## Abstract

Impagliazzo and Wigderson [25] showed that if E = DTIME(2O(n)) requires size 2(n) circuits, then every time T constant-error randomized algorithm can be simulated deterministically in time poly(T). However, such polynomial slowdown is a deal breaker when T = 2α·n, for a constant α> 0, as is the case for some randomized algorithms for NP-complete problems. Paturi and Pudlak [30] observed that many such algorithms are obtained from randomized time T algorithms, for T ϵ2o(n), with large one-sided error 1-ϵ, for ϵ= 2-α·n, that are repeated 1/ϵ times to yield a constant-error randomized algorithm running in time T/ϵ= 2(α+o(1))·n. We show that if E requires size 2(n) nondeterministic circuits, then there is a poly(n)- Time ϵ-HSG (Hitting-Set Generator) H: {0, 1}O(log n)+log(1/ϵ) ! {0, 1}n, implying that time T randomized algorithms with one-sided error 1-ϵ can be simulated in deterministic time poly(T)/ϵ. In particular, under this hardness assumption, the fastest known constant-error randomized algorithm for k-SAT (for k ≥ 4) by Paturi et al. [31] can be made deterministic with essentially the same time bound. This is the first hardness versus randomness tradeoff for algorithms for NP-complete problems. We address the necessity of our assumption by showing that HSGs with very low error imply hardness for nondeterministic circuits with "few" nondeterministic bits. Applebaum et al. [2] showed that "black-box techniques" cannot achieve poly(n)-time computable ϵ-PRGs (Pseudo-Random Generators) for ϵ= n-!(1), even if we assume hardness against circuits with oracle access to an arbitrary language in the polynomial time hierarchy. We introduce weaker variants of PRGs with relative error, that do follow under the latter hardness assumption. Specifically, we say that a function G : {0, 1}r ! {0, 1}n is an (ϵ, δ)-re-PRG for a circuit C if (1 - ϵ) · Pr[C(Un) = 1] - δ ≤ Pr[C(G(Ur) = 1] ≤ (1 + ϵ) ) · Pr[C(Un) = 1] + δ. We construct poly(n)-time computable (ϵ, δ)-re-PRGs with arbitrary polynomial stretch, ϵ = n-O(1) and δ = 2-n(1) . We also construct PRGs with relative error that fool non-boolean distinguishers (in the sense introduced by Dubrov and Ishai [11]). Our techniques use ideas from [30, 43, 2]. Common themes in our proofs are "composing" a PRG/HSG with a combinatorial object such as dispersers and extractors, and the use of nondeterministic reductions in the spirit of Feige and Lund [12].

Original language | English |
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Title of host publication | 31st Conference on Computational Complexity, CCC 2016 |

Editors | Ran Raz |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 9:1-9:35 |

ISBN (Electronic) | 9783959770088 |

DOIs | |

State | Published - 1 May 2016 |

Event | 31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan Duration: 29 May 2016 → 1 Jun 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 50 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Conference on Computational Complexity, CCC 2016 |
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Country/Territory | Japan |

City | Tokyo |

Period | 29/05/16 → 1/06/16 |

### Bibliographical note

Funding Information:Research supported by ERC starting grant 279559. Research supported by the Simons Foundation and NSF grants #CNS-1523467 and CCF-121351.Research supported by an NSERC Discovery grant. Research supported by BSF grant 2010120, ISF grant 864/11, and ERC starting grant 279559.

Publisher Copyright:

© Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, and Ronen Shaltiel.

## Keywords

- Derandomization
- Hitting-set generator
- Pseudorandom generator
- Relative error

## ASJC Scopus subject areas

- Software