### Abstract

We study the task of transforming a hard function f, 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, for δ ε(0,1) and k ε N. We show that this process cannot be carried out in a black-box way by a circuit of depth d and size 2^{o(k1/d)} or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth). In particular, for k = 2 ^{Ω(n)}, such hardness amplification cannot be done in ATIME(O(1), 2^{o(n)}). Therefore, hardness amplification in general requires a high complexity. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently non-uniform in the following sense. Given as an oracle any algorithm which agrees with f′ on (1 - δ^{k})/2 fraction of the input, we still need an additional advice of length Ω(k log(1/δ)) in order to compute f correctly on (1 - δ)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f′, one has to start with a function f which is hard against non-uniform circuits. Finally, we derive similar lower bounds for any black-box construction of pseudorandom generators from hard functions.

Original language | English |
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Pages (from-to) | 170-182 |

Number of pages | 13 |

Journal | Proceedings of the Annual IEEE Conference on Computational Complexity |

DOIs | |

State | Published - 16 Nov 2005 |

Event | 20th Annual IEEE Conference on Computational Complexity - San Jose, CA, United States Duration: 11 Jun 2005 → 15 Jun 2005 |

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## Cite this

*Proceedings of the Annual IEEE Conference on Computational Complexity*, 170-182. https://doi.org/10.1109/CCC.2005.17