## Extracting All the Randomness from a Weakly Random Source (1998)

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### BibTeX

@MISC{Vadhan98extractingall,

author = {Salil Vadhan},

title = {Extracting All the Randomness from a Weakly Random Source},

year = {1998}

}

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### Abstract

In this paper, we give two explicit constructions of extractors, both of which work for a source of any min-entropy on strings of length n. The first extracts any constant fraction of the min-entropy using O(log 2 n) additional random bits. The second extracts all the min-entropy using O(log 3 n) additional random bits. Both 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. The extractors are obtained by observing that a weaker notion of "combinatorial design" suffices for the Nisan--Wigderson pseudorandom generator [NW94], which underlies the recent extractor of Trevisan [Tre98]. We give near-optimal constructions of such "weak designs" which achieve much better parameters than possible with the notion of designs used by Nisan--Wigderson and Trevisan. 1 Introduction Roughly speaking, an extractor is a function which extracts truly random bits from a weakly random source,...

### Citations

1724 | The Probabilistic Method
- Alon, Spencer
- 2000
(Show Context)
Citation Context ...nding designs. The weak designs used in our rst extractor are constructed using an application of the Probabilistic Method, which we then derandomize using the Method of Conditional Expectations (see =-=[ASE92]-=- and [MR95, Ch. 5]). We then apply a simple iteration to these rst weak designs to obtain the weak designs used in our second extractor. We also prove alower bound showing that our weak designs are ne... |

1252 |
Probabilistic encryption
- Goldwasser, Micali
- 1984
(Show Context)
Citation Context ...3 as Pr [D (r1 r1 rm rm) =1], Pr y [D (P (yjS1) P (yjSm)) = 1] >"; where r1;:::;rm are uniformly and independently selected bits and y is uniformly selected fromf0; 1g d . By the \hybrid argument=-=" of [GM84] (cf.-=- [Gol95, Sec. 3.2.3]), there is an i such that Pr y;ri rm D , P (yjS1) P (yjSi,1)ri rm =1, Pr [D (P (yjS1) P (yjSi)ri+1 rm) =1]> "=m y;ri+1 rm Now, renaming ri as b and using the standard transfo... |

760 | A pseudorandom generator from any one-way function
- HĚŠastad, Impagliazzo, et al.
- 1999
(Show Context)
Citation Context ...teady improvement inthe parameters [Zuc96, NZ96, WZ95, GW97, SZ98, SSZ98, NT98, TS98, Tre98]. Most of the work on extractors is based on techniques such ask-wise independence, the Leftover hash lemma =-=[ILL89]-=-, and various forms of composition. A new approach to constructing extractors was recently initiated by Trevisan [Tre98], who discovered that the Nisan{Wigderson pseudorandom generator [NW94] could be... |

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

143 | Foundations of Cryptography (Fragments of a Book). URL: http://www.wisdom.weizmann.ac.il/home/oded/public html/frag.html - Goldreich |

125 | Modern Cryptography, Probabilistic Proofs and Pseudorandomness. Algorithms and Combinatorics - Goldreich - 1998 |

109 | Simulating BPP using a general weak random source
- Zuckerman
- 1996
(Show Context)
Citation Context ...W94] could be used to construct extractors. Explicit constructions of extractors and dispersers have a wide variety ofapplications, including simulating randomized algorithms with weak random sources =-=[Zuc96]-=-; constructing oblivious samplers [Zuc97]; constructive leader election [Zuc97]; randomness e cient error-reduction in randomized algorithms and interactive proofs [Zuc97]; explicit constructions of e... |

102 |
Theory and Applications of Trapdoor Functions (extended abstract
- Yao
- 1982
(Show Context)
Citation Context ...such that Pr y;ri rm D , P (yjS1) P (yjSi,1)ri rm =1, Pr [D (P (yjS1) P (yjSi)ri+1 rm) =1]> "=m y;ri+1 rm Now, renaming ri as b and using the standard transformation from distinguishers to predic=-=tors [Yao82] (cf.-=- [Gol98, Sec. 3.3.3]), we see that Pr y;b;ri+1 rm D , P (yjS1) P (yjSi,1)bri+1 rm b = P (yjSi) > 1 " + 2 m 7sUsing an averaging argument we can x all the bits of y outside Si while preserving the... |

94 | Generating quasi-random sequences from semi-random sources - Santha, Vazirani - 1986 |

89 | Expanders that beat the eigenvalue bound: Explicit construction and applications
- Wigderson, Zuckerman
- 1999
(Show Context)
Citation Context ...truction. This is accomplished by improving the combinatorial construction underlying the Nisan{Wigderson generator used in Trevisan's construction. Applying a construction of Wigderson and Zuckerman =-=[WZ95]-=-, we also obtain improved expanders. Extractors. A distribution X onf0; 1g n is said to have min-entropy k if for all x2f0; 1g n ,Pr[X = x] 2 ,k . Think of this as saying that X has \k bits of randomn... |

