## Constructions of Near-Optimal Extractors Using Pseudo-Random Generators (1998)

Venue: | Electronic Colloquium on Computational Complexity |

Citations: | 20 - 3 self |

### BibTeX

@INPROCEEDINGS{Trevisan98constructionsof,

author = {Luca Trevisan},

title = {Constructions of Near-Optimal Extractors Using Pseudo-Random Generators},

booktitle = {Electronic Colloquium on Computational Complexity},

year = {1998},

pages = {141--148},

publisher = {ACM Press}

}

### OpenURL

### Abstract

We introduce a new approach to construct extractors --- combinatorial objects akin to expander graphs that have several applications. Our approach is based on error correcting codes and on the Nisan-Wigderson pseudorandom generator. A straightforward application of our approach yields a construction that is simple to describe and analyze, does not use any of the standard techniques used in related results, and improves or subsumes almost all the previous constructions. 1 Introduction Informally defined, an extractor is a function that extracts randomness from a weakly random distribution. Explicit constructions of extractors have several applications and are typically very hard to achieve. In this paper we introduce a new approach to the explicit construction of extractors. Our approach yields a construction that improves most of the known results, and that is optimal for certain parameters. Furthermore, our construction is simple and uses techniques that were never used in this field...

### Citations

529 |
Theory and applications of trapdoor functions
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Citation Context ...to be able to use the Chernoff bound, and then we will discard duplicates.) 2 A.2 A Sketch of the Proof of Lemma 5 The following result will be used. Lemma 9 (Distinguishability versus Predictability =-=[Yao82]) Let T : -=-f0; 1g m ! f0; 1g, g : f0; 1g m\Gamma1 ! f0; 1g, f : f0; 1g l ! f0; 1g and " ? 0; if j Pr x2f0;1g l [T (g(x); f(x)) = 1] \Gamma Pr x2f0;1g l ;r2f0;1g [T (g(x); r) = 1]js" then there exists t... |

282 | Hardness vs. Randomness
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Citation Context ...ield: we do not use any of the standard techniques (hash functions in combination with the leftover hash lemma, composition, etc.), whereas our main tool is the Nisan-Wigderson pseudorandom generator =-=[NW94]-=-, which we use for the first time in a framework where information-theoretic randomness is being studied. It is known that a Nisan-Wigderson generator constructed from a fixed hard function transforms... |

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179 | Approximating clique is almost NP-complete - Feige, Goldwasser, et al. - 1991 |

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Citation Context ... random sources. This research area originates from early work by Vazirani and Vazirani [VV85], Santha and Vazirani [SV86], Vazirani [Vaz86, Vaz87], Chor and Goldreich [CG88], and Cohen and Wigderson =-=[CW89]-=- who defined increasingly general models of weak random sources. The recognition of min-entropy as the “right” parameter to measure the amount of algorithmically usable randomness in a source is due t... |

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Citation Context ...ose parameters are optimal for these applications). Construction of extractors (but dispersers would suffice) also yield construction of expander graphs, superconcentrators, and sorting networks. See =-=[WZ93]-=- for results establishing this connection. Constructions of extractors and dispersers yielding tight constructions of expanders, superconcentrators and sorting networks are still not known, though pro... |

88 | Extracting randomness: A survey and new constructions
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Citation Context ...e literature on explicit construction of extractors and dispersers is vast and technically challenging. An excellent and accessible introduction is given by a recent survey by Nisan [Nis96] (see also =-=[NTS98]-=-). In Table 1 we summarize the best known constructions, for different combination of the parameters, and we state the parameters of (a special case of) our construction. Our Main Result. In this pape... |

76 | Computing with Very Weak Random Sources - Srinivasan, Zuckerman - 1994 |

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61 | On extracting randomness from weak random sources
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Citation Context ...r every n an explicit (n; n fl ; n fl 0 ; O(logn); ")-extractor. Reference Min entropy k Output length m Additional randomness Type [Zuc96b] k = \Omega\Gamma n) m = \Omega\Gamma k) O(log n) Extra=-=ctor [TS96]-=- any k k poly log n Extractor [TS96] k = n\Omega\Gamma1/ m = k\Omega\Gamma29 O(log n log \Delta \Delta \Delta log n) Extractor [SSZ98] k = n\Omega\Gamma20 m = k\Omega\Gamma29 O(log n) Disperser [TS98]... |

