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Impossibility Results on Weakly Black-Box Hardness Amplification

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Fundamentals of Computation Theory (FCT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4639))

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Abstract

We study the task of hardness amplification which transforms a hard function into a harder one. It is known that in a high complexity class such as exponential time, one can convert worst-case hardness into average-case hardness. However, in a lower complexity class such as NP or sub-exponential time, the existence of such an amplification procedure remains unclear.

We consider a class of hardness amplifications called weakly black-box hardness amplification, in which the initial hard function is only used as a black box to construct the harder function. We show that if an amplification procedure in TIME(t) can amplify hardness beyond an O(t) factor, then it must basically embed in itself a hard function computable in TIME(t). As a result, it is impossible to have such a hardness amplification with hardness measured against TIME(t). Furthermore, we show that, for any k ∈ ℕ, if an amplification procedure in Σ k P can amplify hardness beyond a polynomial factor, then it must basically embed a hard function in Σ k P. This in turn implies the impossibility of having such hardness amplification with hardness measured against Σ k P/poly.

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Authors and Affiliations

  1. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    Chi-Jen Lu

  2. Department of Computer Science, National Chiao-Tung University, Hsinchu, Taiwan

    Shi-Chun Tsai & Hsin-Lung Wu

Authors
  1. Chi-Jen Lu
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  2. Shi-Chun Tsai
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  3. Hsin-Lung Wu
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Erzsébet Csuhaj-Varjú Zoltán Ésik

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© 2007 Springer-Verlag Berlin Heidelberg

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Lu, CJ., Tsai, SC., Wu, HL. (2007). Impossibility Results on Weakly Black-Box Hardness Amplification. In: Csuhaj-Varjú, E., Ésik, Z. (eds) Fundamentals of Computation Theory. FCT 2007. Lecture Notes in Computer Science, vol 4639. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74240-1_35

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  • DOI: https://doi.org/10.1007/978-3-540-74240-1_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74239-5

  • Online ISBN: 978-3-540-74240-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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