## Extracting randomness using few independent sources (2004)

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Venue: | In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 49 - 6 self |

### BibTeX

@INPROCEEDINGS{Barak04extractingrandomness,

author = {Boaz Barak and Russell Impagliazzo and Avi Wigderson},

title = {Extracting randomness using few independent sources},

booktitle = {In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science},

year = {2004},

pages = {384--393},

publisher = {IEEE Computer Society}

}

### Years of Citing Articles

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### Abstract

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 min-entropy 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 min-entropy Ω(log n) and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the “stepping-up lemma ” used in establishing lower bound

### Citations

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Citation Context ...log |F| tropy larger than . In addition, Salil Vadhan observed that using 2 the fact that the function family {hb,c} (where hb,c(a) = a · b + c) is pairwise independent and the Leftover Hash Lemma of =-=[HILL89]-=-, one can prove that under the conditions of Lemma 3.2 the first 0.8 log |F | bits of A1 · A2 + A3 are within statistical distance |F | −0.01 to the uniform distribution. This is also sufficient to pr... |

259 | Small-bias probability spaces: efficient constructions and applications - Naor, Naor - 1993 |

209 |
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Citation Context ...The first to consider this problem (and indeed, the first to consider the problem of randomness extraction) was von Neumann, who gave a seedless extractor from a stream of biased but independent bits =-=[vN51]-=- (see also [Per92]). Other works, such as [Blu84, SV84, CG85, CW89, 3 We remark that in most of the literature, the name randomness extractor (without any qualifiers) refers to what we call here a see... |

186 | Unbiased bits from sources of weak randomness and probabilistic communication complexity
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Citation Context ... H : {0, 1} 2n → {0, 1} defined as follows H(x, y) = � xiyi (mod 2) (i.e., the adjacency function of the Hadamard graph) is known to be a 1-bit output extractor for sources with more than n/2 entropy =-=[CG85]-=-. This is essentially the best known previous construction in terms of the minimum entropy requirement. 15 There has been some improvement in obtaining variants of this extractor that have a larger ou... |

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134 |
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Citation Context ...mmas hold for any Abelian group, and so the “+” operator may be replaced in them by “·”. Note that we didn’t state these lemmas with the most optimal choice of constants. Lemma 3.5 ([Ruz96] (see also =-=[Nat96]-=-)). Let A, B be subsets of some Abelian group G with |A| = |B| = M and let ρ > 0 be some number. If |A+B| ≤ M 1+ρ then |A+A| ≤ M 1+2ρ . In other words, if A+B is “small” for some B then A+A is small. ... |

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Citation Context ...e form X · Y + Z.) The proof of this lemma is the main technical step in our extractor. The proof uses Theorem 1.5, along with some other additive number-theoretic results of Rusza [Ruz96] and Gowers =-=[Gow98]-=-. We use this lemma to show that the function g sketched above is actually not just a disperser but an extractor. That is, we show that for every random variable X of minentropy at least δ log |F|, g(... |

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Citation Context ...this dependency. In very recent work, [BKS + 05] have achieved this. They give explicit deterministic extractors from 3 independent sources of any linear entropy. This was further improved by Ran Raz =-=[Raz05]-=-, who only needs one of the sources to have linear entropy, while the others can have only logarithmic amount of entropy. Further results in [BKS + 05] give a deterministic disperser from only two ind... |

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Citation Context ...t sources of the same quality. After all, assuming we have such a source in nature does not seem much weaker than assuming we have several. The first to consider this problem were Santha and Vazirani =-=[SV84]-=-, who showed how to use O(log n) independent “semi-random” 5 sources of length n and min-entropy δn for every constant δ > 0. Chor and Goldreich [CG85] were the first to consider general min-entropy s... |

59 |
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Citation Context ...ropy > n/2, and is conjectured to have the same property for entropy δn for all δ > 0. While generally believed, proving this conjecture seems beyond current techniques. Assuming even more, Zuckerman =-=[Zuc90]-=- showed that if this conjecture holds for all multiplicative characters (not just χ2), than a constant (actually poly(1/δ)) sources suffice for extraction of linearly many bits with exponential error.... |

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Citation Context ...racters (not just χ2), than a constant (actually poly(1/δ)) sources suffice for extraction of linearly many bits with exponential error. The extractor we use here is 4 The work of Trevisan and Vadhan =-=[TV00]-=- differs from all the rest, as well as from ours, in that it works in the computational setting; the restriction on the family of sources is computational - they are efficiently sampleable, and the ex... |

54 |
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Citation Context ...t if δ > n/2 than two sources suffice: indeed the Hadamard-Sylvester matrix H : {0,1} n × {0,1} n → {0,1} defined by H(x,y) = 〈x,y〉 (with the inner product in GF(2)) is such an extractor 6 . Vazirani =-=[Vaz85]-=- extended this to show that one can use a similar function to output a linear number of bits which are exponentially close to the uniform distribution. Improving [CG85] seems hard. Even its disperser ... |

53 |
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Citation Context ...ts of {0,1} n of size n O(1) . A central tool we will use is the deterministic coin tossing technique. This technique was used in several contexts in computer science (e.g., the parallel algorithm of =-=[CV86]-=-), and it was also used in a very similar context to ours by Erdos and Hajnal (where it was called the “stepping up lemma”, see Section 4.7 in [GRS80]) and Fiat and Naor [FN93]. By “deterministic coin... |