60 |
General weak random sources
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Citation Context ... increasingly general models of weak random sources. The recognition of min-entropy as the "right" parameter to measure the amount of algorithmically usable randomness in a source is due to =-=Zuckerman [Zuc90]-=-. Extractors allow to use weak random sources in order to simulate every BPP algorithms; dispersers allow for simulation of RP algorithms. Several constructions of extractors and dispersers [NZ93, SZ9... |

58 |
Generating quasirandom sequences from slightly-random sources (extended abstract
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Citation Context ... constructions of such objects is the simulation of randomized algorithms using weak random sources. This research area originates from early work by Vazirani and Vazirani [VV85], Santha and Vazirani =-=[SV86], Vazirani-=- [Vaz86, Vaz87], Chor and Goldreich [CG88], and Cohen and Wigderson [CW89] who defined increasingly general models of weak random sources. The recognition of min-entropy as the "right" param... |

57 |
Random polynomial time is equal to slightly random polynomial time. FOCS
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Citation Context ...ain applications of explicit constructions of such objects is the simulation of randomized algorithms using weak random sources. This research area originates from early work by Vazirani and Vazirani =-=[VV85]-=-, Santha and Vazirani [SV86], Vazirani [Vaz86, Vaz87], Chor and Goldreich [CG88], and Cohen and Wigderson [CW89] who defined increasingly general models of weak random sources. The recognition of min-... |

47 | Two prover protocols - low error at affordable rates - Feige, Kilian - 1994 |

43 | Adversaries and Computation - Vazirani, \Randomness - 1986 |

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40 | Weak random sources, hitting sets, and BPP simulations
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Citation Context ...een achieved in this respect by Saks et al. [SSZ98] for RP algorithms, by finding an explicit construction of (n; n fl ; n fl 0 ; O(logn); 1=2)- dispersers for every 0 ! fl ! fl 0 ! 1. Andreev et al. =-=[ACRT97]-=- showed how to use dispersers in order to simulate BPP algorithms (their result is based on techniques from [ACR98]). The result of [ACRT97], together with the dispersers of [SSZ98] implies an optimal... |

37 |
deterministic amplification, and weak random sources
- Dispersers
- 1989
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Citation Context ... random sources. This research area originates from early work by Vazirani and Vazirani [VV85], Santha and Vazirani [SV86], Vazirani [Vaz86, Vaz87], Chor and Goldreich [CG88], and Cohen and Wigderson =-=[CW89] who defin-=-ed increasingly general models of weak random sources. The recognition of min-entropy as the "right" parameter to measure the amount of algorithmically usable randomness in a source is due t... |

37 | On Unapproximable Versions of NP-Complete Problems - Zuckerman - 1996 |

36 |
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Citation Context ...fl ; n fl 0 ; O(logn); 1=2)- dispersers for every 0 ! fl ! fl 0 ! 1. Andreev et al. [ACRT97] showed how to use dispersers in order to simulate BPP algorithms (their result is based on techniques from =-=[ACR98]-=-). The result of [ACRT97], together with the dispersers of [SSZ98] implies an optimal simulation of BPP algorithms. The existence of extractors strong enough to give directly an optimal simulation of ... |

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29 |
P = BPP unless E has sub-exponential circuits
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Citation Context ...e sampled strings u. One way to improve the construction, and to prove a slightly worse version of the Main Theorem, would be to use a pseudorandom generator construction by Impagliazzo and Wigderson =-=[IW97]-=-. The generator IW \Delta (\Delta) of Impagliazzo and Wigderson has the property that if f is a boolean function and T is a test that distinguishes IW f (\Delta) from the uniform distribution, then f ... |

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Citation Context ...ndreev, Clementi and Rolim [ACR97] use dispersers to prove that certain circuit-complexity assumptions imply P=BPP (without dispersers they would need a stronger assumption) . Goldreich and Zuckerman =-=[GZ97]-=- show how to use constructions of extractors to give a simple proof that MA is in ZPP NP . An open question that may be solved by better construction of dispersers is to prove that Max Clique is not a... |

23 |
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Citation Context ...d sorting networks are still not known, though progress was made in [NZ93, SZ94, SSZ98, TS96, TS98]. Other applications have been found more recently in complexity theory: Andreev, Clementi and Rolim =-=[ACR97]-=- use dispersers to prove that certain circuit-complexity assumptions imply P=BPP (without dispersers they would need a stronger assumption) . Goldreich and Zuckerman [GZ97] show how to use constructio... |

