## Extracting all the Randomness and Reducing the Error in Trevisan's Extractors (1999)

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Venue: | In Proceedings of the 31st Annual ACM Symposium on Theory of Computing |

Citations: | 80 - 17 self |

### BibTeX

@INPROCEEDINGS{Raz99extractingall,

author = {Ran Raz and Omer Reingold and Salil Vadhan},

title = {Extracting all the Randomness and Reducing the Error in Trevisan's Extractors},

booktitle = {In Proceedings of the 31st Annual ACM Symposium on Theory of Computing},

year = {1999},

pages = {149--158}

}

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### Abstract

We give explicit constructions of extractors which work for a source of any min-entropy on strings of length n. These extractors can extract any constant fraction of the min-entropy using O(log² n) additional random bits, and can extract all the min-entropy using O(log³ n) additional random bits. Both of these constructions use fewer truly random bits than any previous construction which works for all min-entropies and extracts a constant fraction of the min-entropy. We then improve our second construction and show that we can reduce the entropy loss to 2 log(1=") +O(1) bits, while still using O(log³ n) truly random bits (where entropy loss is defined as [(source min-entropy) + (# truly random bits used) (# output bits)], and " is the statistical difference from uniform achieved). This entropy loss is optimal up to a constant additive term. our...

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Citation Context ...he weak designs used in the 5srst extractor of Theorem 1 are constructed using an application of the Probabilistic Method, which we then derandomize using the Method of Conditional Expectations (cf., =-=[AS00]-=- and [MR95, Ch. 5]). We then apply a simple iteration to thesesrst weak designs to obtain the weak designs used in the second extractor. We also prove a lower bound showing that our weak designs are n... |

726 | A pseudorandom generator from any one-way function
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Citation Context ...vement in the parameters [Zuc96, NZ96, WZ99, GW97, SZ98, SSZ98, NT98, Zuc97, Ta-98, Tre99]. Most of the work on extractors was based on techniques such as k-wise independence, the Leftover hash lemma =-=[ILL89]-=-, and various forms of composition. A new approach to constructing extractors was recently initiated by Trevisan [Tre99], who discovered a fascinating connection between constructing extractors and co... |

358 | The Complexity of Boolean Functions - Wegener - 1987 |

230 | Randomness is linear in space - Nisan, Zuckerman - 1996 |

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Citation Context ... ) `, and 13 2. For all i, X j 2 H(X j jY i ) (m 1): Then H(X) min ` 2 log 2 ; m` : In Proposition 21, H() denotes the entropy function and H(j) denotes conditional entropy (cf., [CT91]). Impagliazzo and Wigderson [IW97] had previously given a statement like Proposition 21 with the second condition replaced by max i;j H(X j jY i ) log ; ours is a generalization to the analogue of... |

206 | Free bits, PCPs and non-approximability — towards tight results - Bellare, Goldreich, et al. - 1998 |

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Citation Context ...H(X j jY i ) (m 1): Then H(X) min ` 2 log 2 ; m` : In Proposition 21, H() denotes the entropy function and H(j) denotes conditional entropy (cf., [CT91]). Impagliazzo and Wigderson [IW97] had previously given a statement like Proposition 21 with the second condition replaced by max i;j H(X j jY i ) log ; ours is a generalization to the analogue of \weak designs." The proof directly... |

130 | Pseudorandom generators without the XOR lemma - Sudan, Trevisan, et al. - 2001 |

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Citation Context ...convert the extractors of the present paper into ones whose seed length depends additively on O(log 1=") (for extracting up to a constant fraction of the min-entropy). Reingold, Vadhan, and Wigde=-=rson [RVW00] have-=- concentrated on the case of \high min-entropy", giving constructions with seed-length (poly)logarithmic in n k, improving the linear dependence on n k in [GW97] and in contrast to most works on ... |

110 | Simulating BPP using a general weak random source
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Citation Context .... This is the starting point for our work. Explicit constructions of extractors and dispersers have a wide variety of applications, including simulating randomized algorithms with weak random sources =-=[Zuc96]-=-; constructing oblivious samplers [Zuc97]; constructive leader election [Zuc97, RZ98]; randomness-ecient error reduction in randomized algorithms and interactive proofs [Zuc97]; explicit constructions... |

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Citation Context ...tS (u; y) = NWS;u (y) = u(yj S1 ) u(yj Sm ): We will now analyze this extractor. The following lemma, due to Yao, allows us to focus on \next-bit predictors" instead of distinguishers. Lemma 8=-= ([Yao82]) -=-Suppose that hY; Zi is a distribution on f0; 1g d f0; 1g m s.t. Y is uniformly distributed over f0; 1g d yet the statistical dierence of hY; Zi from U d Um is greater than ". Then there is an i ... |

