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On the security of Goldreich’s one-way function

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Abstract

Goldreich (ECCC 2000) suggested a simple construction of a candidate one-way function f : {0, 1}n → {0, 1}m where each bit of output is a fixed predicate P of a constant number d of (random) input bits. We investigate the security of this construction in the regime m = Dn, where D(d) is a sufficiently large constant. We prove that for any predicate P that correlates with either one or two of its inputs, f can be inverted with high probability.

We also prove an amplification claim regarding Goldreich’s construction. Suppose we are given an assignment \({x' \in \{0, 1\}^n}\) that has correlation \({\varepsilon > 0}\) with the hidden assignment \({x \in \{0, 1\}^n}\) . Then, given access to x′, it is possible to invert f on x with high probability, provided \({D = D(d, \varepsilon)}\) is sufficiently large.

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Authors and Affiliations

  1. Department of CSE and ITCSC, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Andrej Bogdanov

  2. ITCS, Tsinghua University, Beijing, 100084, China

    Youming Qiao

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  1. Andrej Bogdanov
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  2. Youming Qiao
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Correspondence to Andrej Bogdanov.

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Bogdanov, A., Qiao, Y. On the security of Goldreich’s one-way function. comput. complex. 21, 83–127 (2012). https://doi.org/10.1007/s00037-011-0034-0

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  • DOI: https://doi.org/10.1007/s00037-011-0034-0

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