23 | Almost optimal dispersers
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Citation Context ...[TS96] any k k poly log n Extractor [TS96] k = n\Omega\Gamma1/ m = k\Omega\Gamma29 O(log n log \Delta \Delta \Delta log n) Extractor [SSZ98] k = n\Omega\Gamma20 m = k\Omega\Gamma29 O(log n) Disperser =-=[TS98]-=- any k m = k 1\Gammao(1) O(log n) Disperser This paper k = n\Omega\Gamma20 m = k\Omega\Gamma29 O(log n) Extractor Table 1: A summary of previous results and our result. Our construction improves on th... |

22 |
Randomness-optimal sampling, extractors, and constructive leader election
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Citation Context ...also yield oblivious samplers, with applications to randomnessefficient reduction of error in randomized algorithms and in interactive proof-systems, and to leader election in anonymous networks (see =-=[Zuc96b]-=- for a construction whose parameters are optimal for these applications). Construction of extractors (but dispersers would suffice) also yield construction of expander graphs, superconcentrators, and ... |

21 |
Extracting randomness: How and why
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- 1996
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Citation Context ... that more applications of extractors will be found in the future. Nisan remarks that extractors "exhibit some of the most `random-like' properties of explicitly constructed combinatorial structu=-=res" [Nis96]-=-. The literature on explicit construction of extractors and dispersers is vast and technically challenging. An excellent and accessible introduction is given by a recent survey by Nisan [Nis96] (see a... |

18 |
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- 1988
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Citation Context ...ce-versa. Indeed, constructions of dispersers are somewhat easier than constructions of extractors with the same parameters. Previous Results and Applications. Dispersers were first defined by Sipser =-=[Sip88]-=-, while extractors were first defined by Nisan and Zuckerman [NZ93]. Extractors and dispersers are useful lucat@dimacs.rutgers.edu. DIMACS Center, Rutgers University, Piscataway, NJ & Columbia Univers... |

13 | Explicit ORdispersers with polylogarithmic degree
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- 1998
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Citation Context ...n of RP algorithms. Several constructions of extractors and dispersers [NZ93, SZ94, SSZ98, TS96] were motivated by this application. An optimal result has been achieved in this respect by Saks et al. =-=[SSZ98]-=- for RP algorithms, by finding an explicit construction of (n; n fl ; n fl 0 ; O(logn); 1=2)- dispersers for every 0 ! fl ! fl 0 ! 1. Andreev et al. [ACRT97] showed how to use dispersers in order to s... |

10 |
Clique is hard to approximate within n 1\Gamma
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- 1999
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Citation Context ...s to prove that Max Clique is not approximable within n 1\Gamma" unless P=NP (the current randomized reduction [FGL + 91, Zuc96a, FK94, BS94] from PCP to Max Clique and the PCP construction of Ha=-=stad [Has97] only-=- imply the somewhat weaker consequence that ZPP=NP). It is likely that more applications of extractors will be found in the future. Nisan remarks that extractors "exhibit some of the most `random... |

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6 | Extracting all the randomness from a weakly random source
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- 1998
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Citation Context ...hat the output is statistically close, rather than computationally indistinguishable, from the uniform distribution). Later results. Shortly after the development of the results of this paper, Vadhan =-=[Vad98]-=- showed how to reduce the number of additional random bits that are used in our construction when the length of the output is required to be very close to the min-entropy of the source. In our constru... |

1 |
Lecture notes for 6.042: Mathematics for computer science, mit, fall'97. Lecture 25. Available at http://theory.lcs.mit.edu/classes/6.042/Fall97-pub
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Citation Context ...s such that E [ P i X i ] = ��. Then, for every ff ? 1 it holds Pr[ X i X isff��]se \Gamma((ln ff)+ 1 ff \Gamma1)ff�� This bound is proved in the standard way, and a proof can be found for=-= example in [LV97]-=-. We can now sketch the proof of Lemma 3 as it was carried on in [NW94]. Proof:[Of Lemma 3] Sequentially choose m subsets of [d] such that any of the chosen subsets intersects the previously chosen on... |

1 |
Another proof that BP P ⊆ P H (and more
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Citation Context ...Andreev, Clementi and Rolim [ACR97] use dispersers to prove that certain circuit-complexity assumptions imply P=BPP (without dispersers they would need a stronger assumption). Goldreich and Zuckerman =-=[GZ97]-=- show how to use constructions of extractors to give a simple proof that MA is in ZPP NP . An open question that may be solved by better construction of dispersers is to prove that Max Clique is not a... |