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On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets

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Complexity and Approximation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12000))

Abstract

We explain our recent results [21] on the computational power of an arbitrary distinguisher for (not necessarily computable) hitting set generators. This work is motivated by the desire of showing the limits of black-box reductions to some distributional \(\mathsf {NP}\) problem. We show that a black-box nonadaptive randomized reduction to any distinguisher for (not only polynomial-time but also) exponential-time computable hitting set generators can be simulated in \(\mathsf {AM}\cap \mathsf {co} \mathsf {AM}\); we also show an upper bound of \(\mathsf {S}_2^\mathsf {NP}\) even if there is no computational bound on a hitting set generator. These results provide additional evidence that the recent worst-case to average-case reductions within \(\mathsf {NP}\) shown by Hirahara (2018, FOCS) are inherently non-black-box. (We omit all detailed arguments and proofs, which can be found in [21].)

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Notes

  1. 1.

    As a black-box reduction to any distinguisher for G, it is required in [15] that there exists a single machine that computes a reduction to every oracle avoiding G. On the other hand, as stated in Theorem 1, we allow reductions to depend on oracles, which makes our results stronger.

  2. 2.

    We emphasize that we are concerned the nonadaptivity of reductions used in the security proof of pseudorandom generators. Several simplified constructions of pseudorandom generators \(G^f\) from one-way functions f (e.g., [16, 23]) are nonadaptive in the sense that \(G^f\) can be efficiently computed with nonadaptive oracle access to f; however, the security reductions of these constructions are adaptive because of the use of Holenstein’s uniform hardcore lemma [22]. Similarly, the reduction of [17, Lemma 6.5] is adaptive. (We note that, in the special case when the degeneracy of a one-way function is efficiently computable, the reduction of [17] is nonadaptive.).

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Authors and Affiliations

  1. National Institute of Informatics, Tokyo, Japan

    Shuichi Hirahara

  2. Tokyo Institute of Technology, Tokyo, Japan

    Osamu Watanabe

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  1. Shuichi Hirahara
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  2. Osamu Watanabe
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Correspondence to Osamu Watanabe .

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  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. University of Massachusetts Lowell, Lowell, MA, USA

    Jie Wang

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Hirahara, S., Watanabe, O. (2020). On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets. In: Du, DZ., Wang, J. (eds) Complexity and Approximation. Lecture Notes in Computer Science(), vol 12000. Springer, Cham. https://doi.org/10.1007/978-3-030-41672-0_6

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  • DOI: https://doi.org/10.1007/978-3-030-41672-0_6

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