## Weak Random Sources, Hitting Sets, and BPP Simulations (1998)

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Citations: | 40 - 5 self |

### BibTeX

@MISC{Andreev98weakrandom,

author = {Alexander E. Andreev and Andrea E. F. Clementi and Jose D. P. Rolim and Luca Trevisan},

title = {Weak Random Sources, Hitting Sets, and BPP Simulations},

year = {1998}

}

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### Abstract

We show how to simulate any BPP algorithm in polynomial time using a weak random source of r bits and min-entropy r fl for any fl ? 0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [Saks, Srinivasan and Zhou 1995] and a quasi-polynomial time simulation of BPP [Ta-Shma 1996]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of [Saks, Srinivasan, and Zhou 1995] in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim 1996]. AMS Subject Classification: 68Q10, 11K45. Key Words and Phrases: Derandomization, Imperfect Sources of Randomness, Hitting Sets, Randomized Computations, Expander Graphs. Abbreviated Title: BPP Simulations using Weak Random Sources. 1 Introduction Randomi...

### Citations

1928 | Randomized Algorithms
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Citation Context ...ct classifications. 68Q10, 11K45 PII. S0097539797325636 1. Introduction. Randomized algorithms are often the simplest ones that can be used to solve a given problem, or the most e#cient, or both (see =-=[MR95]-=-). For some problems, including primality testing and approximation of #P-complete counting problems, only randomized solutions are known. The practical applicability of such randomized methods depend... |

1791 | An introduction to Kolmogorov complexity and its applications 2nd edition
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Citation Context ...acle access to f . For example, KU (f |f) = O(1). As usual, if we fix another universal Turing machine U # , it holds that KU # (g|f) = KU (g|f) + #(1). We will usually omit the subscript. See, e.g., =-=[LV90]-=- for an introduction to Kolmogorov complexity. In this paper we use only the obvious fact that, for any fixed f , the number of functions g such that K(g|f) # k is at most 2 k . Definition 9 (hitting ... |

623 | How to generate cryptographically strong sequences of pseudo-random bits - Blum, Micali - 1984 |

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282 | Hardness vs. Randomness - Nisan, Wigderson - 1988 |

225 | Randomness is linear in space
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Citation Context ...ndomized algorithms are possible [Y82, BM84, N90, BFNW93, NW94, IW97, ACR97]. In some cases, combinatorial objects developed in the study of weak random sources have been used to give derandomization =-=[NZ96]-=-. Here we reverse this connection and use a derandomization method to take full advantage of a weak random source. Two basic combinatorial objects are studied in the theory of derandomization: pseudor... |

184 | Unbiased bits from sources of weak randomness and probabilistic communication complexity - Chor, Goldreich - 1988 |

180 | P = BPP if E requires exponential circuits: derandomizing the XOR lemma - Impagliazzo, Wigderson - 1997 |

145 | A complexity theoretic approach to randomness
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- 1983
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Citation Context ...f length n x # Y # # + y # {0, 1} p(n) .A(x, y) = 1, x # N # #y # {0, 1} p(n) .A(x, y) = 0. We will use the following result of Lautemann [L83] (which is an improvement on a previous result by Sipser =-=[S83]). Theorem 22 (Laute-=-mann [L83]). If L # BPP , then there exists a polynomial time computable Boolean function A(��, ��, ��) and two polynomials p(��) and q(��) such that for any x of length n x # L # ... |

114 | BPP has subexponential time simulations unless EXPTIME has publishable proofs - Babai, Fortnow, et al. - 1993 |

112 |
The complexity of promise problems with applications to public-key cryptography
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- 1984
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Citation Context ...from Fortnow. We first have to introduce some new notation. For a set S and a property # we denote by # + x # S.#(x) the statement "at least half the elements of S have property #." A promis=-=e problem [ESY84]-=- is a pair of disjoint sets of strings (Y, N ). An algorithm A solves a promise problem (Y, N) if A accepts any element of Y and rejects any element of N . Languages can be seen as a special case of p... |

109 | Simulating BPP Using a General Weak Random Source - Zuckerman - 1996 |

92 | Dispersers, deterministic amplification and weak random sources
- Cohen, Wigderson
- 1989
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Citation Context ...random sources of min-entropy r # for any fixed # > 0, was one of the major open questions in the field. It is not di#cult to show that, to simulate RP by means of a weak random source, OR dispersers =-=[CW89]-=- (from now on, we will simply call them dispersers) are su#cient. A (2 r , 2 m , 2 r # )-disperser is again a bipartite graph G = (V, W,E) with parameters r, m, and d as before, but now the property i... |

79 |
BPP and the Polynomial Hierarchy
- Lautemann
- 1983
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Citation Context ...aper [ACRT97] and also in an appendix of the final version of [ACR98]. More recently, Fortnow has observed that an even simpler proof of Theorem 2 can be given by using a previous result of Lautemann =-=[L83]-=-. Fortnow's proof of Theorem 2 does not use the discrepancy test. To the best of our understanding, this new proof does not extend to the context of dispersers and weak random sources, and it seems th... |

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

69 | A sample of samplers – a computational perspective on sampling (survey - Goldreich - 1997 |