87 | Extracting randomness: A survey and new constructions
- Nisan, Ta-Shma
- 1999
(Show Context)
Citation Context ...orandom generators for space-bounded computation [NZ96]; and other problems in complexity theory [Sip88, GZ97, ACR97]. For a detailed survey of previous work on extractors and their applications, see =-=[NT98]. Our resu-=-lts. In this paper, we construct two extractors: Theorem 1 For every n, k, m, and ", such that m k n, there are explicit (k; ")-extractors Ext:f0; 1g n f0; 1g d !f0; 1g m with 1. d = O log2 ... |

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

60 | On extracting randomness from weak random sources - Ta-Shma - 1996 |

57 |
Nisan and Avi Wigderson. Hardness vs. randomness
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- 1994
(Show Context)
Citation Context ... gi2I is explicit if Exti can be evaluated in time poly(ni;di). 3 Combinatorial designs The combinatorial construction underlying the Nisan{Wigderson generator are combinatorial designs. De nition 4 (=-=[NW94]-=-) 2 A family of sets S1;:::;Sm [d] is an (`; )-design if 1. For all i,jSij = `. 2. For all i6= j,jSi\ Sjj log . 2 There is a somewhat related notion in the combinatorics literature known as a 2-design... |

56 | Random polynomial time is equal to slightly-random polynomial time - Vazirani, Vazirani - 1985 |

53 | Extracting randomness: How and why: A survey - Nisan - 1996 |

51 | Strong communication complexity or generating quasirandom sequences from two communicating semirandom sources - Vazirani - 1987 |

50 | and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity - Chor - 1988 |

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

36 | Sorting and selecting in rounds - Pippenger - 1987 |

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

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

24 | Almost optimal dispersers
- Ta-Shma
- 2002
(Show Context)
Citation Context ...m bits in comparison to a polynomial of unspeci ed (and presumably large) degree in log n. Both of our extractors use more truly random bits than the extractors of [Zuc97, Tre98] and the disperser of =-=[TS98]-=-, but our extractors have the advantage that they work for any min-entropy (unlike [Zuc97]) and extract all (or a constant fraction) of the min-entropy (unlike [TS98, Tre98]). The disadvantage of the ... |

20 |
or Time versus Space
- Expanders
- 1988
(Show Context)
Citation Context ...he k bits of hidden randomness, one can hope to have m be close to k + d. This will be discussed in more detail under the heading \Entropy loss." 1sPrevious work. Dispersers were rst de ned by Si=-=pser [Sip88] and -=-extractors were rst de ned 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... |

20 | Constructions of near-optimal extractors using pseudo-random generators
- Trevisan
- 1999
(Show Context)
Citation Context ...ts as a catalyst. A large body of work has focused on giving explicit constructions of extractors, as such constructions have a number of applications. A recent breakthrough was made by Luca Trevisan =-=[Tre98]-=-, who discovered that the Nisan--Wigderson pseudorandom generator [NW94], previously only used in a computational setting, could be used to construct extractors. Trevisan's extractor improves on most ... |

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

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

7 | Worst-case hardness su#ces for derandomization: A new method for hardness vs randomness trade-o#s - Andreev, Clementi, et al. - 1999 |

5 | ciency considerations in using semi-random sources - Vazirani, E - 1987 |

2 |
Designs and their codes. Number 103
- Assmus, Key
- 1992
(Show Context)
Citation Context ... of sets S1;:::;Sm [d] is an (`; )-design if 1. For all i,jSij = `. 2. For all i6= j,jSi\ Sjj log . 2 There is a somewhat related notion in the combinatorics literature known as a 2-design (see, e.g. =-=[AK92]-=-). In 2-designs, strong additional regularity requirements are imposed (such as all the pairwise intersections being exactly the same size and all points being contained in the same number of sets). T... |

2 | Endre Szemeredi. Almost sorting in one round - Ajtai, Komlos, et al. - 1989 |

1 |
Simple and improved construction of extractors. Unpublished manuscript
- Trevisan
- 1998
(Show Context)
Citation Context ... The extractors are obtained by observing that a weaker notion of \combinatorial design" su ces for the Nisan{Wigderson pseudorandom generator [NW94], which underlies the recent extractor of Trev=-=isan [Tre98]. We -=-give near-optimal constructions of such \weak designs" which achieve much better parameters than possible with the notion of designs used by Nisan{Wigderson and Trevisan. 1 Introduction Roughly s... |

1 | Vazirani and Vijay V.Vazirani. Random polynomial time is equal to slightly-random polynomial time - Umesh - 1985 |

1 | AKSS89] Miklos Ajtai - Komlos, Steiger - 1992 |

1 | Another proof that BPP PH (and more). Electronic Colloquium on Computational Complexity - Goldreich, Zuckerman - 1997 |

1 | ciency considerations in using semi-random sources (extended abstract - unknown authors - 1987 |