90 | Extracting randomness: A survey and new constructions
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- 1999
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Citation Context ...mizing BPP under circuit complexity assumptions [ACR97, STV99]; and other problems in complexity theory [Sip88, GZ97]. For a detailed survey of previous work on extractors and their applications, see =-=[NT98]. Main r-=-esults Thesrst family of extractors constructed in this paper is given in the following theorem: Theorem 1 For every n; k; m 2 N and " > 0, such that m k n, there are explicit (k; ")-extra... |

90 | Expanders that beat the eigenvalue bound: Explicit construction and applications
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- 1999
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Citation Context ...ed by replacing the \combinatorial designs" underlying the Nisan{Wigderson generator and Trevisan's construction with a weaker (and more suitable) notion. Applying a result of Wigderson and Zucke=-=rman [WZ99]-=- to these extractors, we also obtain improved constructions of highly expanding graphs and superconcentrators. The second idea improves Trevisan's construction in the case where the output bits are re... |

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- 1999
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Citation Context ...igure 2 indicates; it is a strong extractor with an entropy loss of n k +O(log(1=")) (though this is only interesting when k is very close to n). In addition, the \tiny families of hash functions=-=" of [SZ98] give-=- strong extractors with d = O(k + log n) and entropy loss 2 log(1=") +O(1); these have optimal entropy loss but are only interesting when k is very small (e.g., k = polylog n), as d is linear in ... |

61 |
Nisan and Avi Wigderson. Hardness vs randomness
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Citation Context ...signs. The main combinatorial objects underlying the Nisan{Wigderson pseudorandom generator and subsequently Trevisan's extractors are collections of sets with small pairwise intersections. Following =-=[N-=-W94], we will refer to these as designs, but in the combinatorics literature, they are often called packings (cf., [AS00, Sec. 4.7]). Denition 5 For ` 2 N and 1, a family of sets S 1 ; : : : ; Sm [... |

60 | On extracting randomness from weak random sources - Ta-Shma - 1999 |

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Construction of extractors using pseudorandom generators
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- 1999
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Citation Context ...a catalyst. A large body of work has focused on giving explicit constructions of extractors, as such constructions have a wide variety of applications. A recent breakthrough was made by Luca Trevisan =-=[Tre99]-=-, who discovered that the Nisan{Wigderson pseudorandom generator [NW94], previously only used in a computational setting, could be used to construct extractors. For certain settings of the parameters,... |

54 | Strong communication complexity or generating quasirandom sequences from two communicating semi-random sources - Vazirani - 1987 |

50 |
Randomness-optimal oblivious sampling. Random Structures and Algorithms
- Zuckerman
- 1997
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Citation Context .... Explicit constructions of extractors and dispersers have a wide variety of applications, including simulating randomized algorithms with weak random sources [Zuc96]; constructing oblivious samplers =-=[Zuc97]-=-; constructive leader election [Zuc97, RZ98]; randomness-ecient error reduction in randomized algorithms and interactive proofs [Zuc97]; explicit constructions of expander graphs, superconcentrators, ... |

47 |
and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity
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Citation Context ...lved using standard techniques. Specically, the fact that the i'th component of the output is unpredictable given thesrst i 1 components means that the output is what is known as a block-wise source [=-=CG88]-=-. In our case, the block-wise source has blocks of logarithmic length, and standard techniques can be used to extract truly random bits from such a source using a small number of additional truly rand... |

43 | Extracting randomness via repeated condensing
- Reingold, Shaltiel, et al.
- 1999
(Show Context)
Citation Context ... constructed deterministically in time poly(N). 4 However, subsequent to this work, it was shown how to transform any extractor into a strong extractor, with only a small cost in the other parameters =-=[RSW00-=-]. 4 In particular, in order for the output of the extractor to have statistical dierence :01 from uniform, one need only lose a constant number of bits of entropy. A comparison of this result with pr... |

42 | Salil Vadhan. Pseudorandom generators without the XOR lemma - Sudan, Trevisan - 1999 |

41 | Extractors and pseudo-random generators with optimal seed length. STOC
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- 2000
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Citation Context ... weak designs based on low-degree polynomials. (Our construction of weak designs, based on the Method of Conditional Expectations, appears inherently sequential.) Impagliazzo, Shaltiel, and Wigderson =-=[ISW00]-=- have also given better extractor constructions by improving the Nisan{Wigderson pseudorandom generator and applying Trevisan's connection. They show how to build extractors with seed length O(log n) ... |

