## Abstract

A sampling procedure for a distribution P over {0, 1}^{ℓ}, is a function C : {0, 1}^{n} → {0, 1}^{ℓ} such that the distribution C(U_{n}) (obtained by applying C on the uniform distribution U_{n}) is the "desired distribution" P. Let n > r ≥ ℓ = n^{Ω(1)}. An nb-PRG (defined by Dubrov and Ishai (STOC 2006)) is a function G : {0, 1}^{r} → {0, 1} ^{n} such that for every C : {0, 1}^{n} → {0, 1} ^{ℓ} in some class of "interesting sampling procedures", C′ (U_{r}) = C(G(U_{r})) is close to C(U_{n}) in statistical distance. We construct poly-time computable nb-PRGs with r =Ω(ℓ) (which is best possible) for poly-size circuits. Previous nb-PRGs of Dubrov and Ishai have r = Ω(ℓ^{2}). We rely on the assumption that: there exists β > 0, and a problem L in E = DTIME(2 ^{O(n)}) such that for every large enough n, nondeterministic circuits of size 2^{βn} that have NP-gates cannot solve L on inputs of length n. This assumption is a scaled nonuniform analogue of (the widely believed) EXP ≠ = ∑_{2}^{P}, and similar assumptions appear in various contexts in derandomization. The nb-PRGs of Dubrov and Ishai are based on very strong cryptographic assumptions, or alternatively, on non-standard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits C : {0, 1}^{n} → {0, 1}^{ℓ} with Shannon entropy H(C(U_{n})) ≤ κ, for ℓ > κ = n^{Ω(1)}, our nb-PRGs have r = O(κ) which is best possible. The nb-PRGs of Dubrov and Ishai use seed length r = Ω(κ^{2}) and require that the probability distribution of C(U_{n}) is efficiently computable. Our nb-PRGs follow from a notion of "conditional PRGs" which may be of independent interest. These are PRGs where G(U_{r}) remains pseudorandom even when conditioned on a "large" event {A(G(U_{r})) = 1}, for an arbitrary polysize circuit A. A related notion was considered by Shaltiel and Umans (CCC 2005) in a different setup, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai. We also give an unconditional construction of a poly-time computable nb-PRGs for poly(n)-size, depth d circuits C : {0, 1}^{n} → {0, 1}^{ℓ} with r = O(ℓ · log^{d+O(1)} n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ^{2}. Our nb-PRGs can be implemented by a uniform family of poly-size constant depth circuits (with slightly larger, but still almost linear seed length). The nb-PRG of Dubrov and Ishai computes large parities and cannot be computed in poly-size and constant depth. This result follows by adapting a recent PRG construction of Trevisan and Xue (CCC 2013) to the case of nb-PRGs, and implementing it by constant-depth circuits.

Original language | American English |
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Title of host publication | STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 99-108 |

Number of pages | 10 |

ISBN (Print) | 9781450327107 |

DOIs | |

State | Published - 2014 |

Event | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States Duration: 31 May 2014 → 3 Jun 2014 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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### Conference

Conference | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 |
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Country/Territory | United States |

City | New York, NY |

Period | 31/05/14 → 3/06/14 |

## Keywords

- ACM proceedings
- LATEX
- Text tagging

## All Science Journal Classification (ASJC) codes

- Software