..."
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Cited by 55 (8 self)
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The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, we consider the problem of deterministically converting a weak source of randomness into an almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. In this paper, we look at sources which are samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large minentropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the uniform setting. One of our uniform constructions is as follows: assuming that there are problems in���ÌÁÅ�ÇÒthat are not solvable by subexponentialsize circuits with¦� gates, there is an efficient extractor that transforms any samplable distribution of lengthÒand minentropy Ò into an output distribution of length ÇÒ, whereis any sufficiently small constant. The running time of the extractor is polynomial inÒand the circuit complexity of the sampler. These extractors are based on a connection be
Extracting randomness using few independent sources
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
"... In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, ..."
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Cited by 48 (6 self)
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In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, 1} n, each having minentropy δn. These extractors output n bits, which are 2 −n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with minentropy Ω(log n) and outputs every possible mbit string with positive probability. The main tool we use is a variant of the “steppingup lemma ” used in establishing lower bound
Extracting Randomness via Repeated Condensing
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution ( ..."
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Cited by 43 (16 self)
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On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). In this paper we construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of [NZ96], the observation made in [SZ94, NT99] that a failure of the block extraction scheme is also useful, the recursive "winwin" case analysis of [ISW99, ISW00], and the error correction of random sources used in [Tre99]. As a natural byproduct, (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary minentropy; they...