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37 | deterministic amplification, and weak random sources - Dispersers - 1989 |

34 | On recycling the randomness of the states in space bounded computation
- Raz, Reingold
- 1999
(Show Context)
Citation Context ... quadratically on log(1="). In (nonconstructive) optimal extractors and even some previous constructions (discussed later), this dependence is linear. Indeed, some applications of extractors, suc=-=h as [RR99], requir-=-e a linear dependence. In our second theorem, we improve our extractors to have a linear dependence on log(1="). Theorem 2 For every n, k, m, and ", such that m k n, there are explicit (k;... |

34 | Efficient considerations in using semi-random sources - Vazirani - 1987 |

23 | Error reduction for extractors
- Raz, Reingold, et al.
- 1999
(Show Context)
Citation Context ...y(n)). In this case, Theorem 2 gives d = O (log n log(1=")) ; which is an improvement for small ". We only provide a sketch of Theorem 2, because the results have been superseded by our rec=-=ent work [RRV99a]-=- which gives a general method to reduce the error of any extractor. A summary of our results is given in Figure 1, and a comparison with the best previous constructions is given in Figure 2. Trevisan'... |

23 | Almost optimal dispersers
- Ta-Shma
- 1998
(Show Context)
Citation Context ...put length m additional randomness d type [GW97] any k m = k d = O(n k + log(1=")) extractor [Zuc97] k =sn) m = (1 )k d = O(log(n=")) extractor [NT98] any k m = k d = O(log 9 n log(1="=-=)) extractor [Ta-98] any k m = k -=-polylog(n) d = O(log(n=")) disperser [Tre99] any k m = k 1 d = O(log 2 (n=")= log k) extractor ultimate goal any k m = k d = O(log(n=")) extractor Above, is an arbitrarily small const... |

23 | Worst-case hardness suffices for derandomization: A new method for hardness vs randomness trade-offs - Andreev, Clementi, et al. - 1999 |

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21 | Constructions of near-optimal extractors using pseudo-random generators - Trevisan - 1999 |

21 | Randomness-optimal oblivious sampling - Zuckerman - 1997 |

19 |
randomness, or time versus space
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(Show Context)
Citation Context ...d \ln" denotes the natural logarithm. If X is a probability distribution on asnite set, we write x X to indicate that x is selected according to X . Previous work Dispersers weresrst dened by Sip=-=ser [Sip88] and-=- extractors weresrst dened by Nisan and Zuckerman [NZ96]. Much of the motivation for research on extractors comes from work done on \somewhat random sources" [SV86, CG88, Vaz87b, VV85, Vaz84, Vaz... |

17 | Perfect information leader election in log ∗ n + O(1) rounds - Russell, Zuckerman |

17 | Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR - Russell - 1997 |

14 | On the distribution of the number of roots of polynomials and explicit logspace extractors
- Hartman, Raz
- 2003
(Show Context)
Citation Context ...ctor constructions, which bring us closer to (but not yet at) the ultimate goal of optimal extractors for all settings of parameters. The work most directly related to ours is that of Hartman and Raz =-=[HR00-=-], which is not concerned with improving the parameters but rather the explicitness. Specically, they show how to construct extractors with the same parameters as ours, but which are computable in log... |

13 | Explicit ORdispersers with polylogarithmic degree - Saks, Srinivasan, et al. - 1998 |

9 | Another proof that BPP - Goldreich, Zuckerman - 1997 |

9 | Salil Vadhan. Error reduction for extractors - Raz, Reingold - 1999 |

8 | Santha and Umesh V. Vazirani. Generating quasi-random sequences from semi-random sources - Miklos - 1986 |

7 | Worst-case hardness su ces for derandomization: A new method for hardness-randomness trade-o s - Andreev, Clementi, et al. - 1997 |

7 |
Salil Vadhan. Extracting all the randomness and reducing the error in Trevisan’s extractors
- Raz, Reingold
- 2002
(Show Context)
Citation Context ... by Trevisan. Keywords: Extractors, Combinatorial Designs, Expander Graphs, Probabilistic Method, Pseudorandom Generators Preliminary versions of this work appeared in STOC `99 [RRV99b] and on ECCC [=-=RRV99c]-=-. y Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot, 76100 Israel. E-mail: ranraz@wisdom.weizmann.ac.il Work supported by an American-Israeli BSF grant 95-00238 and... |

5 | Hardness of Approximating # p 2 Minimization Problems - Umans - 1999 |