52 | Extractors: Optimal up to constant factors
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Citation Context ... current state of the art is a construction of extractors that are nearly optimal in the sense that they use a seed Y of length O(log n) bits, extracting essentially all the min-entropy of the source =-=[LRVW03]-=-. This in particular means that using such extractors, together with enumeration over all possible seed values, it is possible to simulate any probabilistic algorithm with polynomial overhead, using o... |

47 | A sum-product estimate in finite fields, and applications
- Bourgain, Katz, et al.
(Show Context)
Citation Context ...tropy 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,... |

43 | Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors - Barak, Kindler, et al. - 2005 |

39 | Extractors for a constant number of polynomially small min-entropy independent sources - Rao - 2006 |

37 | On sums and products of integers
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Citation Context ...main tools for Theorem 1.1 are several results from additive number theory and in particular a relatively recent result by Bourgain, Katz and Tao [BKT03]. They proved an analog of the Erdős-Szemerédi =-=[ES83]-=- theorem for finite prime fields. Let A be a subset of some field F. We define the set A + A to equal {a + b | a, b ∈ A} and the set A · A to equal {a · b | a, b ∈ A}. Note that |A| ≤ |A + A| ≤ |A| 2 ... |

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

31 |
Iterating von Neumann’s procedure for extracting random bits
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Citation Context ...ider this problem (and indeed, the first to consider the problem of randomness extraction) was von Neumann, who gave a seedless extractor from a stream of biased but independent bits [vN51] (see also =-=[Per92]-=-). Other works, such as [Blu84, SV84, CG85, CW89, 3 We remark that in most of the literature, the name randomness extractor (without any qualifiers) refers to what we call here a seeded randomness ext... |

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21 |
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Citation Context ...call the additional input Y the seed of the extractor, and thus we call such constructions seeded extractors. 3 Seeded extractors have been studied extensively in the past two decades (see the survey =-=[Sha02]-=-) with a series of exciting results, techniques, and applications. The current state of the art is a construction of extractors that are nearly optimal in the sense that they use a seed Y of length O(... |

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Citation Context ...e parallel algorithm of [CV86]), and it was also used in a very similar context to ours by Erdos and Hajnal (where it was called the “stepping up lemma”, see Section 4.7 in [GRS80]) and Fiat and Naor =-=[FN93]-=-. By “deterministic coin tossing” we mean the function ct : {0,1} 2n → [n] defined as follows: for every x,y ∈ {0,1} n , ct(x,y) is equal to the first position i such that xi �= yi (if x = y then we l... |

12 |
A sum-product estimate in fields of prime order
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(Show Context)
Citation Context ...rs 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 min-entropy Ω... |

10 |
Sums of finite sets
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- 1991
(Show Context)
Citation Context ...distributions of the form X · Y + Z.) The proof of this lemma is the main technical step in our extractor. The proof uses Theorem 1.5, along with some other additive number-theoretic results of Rusza =-=[Ruz96]-=- and Gowers [Gow98]. We use this lemma to show that the function g sketched above is actually not just a disperser but an extractor. That is, we show that for every random variable X of minentropy at ... |

10 | Eine zahlentheoretische Anwendung der Graphentheorie - Plünnecke - 1970 |

9 | Pseudorandom sets and explicit constructions of Ramsey graphs. In: Quaderni di Matematica
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Citation Context ...stant output for any two sources of entropy less rate below 1/2, is the notorious “Bipartite Ramsey” problem is still open. A slight improvement for this problem was made this year by Pudlak and Rödl =-=[PR04]-=-, who lowered the min-entropy requirement for such a disperser to n/2 − √ n, but getting a constant δ < 1/2 remains a barrier for 2 sources. (Very recently, Barak, Kindler, Shaltiel, Sudakov and Wigde... |

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3 |
More on the Sum-Product Phenomenon
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Citation Context ... method, or enhance it? The same question applies equally well to the problem of extraction from affine sources. An exciting new development in this direction is a very recent result by Jean Bourgain =-=[Bou05]-=-, which gives (again using results from [BKT03]) a simple algebraic construction of an extractor for 2 independent sources requiring ( 1 2 − ɛ)n entropy for some small absolute constant ɛ > 0. To summ... |

2 |
Simulating BPP Using a General Weak Random
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(Show Context)
Citation Context ...he high-entropy distribution to obtain such a prime. To do so, we will use the following result on seeded dispersers (which we state here with the parameters suitable for our purposes): Theorem 3.10 (=-=[Zuc91]-=-). For every δ > 0, there exists a constant d > 1 and a polynomial-time computable function D : {0,1} (10/δ)n × {0,1} d log n → {0,1} n such that for every set A ⊆ {0,1} (10/δ)n with |A| ≥ 2 2n , it h... |

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Citation Context ... that if A = B in the conditions of this lemma, then A ′ = B ′ in its conclusion. A slightly different proof of this lemma (along with an extension to sums of more than two sets) can also be found in =-=[SSV04]-=-. 3.3 Proof of Lemma 3.1 Fixing ɛ. We fix ɛ small enough such that Theorem 1.5 ensures us that if X is a set of size at least M 1−104 ɛ then max{|X · X |, |X + X |} is at least M 1+10 4 ɛ (e.g., we ca... |

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