### Abstract

For δ ∈ (0, 1) and k, n ∈ N, we study the task of transforming a hard function f: {0, 1}^{n} → {0, 1}, with which any small circuit disagrees on (1 - δ)/2 fraction of the input, into a harder function f′, with which any small circuit disagrees on (1 - δ^{k})/2 fraction of the input. First, we show that such hardness amplification, when carried out in some black-box way, must require a high complexity. In particular, it cannot be realized by a circuit of depth d and size 2^{o} (k^{k1/d}) or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth) for any δ ∈ (0, 1). This extends the result of Viola, which only works when (1 - δ)/2 is small enough. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently nonuniform in the following sense. To guarantee the hardness of the resulting function f′, even against uniform machines, one has to start with a function f, which is hard against nonuniform algorithms with Ω (k log (1/δ)) bits of advice. This extends the result of Trevisan and Vadhan, which only addresses the case with (1 - δ)/2 = 2^{-n}. Finally, we derive similar lower bounds for any black-box construction of a pseudorandom generator (PRG) from a hard function. To prove our results, we link the task of hardness amplifications and PRG constructions, respectively, to some type of error-reduction codes, and then we establish lower bounds for such codes, which we hope could find interest in both coding theory and complexity theory.

Original language | English |
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Pages (from-to) | 4575-4586 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 54 |

Issue number | 10 |

DOIs | |

State | Published - 9 Oct 2008 |

### Keywords

- Complexity theory
- Computational complexity
- Construction industry
- Decoding
- Encoding
- Hardness amplification
- Integrated circuit modeling
- List-decodable code
- Noise
- Pseudorandom generator
- Transforms

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## Cite this

*IEEE Transactions on Information Theory*,

*54*(10), 4575-4586. https://doi.org/10.1109/TIT.2008.928988