64 |
Randomness-ecient oblivious sampling
- Bellare, Rompel
- 1994
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Citation Context ...ve a seemingly weaker (but in fact equivalent) formal definition. 2106 ANDREEV, CLEMENTI, ROLIM, AND TREVISAN our sampling algorithm is not oblivious according to the definition of Bellare and Rompel =-=[BR94]-=-; however, it is nonadaptive. See [G97] for definitions of these notions and for a survey on sampling. Our main result can be stated in the following way. Theorem 3 (main theorem). For any # > 0, ther... |

61 | On extracting randomness from weak random sources
- Ta-Shma
- 1996
(Show Context)
Citation Context ...l in m, then a polynomial time simulation of BPP would be possible, using an (r, r # )-source. However, the best present construction of extractors for fixed # > 0 and r = poly(m) has d = n log (k) n =-=[T96]-=-. This implies a quasi-polynomial time simulation of BPP. A polynomial-time simulation of BPP, using weak random sources of min-entropy r # for any fixed # > 0, was one of the major open questions in ... |

60 | General weak random sources - Zuckerman - 1990 |

58 | Generating quasirandom sequences from slightly-random sources (extended abstract - Santha, Vazirani - 1984 |

57 | Random polynomial time is equal to slightly random polynomial time. FOCS - Vazirani, Vazirani - 1985 |

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

36 |
A new general derandomization method
- Andreev, Clementi, et al.
- 1998
(Show Context)
Citation Context ...ly speaking, in the context of derandomization, pseudorandom generators play the role of extractors and hitting-set generators play that of dispersers. A recent result of Andreev, Clementi, and Rolim =-=[ACR98]-=- shows how to deterministically simulate BPP algorithms by using hitting set generators. This suggests that perhaps dispersers could be used to simulate BPP with weak random sources. A quick #-hitting... |

34 | Efficiency Considerations in Using Semi-Random Sources - Vazirani - 1987 |

27 | Hitting sets derandomize BPP - Andreev, Clementi, et al. - 1996 |

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

22 | Randomness-optimal sampling, extractors, and constructive leader election - Zuckerman |

21 |
Extracting randomness: How and why
- Nisan
- 1996
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Citation Context ...onstruction is somewhat easier to obtain, and indeed Saks, Srinivasan, and Zhou [SSZ95] give a disperser with d = poly(n), for any constant # > 0, allowing for a polynomial time simulation of RP. See =-=[N96]-=- for a complete survey on extractors, dispersers, and weak random sources. Pseudorandom generators and hitting sets. A more ambitious goal than simulating BPP with weak random sources is the determini... |

14 | Using Hard Problems to Create Pseudorandom Generators - Nisan - 1992 |

14 |
Private communication
- Saks
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Citation Context ...ications. We also emphasize that our simulation runs in NC. This is due to the parallel nature of our construction and to the fact that it is possible to give an NC construction of the SSZ-dispersers =-=[SSZ97]-=-. Thus, our method provides also an e#cient simulation of BPNC algorithms using weak random sources. Likewise, the proof of Lemma 1 as appeared in a preliminary version of this paper [ACRT97], as well... |

12 |
Explicit dispersers with polylog degree
- Saks, Srinivasan, et al.
- 1995
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Citation Context ... any set W # # W of more than 2 m /2 vertices on the right side, there is at least one edge joining V # and W # . This construction is somewhat easier to obtain, and indeed Saks, Srinivasan, and Zhou =-=[SSZ95]-=- give a disperser with d = poly(n), for any constant # > 0, allowing for a polynomial time simulation of RP. See [N96] for a complete survey on extractors, dispersers, and weak random sources. Pseudor... |

7 |
Worst-case hardness su#ces for derandomization: A new method for hardness vs randomness trade-o#s
- Andreev, Clementi, et al.
- 1999
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Citation Context ...y 0s1, the existence of a quick (1-2 -n 1-# )-HSG implies the existence of a quick (1/poly(n))-HSG. By using random walks on expander graphs instead of simple repetition, Andreev, Clementi, and Rolim =-=[ACR97]-=- show that, for c > 1/2, even the existence of a (1 - 2 -cn )-HSG is an equivalent condition. 3. The discrepancy test. In this section we describe the discrepancy test from [ACR98]. We present a sligh... |

5 |
Weak random sources, hitting sets
- Andreev, Clementi, et al.
- 1999
(Show Context)
Citation Context ...date discrepancy sets S 1 , . . . , S k and then applying the discrepancy test to them also yields a simple proof of Lemma 1. This simplified proof is presented in a preliminary version of this paper =-=[ACRT97]-=- and also in an appendix of the final version of [ACR98]. More recently, Fortnow has observed that an even simpler proof of Theorem 2 can be given by using a previous result of Lautemann [L83]. Fortno... |

5 |
A Sample of Samplers—a Computational Perspective on Sampling
- Goldreich
- 1997
(Show Context)
Citation Context ...alent) formal definition. 2106 ANDREEV, CLEMENTI, ROLIM, AND TREVISAN our sampling algorithm is not oblivious according to the definition of Bellare and Rompel [BR94]; however, it is nonadaptive. See =-=[G97]-=- for definitions of these notions and for a survey on sampling. Our main result can be stated in the following way. Theorem 3 (main theorem). For any # > 0, there exist a polynomial p and a determinis... |

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