## On Pseudorandomness with respect to Deterministic Observers (2000)

Venue: | ICALP Satellite Workshops |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Goldreich00onpseudorandomness,

author = {Oded Goldreich and Rehovot Israel and Avi Wigderson},

title = {On Pseudorandomness with respect to Deterministic Observers},

booktitle = {ICALP Satellite Workshops},

year = {2000},

pages = {77--84}

}

### OpenURL

### Abstract

In the theory of pseudorandomness, potential (uniform) observers are modeled as probabilistic polynomial-time machines. In fact many of the central results in that theory are proven via probabilistic polynomial-time reductions. In this paper we show that analogous deterministic reductions are unlikely to hold. We conclude that randomness of the observer is essential to the theory of pseudorandomness. What we actually prove is that the hypotheses of two central theorems (in the theory of pseudorandomness) hold unconditionally when stated with respect to deterministic polynomialtime algorithms. Thus, if these theorems were true for deterministic observers, then their conclusions would hold unconditionally, which we consider unlikely. For example, it would imply (unconditionally) that any unary language in BPP is in P. The results are proven using diagonalization and pairwise independent sample spaces.

### Citations

1241 |
Probabilistic encryption
- Goldwasser, Micali
- 1984
(Show Context)
Citation Context ...Languages, Diagonalization, Pairwise Independent Sample Spaces. Supported by MINERVA Foundation, Germany. 0 Introduction The theory of pseudorandomness, initiated by Blum, Goldwasser, Micali, and Yao =-=[6, 2, 12]-=-, is one of the fundamental achievements of complexity theory. The pivot of this approach is the suggestion to view objects as equal if they cannot be told apart by any efficient procedure (i.e., poly... |

623 |
How to generate cryptographically strong sequences of pseudorandom bits
- Blum, Micali
- 1984
(Show Context)
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529 |
Theory and applications of trapdoor functions
- Yao
- 1982
(Show Context)
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282 | Hardness vs. Randomness
- Nisan, Wigderson
- 1988
(Show Context)
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197 |
Probabilistic algorithm for testing primality
- Rabin
- 1980
(Show Context)
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180 |
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
- Impagliazzo, Wigderson
- 1997
(Show Context)
Citation Context ...stic complexity classes, which is our main focus here. Comment: Derandomization of randomized complexity classes is typically acheived by pseudorandomness w.r.t nonuniform circuits (cf., for example, =-=[12, 9, 7]-=-). In such a case one can (non-trivially) simulate the randomized algorithm by a deterministic one so that the latter yields the correct output on every input. When using pseudorandomness w.r.t unifor... |

142 |
A fast monte-carlo test for primality
- Solovay, Strassen
- 1977
(Show Context)
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125 | Modern Cryptography, Probabilistic Proofs and Pseudorandomness - Goldreich - 1998 |

95 | On the power of two-point based sampling
- Chor, Goldreich
- 1989
(Show Context)
Citation Context ...lows. We will select u from a small sample space of pairwise-independent N-bit sequences. We use the fact that there exists such sample spaces of size N 2 that can be constructed in poly(N)-time (see =-=[3]-=-). Going over all elements of the sample space, we may check whether each element u satisfies Eq. (1), for each of the m machines, within poly(N)-time. 7 Thus, it is only left to show that a pairwise-... |

73 | Randomness vs. time: De-randomization under a uniform assumption
- Impagliazzo, Wigderson
- 1998
(Show Context)
Citation Context ... a distribution is unpredictable by efficient procedures then the distribution is pseudorandom (i.e., indistinguishable from random by efficient procedures). Thm. B: hardness implies pseudorandomness =-=[8]-=- 2 . If there exists an exponential-time computable predicate that is in-approximable by efficient procedures then every language in BPP has a deterministic sub-exponential simulation that is correct ... |

48 |
Primality testing and abelian varieties over finite fields
- Adleman, Huang
- 1992
(Show Context)
Citation Context ...r infinitely many n's PrimePar(n) = 1 (resp., PrimePar(n) = 0). Deciding membership in the set f1 n : PrimePar(n)g is Cook-reducible to testing primality, and thus the former set is in ZPP ` BPP (see =-=[1, 5, 10, 11]-=-). On the other hand, we know of no other way of deciding membership in the former set. 5 Here and below, the machine trying to fail the simulator is given 1 n as input and is required to generate an ... |

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- Goldwasser, Kilian
- 1999
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